### Revision history for GeneralRelativity

##### Additions:

Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called [[http://en.wikipedia.org/wiki/Manifold manifolds]]. Spacetime is described as a [[http://en.wikipedia.org/wiki/Lorentzian_manifold#Lorentzian_manifold Lorentzian manifold]]. By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#The_Pound-Rebka_Experiment"> Pound-Rebka experiment</a>"". In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys [[FoundationsOfSpecialRelativity special relativity]]. Together with ""<a href=http://www.rqgravity.net/Gravitation#Einsteins_Law_of_Gravitation>Einstein’s equation</a>"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.

##### Deletions:

##### Additions:

""<a href="GeneralRelativity#The_Principle_of_General_Covariance">The Principle of General Covariance </a>""

""<a href="GeneralRelativity#The_Special_Principle_of_Relativity">The Special Principle of Relativity</a>""

""<a href="GeneralRelativity#The_Equivalence_Principle">The Equivalence Principle</a>""

""<a href="GeneralRelativity#The_Spacetime_Manifold">The Spacetime Manifold</a>""

""<a href="GeneralRelativity#What_is_Spacetime">What is Spacetime?</a>""

""<a href="GeneralRelativity#Tangent_Charts">Tangent Charts</a>""

""<a href="GeneralRelativity#Coordinate_Quantities_and_Proper_Quantities">Coordinate Quantities and Proper Quantities</a>""

""<a href="GeneralRelativity#Schwarzschild_Coordinates">Schwarzschild Coordinates</a>""

""<a href="GeneralRelativity#The_Levi-Civita_Connection">The Levi-Civita Connection</a>""

""<a href="GeneralRelativity#Differentiability">Differentiability</a>""

====""<a name="The_Principle_of_General_Covariance"></a>""The Principle of General Covariance ====

General relativity is based on ""<a href="PhilosophicalBackground#The_General_Principle_of_Relativity">the principle of uniformity in nature</a>"",

together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy) and the empirical fact that we do not have a universe obeying [[https://en.wikipedia.org/wiki/Galilean_invariance Newtonian relativity]]. These notions are encapsulated in the ""<a href="PhilosophicalBackground#The_General_Principle_of_Relativity">general principle of relativity</a>"",

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://www.rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#Coordinates>coordinate representation</a>"" changes with the coordinate system, but relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#Contravariant_and_Covariant_Vectors>covariant</a>"". [[http://www.rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

====""<a name="The_Special_Principle_of_Relativity"></a>""The Special Principle of Relativity ====

<<""<span class="math"><b>Special principle of relativity:</b> If a system of coordinates <i>K</i> is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates <i>K'</i> moving in uniform translation relatively to <i>K</i> — Albert Einstein, The foundation of the general theory of relativity, 1916.</span>""<<

The principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the principle in two ways, using it to establish the ""<a href="http://www.rqgravity.net/FoundationsOfSpecialRelativity#Minkowski_Coordinates">coordinate system</a>"" and the constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#Maximum_Signal_Speed">speed of light</a>"".

With [[http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity general relavity]], Einstein further extended the special principle, formulating the general principle and the principle of general covariance, but the special principle remains significant, as the means by which we use Newton’s laws to identify [[http://www.rqgravity.net/TheEquivalencePrinciple#Inertial_Matter inertial frames]] as defining a special class of coordinate systems in which there is a natural correspondence between mathematical structure and physical behaviour.

====""<a name="The_Equivalence_Principle"></a>""The Equivalence Principle====

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://www.rqgravity.net/TheEquivalencePrinciple#Inertial_Matter inertial reference frame]]. Since objects always move in the direction of their velocity vectore, the immediate consequence is [[http://www.rqgravity.net/BasicsOfCurvature#Geodesics geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#The_Equivalence_Principle equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a [[http://www.rqgravity.net/TheEquivalencePrinciple#Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

<<""<span class="math"><b>The Equivalence Principle:</b>"" ""<span class="math">We ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system — Einstein, 1907.""<<

====""<a name="The_Spacetime_Manifold"></a>""The Spacetime Manifold====

According to the general principle, an observer anywhere can use the radar method to define locally ""<a href="FoundationsOfSpecialRelativity#Minkowski_Coordinates">Minkowski Coordinates</a>"", but there is no guarantee that a mapping of distant points to these coordinates can be made without distortion of the map. The situation is analogous to mapping the surface of the Earth. At any point of the Earth’s surface, a cartographer can make a locally flat map. He cannot extend the map without distortion, but this does not mean that geometry at other points of the Earth surface is different from the geometry seen by the cartographer.

Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called [[http://en.wikipedia.org/wiki/Manifold manifolds]]. Spacetime is described as a [[http://en.wikipedia.org/wiki/Lorentzian_manifold#Lorentzian_manifold Lorentzian manifold]]. By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#The_Pound-Rebka_Experiment"> Pound-Rebka experiment</a>"". In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys [[FoundationsOfSpecialRelativity special relativity]]. Together with ""<a href=http://www.rqgravity.net/Gravitation#Einsteins_Law_of_Gravitation_1>Einstein’s equation</a>"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.

<<""<span class="math"><b>Definition:</b> A <i>manifold</i> is a structure in which any point has a neighbourhood which can be described by a <a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>coordinate system</a> or <a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>chart</a>.""<<

Typically a single coordinate system cannot be used to give a full description of a manifold. A collection of charts which describes the whole manifold is an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>atlas</a>"".

====""<a name="What_is_Spacetime"></a>""What is Spacetime?====

We can describe spacetime as a manifold, a geometrical structure which can be mapped onto an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>atlas</a>"", or collection of ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>charts</a>"". In common with many definitions of mathematical structures, this does not tell us what the manifold actually is. Instead it tells us what properties a manifold has, how a manifold behaves.

Here the manifold simply replaces Newton’s conception of ""<a href=http://www.rqgravity.net/PhilosophicalBackground#Space_and_Time_Absolute_or_Relative?>absolute space</a>"" and ""<a href=http://www.rqgravity.net/PhilosophicalBackground#Space_and_Time_Absolute_or_Relative?>absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

<<""<span class="math"><b>Answer 3.</b> I observe that I can, in principle, choose <a href="PhilosophicalBackground#Reference_Frames">reference matter</a> anywhere I wish, and that I can define <a href="FoundationsOfSpecialRelativity#Minkowski_Coordinates">Minkowski coordinates</a> relative to that matter. I now define spacetime by <i>imagining</i> all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.</span>""<<

====""<a name="Tangent_Charts"></a>""Tangent Charts====

In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, ""<a href="http://www.rqgravity.net/GeneralRelativity#The_Principle_Of_General_Covariance" >general covariance</a>"" automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in [[FoundationsOfSpecialRelativity special relativity]], using the radar method. In this case the chart is made on Minkowski spacetime, which has constant ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#3_Dimensions_plus_Time>Minkowski metric</a>"", ""<span class="math"><i>h</i>"". ""<span class="math"><i>h</i>"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""<span class="math"><i>h</i>"" does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Tangent_Space>tangent chart</a>"", an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.

For an inertial observer, metre sticks give the same result, but I have elected to use radar because it leads to a simpler analysis, and is applicable to measurements generally in the Solar System. A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#Coordinate_Transformation">transformed</a>"" into coordinates with Minkowski metric at the position of the observer.

====""<a name="Coordinate_Quantities_and_Proper_Quantities"></a>""Coordinate Quantities and Proper Quantities====

According to ""<a href="GeneralRelativity#The_Principle_of_General_Covariance">general covariance"" physical quantities are scalars, vectors and tensors. Vectors and tensors have different representations, dependent on the coordinate system being used. A proper quantity is the representation of a physical quantity as seen in locally Minkowski coordinates moving with the quantity being measured.

====""<a name="Schwarzschild_Coordinates"></a>""Schwarzschild Coordinates====

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]], that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#The_Equivalence_Principle equivalence principle]] and the [[http://www.rqgravity.net/TheEquivalencePrinciple#The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, relative to coordinates determined by radar, proper distances at the surface of the planet are greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#Maximum_Signal_Speed speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

====""<a name="The_Levi-Civita_Connection"></a>""The Levi-Civita Connection====

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#Orthonormal_Bases>orthogonal</a>"". In this case the metric is diagonal (as seen in ""<a href="GeneralRelativity#Schwarzschild_Coordinates">Schwarzschild coordinates</a>"") and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the ""<a href= BasicsOfCurvature#Lens>lensed and mirrored </a>"" geometries, which are actually flat. To describe curvature requires a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Parallel_Displacement>connection</a>"" in addition to the metric field. Given the metric field, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""<span class="math"><i>x</i>"", say, and another set, at ""<span class="math"><i>x</i> +<i>dx</i>"", where ""<span class="math"><i>dx</i>"" is a small displacement, such that we can meaningfully describe a vector at ""<span class="math"><i>x</i>"" as being parallel to one at ""<span class="math"><i>x</i> +<i>dx</i>"" (other types of [[http://en.wikipedia.org/wiki/Connection_(mathematics) connection]] are used to transport other types of data).

In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between vectors at nearby points using ""<a href="http://www.rqgravity.net/BasicsOfCurvature#Parallel_Displacement">parallel displacement</a>"" in tangent space, and projecting back into the curved surface. This is the [[http://en.wikipedia.org/wiki/Levi-Civita_connection Levi-Civita connection]], defined in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through ""<a href="http://www.rqgravity.net/BasicsOfCurvature#Parallel_Transport">parallel transport</a>"", and is path dependent. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.

A ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>chart</a>"" of spacetime need not be a physical map, like the maps in a world atlas. A mathematical idealisation suffices just as well — that is to say, the map may consist of tables of data and/or formulae. We may imagine, for example, the numbers, or coordinates, describing the times and positions of physical events mapped into a bank of computer memory. In principle, using a large enough bank of computer memory, this could be done to any precision, for as many points as one requires, and a map of a region of spacetime could be produced with any required level of detail, up to the limit of accuracy of measurement and the size of available computer memory.

""<a href="GeneralRelativity#The_Special_Principle_of_Relativity">The Special Principle of Relativity</a>""

""<a href="GeneralRelativity#The_Equivalence_Principle">The Equivalence Principle</a>""

""<a href="GeneralRelativity#The_Spacetime_Manifold">The Spacetime Manifold</a>""

""<a href="GeneralRelativity#What_is_Spacetime">What is Spacetime?</a>""

""<a href="GeneralRelativity#Tangent_Charts">Tangent Charts</a>""

""<a href="GeneralRelativity#Coordinate_Quantities_and_Proper_Quantities">Coordinate Quantities and Proper Quantities</a>""

""<a href="GeneralRelativity#Schwarzschild_Coordinates">Schwarzschild Coordinates</a>""

""<a href="GeneralRelativity#The_Levi-Civita_Connection">The Levi-Civita Connection</a>""

""<a href="GeneralRelativity#Differentiability">Differentiability</a>""

====""<a name="The_Principle_of_General_Covariance"></a>""The Principle of General Covariance ====

General relativity is based on ""<a href="PhilosophicalBackground#The_General_Principle_of_Relativity">the principle of uniformity in nature</a>"",

together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy) and the empirical fact that we do not have a universe obeying [[https://en.wikipedia.org/wiki/Galilean_invariance Newtonian relativity]]. These notions are encapsulated in the ""<a href="PhilosophicalBackground#The_General_Principle_of_Relativity">general principle of relativity</a>"",

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://www.rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#Coordinates>coordinate representation</a>"" changes with the coordinate system, but relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#Contravariant_and_Covariant_Vectors>covariant</a>"". [[http://www.rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

====""<a name="The_Special_Principle_of_Relativity"></a>""The Special Principle of Relativity ====

<<""<span class="math"><b>Special principle of relativity:</b> If a system of coordinates <i>K</i> is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates <i>K'</i> moving in uniform translation relatively to <i>K</i> — Albert Einstein, The foundation of the general theory of relativity, 1916.</span>""<<

The principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the principle in two ways, using it to establish the ""<a href="http://www.rqgravity.net/FoundationsOfSpecialRelativity#Minkowski_Coordinates">coordinate system</a>"" and the constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#Maximum_Signal_Speed">speed of light</a>"".

With [[http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity general relavity]], Einstein further extended the special principle, formulating the general principle and the principle of general covariance, but the special principle remains significant, as the means by which we use Newton’s laws to identify [[http://www.rqgravity.net/TheEquivalencePrinciple#Inertial_Matter inertial frames]] as defining a special class of coordinate systems in which there is a natural correspondence between mathematical structure and physical behaviour.

====""<a name="The_Equivalence_Principle"></a>""The Equivalence Principle====

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://www.rqgravity.net/TheEquivalencePrinciple#Inertial_Matter inertial reference frame]]. Since objects always move in the direction of their velocity vectore, the immediate consequence is [[http://www.rqgravity.net/BasicsOfCurvature#Geodesics geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#The_Equivalence_Principle equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a [[http://www.rqgravity.net/TheEquivalencePrinciple#Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

<<""<span class="math"><b>The Equivalence Principle:</b>"" ""<span class="math">We ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system — Einstein, 1907.""<<

====""<a name="The_Spacetime_Manifold"></a>""The Spacetime Manifold====

According to the general principle, an observer anywhere can use the radar method to define locally ""<a href="FoundationsOfSpecialRelativity#Minkowski_Coordinates">Minkowski Coordinates</a>"", but there is no guarantee that a mapping of distant points to these coordinates can be made without distortion of the map. The situation is analogous to mapping the surface of the Earth. At any point of the Earth’s surface, a cartographer can make a locally flat map. He cannot extend the map without distortion, but this does not mean that geometry at other points of the Earth surface is different from the geometry seen by the cartographer.

Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called [[http://en.wikipedia.org/wiki/Manifold manifolds]]. Spacetime is described as a [[http://en.wikipedia.org/wiki/Lorentzian_manifold#Lorentzian_manifold Lorentzian manifold]]. By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#The_Pound-Rebka_Experiment"> Pound-Rebka experiment</a>"". In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys [[FoundationsOfSpecialRelativity special relativity]]. Together with ""<a href=http://www.rqgravity.net/Gravitation#Einsteins_Law_of_Gravitation_1>Einstein’s equation</a>"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.

<<""<span class="math"><b>Definition:</b> A <i>manifold</i> is a structure in which any point has a neighbourhood which can be described by a <a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>coordinate system</a> or <a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>chart</a>.""<<

Typically a single coordinate system cannot be used to give a full description of a manifold. A collection of charts which describes the whole manifold is an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>atlas</a>"".

====""<a name="What_is_Spacetime"></a>""What is Spacetime?====

We can describe spacetime as a manifold, a geometrical structure which can be mapped onto an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>atlas</a>"", or collection of ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>charts</a>"". In common with many definitions of mathematical structures, this does not tell us what the manifold actually is. Instead it tells us what properties a manifold has, how a manifold behaves.

Here the manifold simply replaces Newton’s conception of ""<a href=http://www.rqgravity.net/PhilosophicalBackground#Space_and_Time_Absolute_or_Relative?>absolute space</a>"" and ""<a href=http://www.rqgravity.net/PhilosophicalBackground#Space_and_Time_Absolute_or_Relative?>absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

<<""<span class="math"><b>Answer 3.</b> I observe that I can, in principle, choose <a href="PhilosophicalBackground#Reference_Frames">reference matter</a> anywhere I wish, and that I can define <a href="FoundationsOfSpecialRelativity#Minkowski_Coordinates">Minkowski coordinates</a> relative to that matter. I now define spacetime by <i>imagining</i> all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.</span>""<<

====""<a name="Tangent_Charts"></a>""Tangent Charts====

In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, ""<a href="http://www.rqgravity.net/GeneralRelativity#The_Principle_Of_General_Covariance" >general covariance</a>"" automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in [[FoundationsOfSpecialRelativity special relativity]], using the radar method. In this case the chart is made on Minkowski spacetime, which has constant ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#3_Dimensions_plus_Time>Minkowski metric</a>"", ""<span class="math"><i>h</i>"". ""<span class="math"><i>h</i>"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""<span class="math"><i>h</i>"" does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Tangent_Space>tangent chart</a>"", an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.

For an inertial observer, metre sticks give the same result, but I have elected to use radar because it leads to a simpler analysis, and is applicable to measurements generally in the Solar System. A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#Coordinate_Transformation">transformed</a>"" into coordinates with Minkowski metric at the position of the observer.

====""<a name="Coordinate_Quantities_and_Proper_Quantities"></a>""Coordinate Quantities and Proper Quantities====

According to ""<a href="GeneralRelativity#The_Principle_of_General_Covariance">general covariance"" physical quantities are scalars, vectors and tensors. Vectors and tensors have different representations, dependent on the coordinate system being used. A proper quantity is the representation of a physical quantity as seen in locally Minkowski coordinates moving with the quantity being measured.

====""<a name="Schwarzschild_Coordinates"></a>""Schwarzschild Coordinates====

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]], that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#The_Equivalence_Principle equivalence principle]] and the [[http://www.rqgravity.net/TheEquivalencePrinciple#The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, relative to coordinates determined by radar, proper distances at the surface of the planet are greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#Maximum_Signal_Speed speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

====""<a name="The_Levi-Civita_Connection"></a>""The Levi-Civita Connection====

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#Orthonormal_Bases>orthogonal</a>"". In this case the metric is diagonal (as seen in ""<a href="GeneralRelativity#Schwarzschild_Coordinates">Schwarzschild coordinates</a>"") and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the ""<a href= BasicsOfCurvature#Lens>lensed and mirrored </a>"" geometries, which are actually flat. To describe curvature requires a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Parallel_Displacement>connection</a>"" in addition to the metric field. Given the metric field, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""<span class="math"><i>x</i>"", say, and another set, at ""<span class="math"><i>x</i> +<i>dx</i>"", where ""<span class="math"><i>dx</i>"" is a small displacement, such that we can meaningfully describe a vector at ""<span class="math"><i>x</i>"" as being parallel to one at ""<span class="math"><i>x</i> +<i>dx</i>"" (other types of [[http://en.wikipedia.org/wiki/Connection_(mathematics) connection]] are used to transport other types of data).

In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between vectors at nearby points using ""<a href="http://www.rqgravity.net/BasicsOfCurvature#Parallel_Displacement">parallel displacement</a>"" in tangent space, and projecting back into the curved surface. This is the [[http://en.wikipedia.org/wiki/Levi-Civita_connection Levi-Civita connection]], defined in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through ""<a href="http://www.rqgravity.net/BasicsOfCurvature#Parallel_Transport">parallel transport</a>"", and is path dependent. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.

A ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts_or_Coordinate_Systems>chart</a>"" of spacetime need not be a physical map, like the maps in a world atlas. A mathematical idealisation suffices just as well — that is to say, the map may consist of tables of data and/or formulae. We may imagine, for example, the numbers, or coordinates, describing the times and positions of physical events mapped into a bank of computer memory. In principle, using a large enough bank of computer memory, this could be done to any precision, for as many points as one requires, and a map of a region of spacetime could be produced with any required level of detail, up to the limit of accuracy of measurement and the size of available computer memory.

##### Deletions:

""<a href="GeneralRelativity#hn_The_Special_Principle_of_Relativity">The Special Principle of Relativity</a>""

""<a href="GeneralRelativity#hn_The_Equivalence_Principle">The Equivalence Principle</a>""

""<a href="GeneralRelativity#hn_The_Spacetime_Manifold">The Spacetime Manifold</a>""

""<a href="GeneralRelativity#hn_What_is_Spacetime">What is Spacetime?</a>""

""<a href="GeneralRelativity#hn_Tangent_Charts">Tangent Charts</a>""

""<a href="GeneralRelativity#hn_Coordinate_Quantities_and_Proper_Quantities">Coordinate Quantities and Proper Quantities</a>""

""<a href="GeneralRelativity#hn_Schwarzschild_Coordinates">Schwarzschild Coordinates</a>""

""<a href="GeneralRelativity#hn_The_Levi-Civita_Connection">The Levi-Civita Connection</a>""

""<a href="GeneralRelativity#hn_Differentiability">Differentiability</a>""

====The Principle of General Covariance ====

General relativity is based on ""<a href="PhilosophicalBackground#hn_The_General_Principle_of_Relativity">the principle of uniformity in nature</a>"",

together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy) and the empirical fact that we do not have a universe obeying [[https://en.wikipedia.org/wiki/Galilean_invariance Newtonian relativity]]. These notions are encapsulated in the ""<a href="PhilosophicalBackground#hn_The_General_Principle_of_Relativity">general principle of relativity</a>"",

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://www.rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinates>coordinate representation</a>"" changes with the coordinate system, but relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_Contravariant_and_Covariant_Vectors>covariant</a>"". [[http://www.rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

====The Special Principle of Relativity ====

<<""<span class="math"><b>Special principle of relativity:</b> If a system of coordinates <i>K</i> is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates <i>K'</i> moving in uniform translation relatively to <i>K</i>. —Albert Einstein: The foundation of the general theory of relativity</span>""<<

The principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the principle in two ways, using it to establish the ""<a href="http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and the constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximum_Signal_Speed">speed of light</a>"".

With [[http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity general relavity]], Einstein further extended the special principle, formulating the general principle and the principle of general covariance, but the special principle remains significant, as the means by which we use Newton’s laws to identify [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial frames]] as defining a special class of coordinate systems in which there is a natural correspondence between mathematical structure and physical behaviour.

====The Equivalence Principle====

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. Since objects always move in the direction of their velocity vectore, the immediate consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesics geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

<<""<span class="math"><b>The Equivalence Principle:</b>"" ""<span class="math">We ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system. (Einstein 1907).""<<

====The Spacetime Manifold====

According to the general principle, an observer anywhere can use the radar method to define locally ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski Coordinates</a>"", but there is no guarantee that a mapping of distant points to these coordinates can be made without distortion of the map. The situation is analogous to mapping the surface of the Earth. At any point of the Earth’s surface, a cartographer can make a locally flat map. He cannot extend the map without distortion, but this does not mean that geometry at other points of the Earth surface is different from the geometry seen by the cartographer.

Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called [[http://en.wikipedia.org/wiki/Manifold manifolds]]. Spacetime is described as a [[http://en.wikipedia.org/wiki/Lorentzian_manifold#Lorentzian_manifold Lorentzian manifold]]. By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment"> Pound-Rebka experiment</a>"". In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys [[FoundationsOfSpecialRelativity special relativity]]. Together with ""<a href=http://www.rqgravity.net/Gravitation#hn_Einsteins_Law_of_Gravitation_1>Einstein’s equation</a>"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.

<<""<span class="math"><b>Definition:</b> A <i>manifold</i> is a structure in which any point has a neighbourhood which can be described by a <a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>coordinate system</a> or <a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>chart</a>.""<<

Typically a single coordinate system cannot be used to give a full description of a manifold. A collection of charts which describes the whole manifold is an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>atlas</a>"".

====""<a name="WhatIsSpacetime"></a>""What is Spacetime?====

We can describe spacetime as a manifold, a geometrical structure which can be mapped onto an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>atlas</a>"", or collection of ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>charts</a>"". In common with many definitions of mathematical structures, this does not tell us what the manifold actually is. Instead it tells us what properties a manifold has, how a manifold behaves.

Here the manifold simply replaces Newton’s conception of ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Space_and_Time_Absolute_or_Relative?>absolute space</a>"" and ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Space_and_Time_Absolute_or_Relative?>absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

<<""<span class="math"><b>Answer 3.</b> I observe that I can, in principle, choose <a href="PhilosophicalBackground#hn_Reference_Frames">reference matter</a> anywhere I wish, and that I can define <a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski coordinates</a> relative to that matter. I now define spacetime by <i>imagining</i> all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.</span>""<<

====""<a name="TangentCharts"></a>""Tangent Charts====

In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, ""<a href="http://www.rqgravity.net/GeneralRelativity#hn_The_Principle_Of_General_Covariance" >general covariance</a>"" automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in [[FoundationsOfSpecialRelativity special relativity]], using the radar method. In this case the chart is made on Minkowski spacetime, which has constant ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_3_Dimensions_plus_Time>Minkowski metric</a>"", ""<span class="math"><i>h</i>"". ""<span class="math"><i>h</i>"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""<span class="math"><i>h</i>"" does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Tangent_Space>tangent chart</a>"", an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.

For an inertial observer, metre sticks give the same result, but I have elected to use radar because it leads to a simpler analysis, and is applicable to measurements generally in the Solar System. A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation">transformed</a>"" into coordinates with Minkowski metric at the position of the observer.

====Coordinate Quantities and Proper Quantities====

According to ""<a href="GeneralRelativity#hn_The_Principle_of_General_Covariance">general covariance"" physical quantities are scalars, vectors and tensors. Vectors and tensors have different representations, dependent on the coordinate system being used. A proper quantity is the representation of a physical quantity as seen in locally Minkowski coordinates moving with the quantity being measured.

====Schwarzschild Coordinates====

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]], that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] and the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, relative to coordinates determined by radar, proper distances at the surface of the planet are greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximum_Signal_Speed speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#hn_Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

====""<a name="TheLevi-CivitaConnection"></a>""The Levi-Civita Connection====

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_Orthonormal_Bases>orthogonal</a>"". In this case the metric is diagonal (as seen in ""<a href="GeneralRelativity#hn_Schwarzschild_Coordinates">Schwarzschild coordinates</a>"") and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the ""<a href= BasicsOfCurvature#Lens>lensed and mirrored </a>"" geometries, which are actually flat. To describe curvature requires a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement>connection</a>"" in addition to the metric field. Given the metric field, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""<span class="math"><i>x</i>"", say, and another set, at ""<span class="math"><i>x</i> +<i>dx</i>"", where ""<span class="math"><i>dx</i>"" is a small displacement, such that we can meaningfully describe a vector at ""<span class="math"><i>x</i>"" as being parallel to one at ""<span class="math"><i>x</i> +<i>dx</i>"" (other types of [[http://en.wikipedia.org/wiki/Connection_(mathematics) connection]] are used to transport other types of data).

In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between vectors at nearby points using ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement">parallel displacement</a>"" in tangent space, and projecting back into the curved surface. This is the [[http://en.wikipedia.org/wiki/Levi-Civita_connection Levi-Civita connection]], defined in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Transport">parallel transport</a>"", and is path dependent. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.

A ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>chart</a>"" of spacetime need not be a physical map, like the maps in a world atlas. A mathematical idealisation suffices just as well — that is to say, the map may consist of tables of data and/or formulae. We may imagine, for example, the numbers, or coordinates, describing the times and positions of physical events mapped into a bank of computer memory. In principle, using a large enough bank of computer memory, this could be done to any precision, for as many points as one requires, and a map of a region of spacetime could be produced with any required level of detail, up to the limit of accuracy of measurement and the size of available computer memory.

##### Additions:

Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called [[http://en.wikipedia.org/wiki/Manifold manifolds]]. Spacetime is described as a [[http://en.wikipedia.org/wiki/Lorentzian_manifold#Lorentzian_manifold Lorentzian manifold]]. By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment"> Pound-Rebka experiment</a>"". In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys [[FoundationsOfSpecialRelativity special relativity]]. Together with ""<a href=http://www.rqgravity.net/Gravitation#hn_Einsteins_Law_of_Gravitation_1>Einstein’s equation</a>"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.

##### Deletions:

##### Additions:

The principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the principle in two ways, using it to establish the ""<a href="http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and the constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximum_Signal_Speed">speed of light</a>"".

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]], that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] and the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, relative to coordinates determined by radar, proper distances at the surface of the planet are greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximum_Signal_Speed speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#hn_Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]], that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] and the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, relative to coordinates determined by radar, proper distances at the surface of the planet are greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximum_Signal_Speed speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#hn_Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

##### Deletions:

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]], that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] and the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, relative to coordinates determined by radar, proper distances at the surface of the planet are greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#hn_Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

##### Additions:

Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called [[http://en.wikipedia.org/wiki/Manifold manifolds]]. Spacetime is described as a [[http://en.wikipedia.org/wiki/Lorentzian_manifold#Lorentzian_manifold Lorentzian manifold]]. By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment"> Pound-Rebka experiment</a>"". In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys [[FoundationsOfSpecialRelativity special relativity]]. Together with ""<a href=http://www.rqgravity.net/Gravitation#hn_Einsteins_Law_of_Gravitation>Einstein’s equation</a>"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.

Here the manifold simply replaces Newton’s conception of ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Space_and_Time_Absolute_or_Relative?>absolute space</a>"" and ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Space_and_Time_Absolute_or_Relative?>absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

>>""<span class="math"><i>“A physical quantity is defined by the series of operations and calculations of which it is the result”</i> — <a href=http://en.wikipedia.org/wiki/Arthur_Eddington>Sir Arthur Stanley Eddington</a>, The Mathematical Theory of Relativity, 2nd ed., p. 3, 1923</span>"">>Only measurements which are actually carried out have physical reality, and generate coordinates for physical events, whereas spacetime consists of all the ways in which this can conceivably be done. So, spacetime does not model reality — only the small subset of spacetime for which there are actual measurements corresponds to reality. Spacetime models our //conception// of reality, not reality itself. At the same time, spacetime does contain the real observational results required for comparison between theory and experiments.

""<span class="math">or, equivalently,</span>""

Here the manifold simply replaces Newton’s conception of ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Space_and_Time_Absolute_or_Relative?>absolute space</a>"" and ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Space_and_Time_Absolute_or_Relative?>absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

>>""<span class="math"><i>“A physical quantity is defined by the series of operations and calculations of which it is the result”</i> — <a href=http://en.wikipedia.org/wiki/Arthur_Eddington>Sir Arthur Stanley Eddington</a>, The Mathematical Theory of Relativity, 2nd ed., p. 3, 1923</span>"">>Only measurements which are actually carried out have physical reality, and generate coordinates for physical events, whereas spacetime consists of all the ways in which this can conceivably be done. So, spacetime does not model reality — only the small subset of spacetime for which there are actual measurements corresponds to reality. Spacetime models our //conception// of reality, not reality itself. At the same time, spacetime does contain the real observational results required for comparison between theory and experiments.

""<span class="math">or, equivalently,</span>""

##### Deletions:

Here the manifold simply replaces Newton’s conception of ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute space</a>"" and ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

>>""<span class="math"><i>“A physical quantity is defined by the series of operations and calculations of which it is the result”</i> — <a href=http://en.wikipedia.org/wiki/Arthur_Eddington>Sir Arthur Stanley Eddington</a>, The Mathematical Theory of Relativity, 2nd ed., p. 3, 1923</span."">>Only measurements which are actually carried out have physical reality, and generate coordinates for physical events, whereas spacetime consists of all the ways in which this can conceivably be done. So, spacetime does not model reality — only the small subset of spacetime for which there are actual measurements corresponds to reality. Spacetime models our //conception// of reality, not reality itself. At the same time, spacetime does contain the real observational results required for comparison between theory and experiments.

""<span class="math">or</span>""

##### Additions:

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]], that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] and the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, relative to coordinates determined by radar, proper distances at the surface of the planet are greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#hn_Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

""<span class="math">where <i>k</i> and κ are functions of position, <i>x</i>. <i>k</i>(<i>x</i>) is the redshift of light passing from the origin to <i>x</i>.</span>"" <<

""<span class="math">where <i>k</i> and κ are functions of position, <i>x</i>. <i>k</i>(<i>x</i>) is the redshift of light passing from the origin to <i>x</i>.</span>"" <<

##### Deletions:

""<span class="math">where <i>k</i> and κ are functions of position, <i>x</i>, and <i>k</i>(<i>x</i>) is the redshift of light from the origin to <i>x</i>.</span>"" <<

##### Additions:

<<""<span class="math"><b>Definition:</b> The <i>Schwarzschild radial coordinate</i> <i>r</i> is given by the proper arc length <i>l</i> of a small arc on a sphere with centre at the origin divided by the angle <i>d</i>θ subtended by that arc: <i>r</i> = <i>l</i> ⁄ <i>d</i>θ = <i>C</i> ⁄ 2π</span>"" <<

""<span class="math">where <i>k</i> and κ are functions of position, <i>x</i>, and <i>k</i>(<i>x</i>) is the redshift of light from the origin to <i>x</i>.</span>"" <<

It is is necessary to solve Einstein’s equation to determine the functions ""<span class="math"><i>k</i></span>"" and ""<span class="math">κ</span>"".

""<span class="math">where <i>k</i> and κ are functions of position, <i>x</i>, and <i>k</i>(<i>x</i>) is the redshift of light from the origin to <i>x</i>.</span>"" <<

It is is necessary to solve Einstein’s equation to determine the functions ""<span class="math"><i>k</i></span>"" and ""<span class="math">κ</span>"".

##### Deletions:

""<span class="math">where <i>k</i> and <i>k</i> are functions of positon, <i>x</i>, and <i>k</i>(<i>x</i>) is the redshift of light from the origin to <i>x</i>.</span>"" <<

##### Additions:

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-7" title="Schwarzschild coordinates" src="images/gtr/GTR-7.gif">Proper radius, <span class="math"><i>R</i></span>, is not easily measured, and is not a convenient quantity for the definition of coordinates. A commonly used choice is to define the Schwarzschild radial coordinate, <span class="math"><i>r</i></span>, such that <span class="math"><i>r</i> = <i>C</i> ⁄ 2π</span>.</table>""

<<""<span class="math"><b>Definition:</b> The <i>Schwarzschild radial coordinate</i> <i>r</i> is given by the proper arc length <i>r</i> of a small arc on a sphere at the origin divided by the angle <i>d</i>θ subtended by that arc. : <i>r</i> = <i>l</i> ⁄ <i>d</i>θ = <i>C</i> /⁄ 2π</span>"" <<

<<""<span class="math"><b>Definition:</b> The <i>Schwarzschild radial coordinate</i> <i>r</i> is given by the proper arc length <i>r</i> of a small arc on a sphere at the origin divided by the angle <i>d</i>θ subtended by that arc. : <i>r</i> = <i>l</i> ⁄ <i>d</i>θ = <i>C</i> /⁄ 2π</span>"" <<

##### Deletions:

<<""<span class="math"><b>Definition:</b> The <i>Schwarzschild radial coordinate</i> is given by the proper arc length of a small arc on a sphere at the origin divided by the angle subtended by that arc. :"" <<

##### Additions:

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-7" title="Schwarzschild coordinates" src="images/gtr/GTR-7.gif">Proper radius, <span class="math"><i>R</i></span>, is not easily measured, and is not a convenient quantity for the definition of coordinates. A commonly used choice is to define the Schwarzschild radial coordinate <span class="math"><i>r</i></span> such that <span class="math"><i>i</i> = <i>C</i> ⁄ 2π</span>.</table>""

##### Deletions:

##### Additions:

Consider a static spherical geometry, such as the region outside an isolated star or planet. An observer in a circular orbit measures the proper circumference, ""<span class="math"><i>C</i></span"", of the orbit, that is the sum of the lengths of small sections as they would be measured by stationary observers at each point. Similarly the proper radius, ""<span class="math"><i>R</i></span"", is sum of the lengths of small sections along a radius as they would be measured by stationary observers at each point on the radius.

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]] that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] and the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, in coordinates determined by radar, proper distances at the surface appear greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#hn_Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-7" title="The radar method, as used by Alf and Beth" src="images/gtr/GTR-7.gif">Proper radius, <span class="math"><i>R</i></span"", is not easily measured, and is not a convenient quantity for the definition of coordinates. A commonly used choice is to define the Schwarzschild radial coordinate <span class="math"><i>r</i></span> such that <span class="math"><i>i</i> = <i>C</i> ⁄ 2π</span>.</table>""

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]] that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] and the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, in coordinates determined by radar, proper distances at the surface appear greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#hn_Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-7" title="The radar method, as used by Alf and Beth" src="images/gtr/GTR-7.gif">Proper radius, <span class="math"><i>R</i></span"", is not easily measured, and is not a convenient quantity for the definition of coordinates. A commonly used choice is to define the Schwarzschild radial coordinate <span class="math"><i>r</i></span> such that <span class="math"><i>i</i> = <i>C</i> ⁄ 2π</span>.</table>""

##### Deletions:

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]] that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle and the [http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, in coordinates determined by radar, proper distances at the surface appear greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#hn_Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-6" title="The radar method, as used by Alf and Beth" src="images/gtr/GTR-6.gif">Proper radius, ""<span class="math"><i>R</i></span"", is not easily measured, and is not a convenient quantity for the definition of coordinates. A commonly used choice is to define the Schwarzschild radial coordinate ""<span class="math"><i>r</i></span"" such that ""<span class="math"><i>i</i> = <i>C</i> ⁄ 2π</span>"".

##### Additions:

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]] that clocks at a height above a planet run fast compared to clocks at the surface of the planet. We also know this directly from the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle and the [http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment Pound-Rebka experiment]]. Consequently, in coordinates determined by radar, proper distances at the surface appear greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>"". This means that space has a [[http://www.rqgravity.net/BasicsOfCurvature#hn_Intrinsic_and_Extrinsic_Curvature curved geometry]] in the region of a gravitating body.

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-6" title="The radar method, as used by Alf and Beth" src="images/gtr/GTR-6.gif">Proper radius, ""<span class="math"><i>R</i></span"", is not easily measured, and is not a convenient quantity for the definition of coordinates. A commonly used choice is to define the Schwarzschild radial coordinate ""<span class="math"><i>r</i></span"" such that ""<span class="math"><i>i</i> = <i>C</i> ⁄ 2π</span>"".

<<""<span class="math"><b>Definition:</b> The <i>Schwarzschild radial coordinate</i> is given by the proper arc length of a small arc on a sphere at the origin divided by the angle subtended by that arc. :"" <<

Equivalently,

<<""<span class="math"><b>Definition:</b> <i>Schwarzschild coordinates</i> have spacetime metric given by:""

""<img alt="GTR-9g" title="The metric in Schwarzschild coordinates" src="images/gtr/GTR-9g.gif" align="texttop" vspace="0">""

""<span class="math">or</span>""

""<img alt="GTR-8g" title="The line-element (metric) in Schwarzschild coordinates" src="images/gtr/GTR-8g.gif" align="texttop" vspace="0">""

""<span class="math">where <i>k</i> and <i>k</i> are functions of positon, <i>x</i>, and <i>k</i>(<i>x</i>) is the redshift of light from the origin to <i>x</i>.</span>"" <<

====""<a name="TheLevi-CivitaConnection"></a>""The Levi-Civita Connection====

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_Orthonormal_Bases>orthogonal</a>"". In this case the metric is diagonal (as seen in ""<a href="GeneralRelativity#hn_Schwarzschild_Coordinates">Schwarzschild coordinates</a>"") and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the ""<a href= BasicsOfCurvature#Lens>lensed and mirrored </a>"" geometries, which are actually flat. To describe curvature requires a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement>connection</a>"" in addition to the metric field. Given the metric field, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""<span class="math"><i>x</i>"", say, and another set, at ""<span class="math"><i>x</i> +<i>dx</i>"", where ""<span class="math"><i>dx</i>"" is a small displacement, such that we can meaningfully describe a vector at ""<span class="math"><i>x</i>"" as being parallel to one at ""<span class="math"><i>x</i> +<i>dx</i>"" (other types of [[http://en.wikipedia.org/wiki/Connection_(mathematics) connection]] are used to transport other types of data).

In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between vectors at nearby points using ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement">parallel displacement</a>"" in tangent space, and projecting back into the curved surface. This is the [[http://en.wikipedia.org/wiki/Levi-Civita_connection Levi-Civita connection]], defined in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Transport">parallel transport</a>"", and is path dependent. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-6" title="The radar method, as used by Alf and Beth" src="images/gtr/GTR-6.gif">Proper radius, ""<span class="math"><i>R</i></span"", is not easily measured, and is not a convenient quantity for the definition of coordinates. A commonly used choice is to define the Schwarzschild radial coordinate ""<span class="math"><i>r</i></span"" such that ""<span class="math"><i>i</i> = <i>C</i> ⁄ 2π</span>"".

<<""<span class="math"><b>Definition:</b> The <i>Schwarzschild radial coordinate</i> is given by the proper arc length of a small arc on a sphere at the origin divided by the angle subtended by that arc. :"" <<

Equivalently,

<<""<span class="math"><b>Definition:</b> <i>Schwarzschild coordinates</i> have spacetime metric given by:""

""<img alt="GTR-9g" title="The metric in Schwarzschild coordinates" src="images/gtr/GTR-9g.gif" align="texttop" vspace="0">""

""<span class="math">or</span>""

""<img alt="GTR-8g" title="The line-element (metric) in Schwarzschild coordinates" src="images/gtr/GTR-8g.gif" align="texttop" vspace="0">""

""<span class="math">where <i>k</i> and <i>k</i> are functions of positon, <i>x</i>, and <i>k</i>(<i>x</i>) is the redshift of light from the origin to <i>x</i>.</span>"" <<

====""<a name="TheLevi-CivitaConnection"></a>""The Levi-Civita Connection====

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_Orthonormal_Bases>orthogonal</a>"". In this case the metric is diagonal (as seen in ""<a href="GeneralRelativity#hn_Schwarzschild_Coordinates">Schwarzschild coordinates</a>"") and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the ""<a href= BasicsOfCurvature#Lens>lensed and mirrored </a>"" geometries, which are actually flat. To describe curvature requires a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement>connection</a>"" in addition to the metric field. Given the metric field, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""<span class="math"><i>x</i>"", say, and another set, at ""<span class="math"><i>x</i> +<i>dx</i>"", where ""<span class="math"><i>dx</i>"" is a small displacement, such that we can meaningfully describe a vector at ""<span class="math"><i>x</i>"" as being parallel to one at ""<span class="math"><i>x</i> +<i>dx</i>"" (other types of [[http://en.wikipedia.org/wiki/Connection_(mathematics) connection]] are used to transport other types of data).

In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between vectors at nearby points using ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement">parallel displacement</a>"" in tangent space, and projecting back into the curved surface. This is the [[http://en.wikipedia.org/wiki/Levi-Civita_connection Levi-Civita connection]], defined in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Transport">parallel transport</a>"", and is path dependent. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.

##### Deletions:

##### Additions:

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]] that clocks at a height above a planet run fast compared to clocks at the surface of the planet. Consequently, in coordinates determined by radar, proper distances at the surface appear greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>"" times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span>""

##### Deletions:

##### Additions:

Consider a static spherical geometry, such as the region outside an isolated star or planet. An observer in a circular orbit measures the proper circumference, ""<span class="math"><i>C</i></span"", of the orbit, that is the sum of the lengths of small sections as they would be measured by stationary observers at each point. Similarly the proper radius, ""<span class="math"><i>R</i></span"", is sum of the lengths of small sections along a radius as they would be measured by stationary observers at each point on the radius.

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]] that clocks at a height above a planet run fast compared to clocks at the surface of the planet. Consequently, in coordinates determined by radar, proper distances at the surface appear greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>2 times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span""

It is observed, for example using clocks on [[http://en.wikipedia.org/wiki/GPS_(satellite) GPS satellites]] that clocks at a height above a planet run fast compared to clocks at the surface of the planet. Consequently, in coordinates determined by radar, proper distances at the surface appear greater than proper distances in orbit (to preserve the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information speed of light]] which is used for the [[http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates definition of distances]] locally). Consequently the proper length of the circumference is greater than ""<span class="math">2π</span>2 times the proper radius, ""<span class="math"><i>C</i> > 2π<i>R</i></span""

##### Additions:

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://www.rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinates>coordinate representation</a>"" changes with the coordinate system, but relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_Contravariant_and_Covariant_Vectors>covariant</a>"". [[http://www.rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

The principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the principle in two ways, using it to establish the ""<a href="http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and the constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information">speed of light</a>"".

With [[http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity general relavity]], Einstein further extended the special principle, formulating the general principle and the principle of general covariance, but the special principle remains significant, as the means by which we use Newton’s laws to identify [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial frames]] as defining a special class of coordinate systems in which there is a natural correspondence between mathematical structure and physical behaviour.

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. Since objects always move in the direction of their velocity vectore, the immediate consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesics geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

<<""<span class="math"><b>The Equivalence Principle:</b>"" ""<span class="math">We ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system. (Einstein 1907).""<<

Here the manifold simply replaces Newton’s conception of ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute space</a>"" and ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

====Coordinate Quantities and Proper Quantities====

In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer (typically) at a distance, and the same quantities as they would be described by a inertial observer who determines them locally in Minkowski coordinates.

<<""<span class="math"><b>Definition:</b> A <i>Proper quantity</i> is a physical quantities as it would be measured in Minkowski coordinates by an inertial observer moving with that quantity.""<<

For example:

According to ""<a href="GeneralRelativity#hn_The_Principle_of_General_Covariance">general covariance"" physical quantities are scalars, vectors and tensors. Vectors and tensors have different representations, dependent on the coordinate system being used. A proper quantity is the representation of a physical quantity as seen in locally Minkowski coordinates moving with the quantity being measured.

====Schwarzschild Coordinates====

The principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the principle in two ways, using it to establish the ""<a href="http://www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and the constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information">speed of light</a>"".

With [[http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity general relavity]], Einstein further extended the special principle, formulating the general principle and the principle of general covariance, but the special principle remains significant, as the means by which we use Newton’s laws to identify [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial frames]] as defining a special class of coordinate systems in which there is a natural correspondence between mathematical structure and physical behaviour.

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. Since objects always move in the direction of their velocity vectore, the immediate consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesics geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

<<""<span class="math"><b>The Equivalence Principle:</b>"" ""<span class="math">We ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system. (Einstein 1907).""<<

Here the manifold simply replaces Newton’s conception of ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute space</a>"" and ""<a href=http://www.rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

====Coordinate Quantities and Proper Quantities====

In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer (typically) at a distance, and the same quantities as they would be described by a inertial observer who determines them locally in Minkowski coordinates.

<<""<span class="math"><b>Definition:</b> A <i>Proper quantity</i> is a physical quantities as it would be measured in Minkowski coordinates by an inertial observer moving with that quantity.""<<

For example:

According to ""<a href="GeneralRelativity#hn_The_Principle_of_General_Covariance">general covariance"" physical quantities are scalars, vectors and tensors. Vectors and tensors have different representations, dependent on the coordinate system being used. A proper quantity is the representation of a physical quantity as seen in locally Minkowski coordinates moving with the quantity being measured.

====Schwarzschild Coordinates====

##### Deletions:

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Coordinates>coordinate representation</a>"" changes with the coordinate system, but relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Contravariant_and_Covariant_Vectors>covariant</a>"". [[http://rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

The principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the principle in two ways, using it to establish the ""<a href="http://rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and the constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information">speed of light</a>"".

With [[http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity general relavity]], Einstein further extended the special principle, formulating the general principle and the principle of general covariance, but the special principle remains significant, as the means by which we use Newton’s laws to identify [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial frames]] as defining a special class of coordinate systems in which there is a natural correspondence between mathematical structure and physical behaviour.

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. Since objects always move in the direction of their velocity vectore, the immediate consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesics geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a [[http://rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

<<""<span class="math"><b>The Equivalence Principle:</b>"" ""<span class="math">We ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system" (Einstein 1907).""<<

Here the manifold simply replaces Newton’s conception of ""<a href=http://rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute space</a>"" and ""<a href=http://rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

====""<a name="CoordinateTimeAndProperTime"></a>""Coordinate Quantities and Proper Quantities====

In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer at a distance, and the same quantities as they would be described by a observer who determines them locally in Minkowski coordinates. Let Alf be an observer, with a clock at some point, ""<span class="math">A</span>"", the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at ""<span class="math">B</span>"", the origin of Beth’s coordinates. Alf and Beth both determine locally ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski Coordinates</a>"". Alf’s time axis is denoted the 0-axis, and his space axes are labelled ""<span class="math">1"", ""<span class="math">2"", ""<span class="math">3"". Beth’s coordinates are denoted with primes, ""<span class="math">0'"", ""<span class="math">1'"", ""<span class="math">2'"", ""<span class="math">3'"".

Using ""<a href="http://www.rqgravity.net/GeneralRelativity#hn_The_Principle_Of_General_Covariance" >general covariance</a>"", each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing ""<span class="math"><i>t</i>'"" seconds for a stationary object at the origin of Beth’s coordinates ""<span class="math">τ = τ<sup><i>i'</i></sup> = (<i>t'</i>, 0, 0, 0)"". ""<span class="math"><i>t'</i>"" is the actual amount of time measured by Beth using her own clock, and is known as //proper time//. In Alf’s coordinates, ""<img alt="GTR-1" title="coordinate transformation" src="images/gtr/GTR-1.gif" align="texttop" vspace="0">"" is found by ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation">coordinate transformation</a>"". If Beth is stationary in Alf’s coordinates, ""<span class="math">τ = τ<sup><i>i</i>'</sup> = (<i>kt'</i>, 0, 0, 0)"", where ""<span class="math"><i>k</i>"" is the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#hn_Gravitational_Red_Shift">gravitational redshift</a>"" factor, ""<span class="math"><i>k</i> = 1 + <i>z</i>"".

====""<a name="SchwarzschildCoordinates"></a>""Schwarzschild Coordinates====

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-5" title="rate of clocks and gravitational redshift" src="images/gtr/GTR-5.gif">In a static geometry, an observer, Alf, defines <a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_Spherical_Coordinates>spherical coordinates</a> by the radar method, using time <span class="math"><i>t</i></span>, determined from a clock at an origin at <span class="math">A</span>. Coordinate distance, <span class="math"><i>r*</i></span>, from <span class="math">A</span> is defined by setting the radial speed of light to unity. Spherical coordinates are <a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Orthonormal_Bases>orthogonal</a> so that the <a href=http://rqgravity.net/GeneralRelativity#hn_The_Spacetime_Metric>spacetime metric</a> is diagonal. A second observer, Beth, is at a constant position, <span class="math">B</span>, at radial coordinate <span class="math"><i>r</i></span> in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor <span class="math"><i>k</i> = 1 + <i>z</i> > 1</span> (the argument below also holds for <span class="math"><i>k</i> < 1</span>). Then light transmitted from Alf to Beth is redshifted by factor <span class="math"><i>k</i></span> and an interval <span class="math"><i>t</i></span> of Alf’s coordinate time at <span class="math">B</span> is measured by Beth as proper time interval <span class="math"><i>t' = kt</i></span>. Then the <a href=http://www.rqgravity.net/GeneralRelativity#hn_The_Spacetime_Metric>spacetime metric</a> has <span class="math"><i>g</i><sub>00</sub> = <i>k</i><sup>2</sup></i></span>.</td></table>""

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-6" title="The radar method, as used by Alf and Beth" src="images/gtr/GTR-6.gif">Beth determines proper distances local to <span class="math">B</span> using the radar method, with lightspeed equal to unity. Since Beth’s clock runs faster than Alf’s, proper distances local to Beth are greater than corresponding coordinate distances in Alf’s coordinates by a factor <span class="math"><i>k</i></span>. Using unit light speed Beth calculates coordinate distance <span class="math"><i>r = kr*</i></span> to Alf.</td></table>""""Using <span class="math"><i>r</i></span> as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of <span class="math">B</span>, so that the angle subtended at <span class="math">A</span> by a small rod of proper length <span class="math"><i>l</i></span> at <span class="math">B</span>, perpendicular to <span class="math">AB</span> is <span class="math"><i>l / r</i></span>, as it would be in flat space. This is the defining condition for <i>Schwarzschild coordinates</i>.""

<<""<span class="math"><b>Definition:</b> The <i>Schwarzschild radial coordinate</i> is given by the proper arc length of a small arc on a sphere at the origin divided by the angle subtended by that arc. :"" <<

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-7" title="Schwarzschild coordinates use time from Alf’s clock, together with radial coordinate, r, as determined by Beth" src="images/gtr/GTR-7.gif">In Schwarzschild coordinates a ring of short rods at radial distance <span class="math"><i>r</i></span> from <span class="math">A</span> can be drawn on Beth’s map to form a continuous circle, centre <span class="math">A</span>, without overlaps. It follows that, in spherical coordinates with origin <span class="math">A</span> and radial distance <span class="math"><i>r</i></span>, <span class="math"><i>g</i><sub>22</sub> = −<i>r</i><sup>2</sup></span> and <span class="math"><i>g</i><sub>33</sub> = −<i>r</i><sup>2</sup>sin<sup>2</sup>θ</span>, and that, since Beth has increased the scale of local distances by a factor <span class="math"><i>k</i></span>, <span class="math"><i>g</i><sub>11</sub> = −<i>k</i><sup>−2</sup></span>.</td></table>""

<<""<span class="math"><b>Theorem:</b> Schwarzschild coordinates in vacuum have spacetime metric given by:""

""<img alt="GTR-9g" title="The metric in Schwarzschild coordinates" src="images/gtr/GTR-9g.gif" align="texttop" vspace="0">""

""<span class="math">or</span>""

""<img alt="GTR-8g" title="The line-element (metric) in Schwarzschild coordinates" src="images/gtr/GTR-8g.gif" align="texttop" vspace="0">""

""<span class="math">where <i>k</i> and <i>k</i> are functions of positon, <i>x</i>, and <i>k</i>(<i>x</i>) is the redshift of light from the origin to <i>x</i>.</span>"" <<

This result is usually found by solving ""<a href=rqgravity.net/Gravitation#hn_Einsteins_Law_of_Gravitation>Einstein’s field equation</a>"". Here I have established it purely geometrically, because I think this gives greater insight and because it will simplify (slightly) the [[Schwarzschild calculation]] of the ""<a href=rqgravity.net/Gravitation#hn_The_Schwarzschild_Solution>Schwarzschild solution</a>"".

====""<a name="TheLevi-CivitaConnection"></a>""The Levi-Civita Connection====

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Orthonormal_Bases>orthogonal</a>"". In this case the metric is diagonal (as seen in ""<a href="GeneralRelativity#hn_Schwarzschild_Coordinates">Schwarzschild coordinates</a>"") and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the ""<a href= BasicsOfCurvature#Lens>lensed and mirrored </a>"" geometries, which are actually flat. To describe curvature requires a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement>connection</a>"" in addition to the metric field. Given the metric field, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""<span class="math"><i>x</i>"", say, and another set, at ""<span class="math"><i>x</i> +<i>dx</i>"", where ""<span class="math"><i>dx</i>"" is a small displacement, such that we can meaningfully describe a vector at ""<span class="math"><i>x</i>"" as being parallel to one at ""<span class="math"><i>x</i> +<i>dx</i>"" (other types of [[http://en.wikipedia.org/wiki/Connection_(mathematics) connection]] are used to transport other types of data).

In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between vectors at nearby points using ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement">parallel displacement</a>"" in tangent space, and projecting back into the curved surface. This is the [[http://en.wikipedia.org/wiki/Levi-Civita_connection Levi-Civita connection]], defined in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Transport">parallel transport</a>"", and is path dependent. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.

##### Additions:

""<a href="GeneralRelativity#hn_Coordinate_Quantities_and_Proper_Quantities">Coordinate Quantities and Proper Quantities</a>""

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. Since objects always move in the direction of their velocity vectore, the immediate consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesics geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a [[http://rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

For an inertial observer, metre sticks give the same result, but I have elected to use radar because it leads to a simpler analysis, and is applicable to measurements generally in the Solar System. A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation">transformed</a>"" into coordinates with Minkowski metric at the position of the observer.

====""<a name="CoordinateTimeAndProperTime"></a>""Coordinate Quantities and Proper Quantities====

In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer at a distance, and the same quantities as they would be described by a observer who determines them locally in Minkowski coordinates. Let Alf be an observer, with a clock at some point, ""<span class="math">A</span>"", the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at ""<span class="math">B</span>"", the origin of Beth’s coordinates. Alf and Beth both determine locally ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski Coordinates</a>"". Alf’s time axis is denoted the 0-axis, and his space axes are labelled ""<span class="math">1"", ""<span class="math">2"", ""<span class="math">3"". Beth’s coordinates are denoted with primes, ""<span class="math">0'"", ""<span class="math">1'"", ""<span class="math">2'"", ""<span class="math">3'"".

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. Since objects always move in the direction of their velocity vectore, the immediate consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesics geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a [[http://rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

For an inertial observer, metre sticks give the same result, but I have elected to use radar because it leads to a simpler analysis, and is applicable to measurements generally in the Solar System. A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation">transformed</a>"" into coordinates with Minkowski metric at the position of the observer.

====""<a name="CoordinateTimeAndProperTime"></a>""Coordinate Quantities and Proper Quantities====

In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer at a distance, and the same quantities as they would be described by a observer who determines them locally in Minkowski coordinates. Let Alf be an observer, with a clock at some point, ""<span class="math">A</span>"", the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at ""<span class="math">B</span>"", the origin of Beth’s coordinates. Alf and Beth both determine locally ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski Coordinates</a>"". Alf’s time axis is denoted the 0-axis, and his space axes are labelled ""<span class="math">1"", ""<span class="math">2"", ""<span class="math">3"". Beth’s coordinates are denoted with primes, ""<span class="math">0'"", ""<span class="math">1'"", ""<span class="math">2'"", ""<span class="math">3'"".

##### Deletions:

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesic_Motion geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a [[http://rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation">transformed</a>"" into coordinates with Minkowski metric at the position of the observer.

====""<a name="CoordinateTimeAndProperTime"></a>""Coordinate Time and Proper Time====

In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer at a distance, and the same quantities as they would be described by a observer who determines them locally. Let Alf be an observer, with a clock at some point, ""<span class="math">A</span>"", the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at ""<span class="math">B</span>"", the origin of Beth’s coordinates. Alf and Beth both determine locally ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski Coordinates</a>"". Alf’s time axis is denoted the 0-axis, and his space axes are labelled ""<span class="math">1"", ""<span class="math">2"", ""<span class="math">3"". Beth’s coordinates are denoted with primes, ""<span class="math">0'"", ""<span class="math">1'"", ""<span class="math">2'"", ""<span class="math">3'"".

====""<a name="TheSpacetimeMetric"></a>""The Spacetime Metric====

The ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_The_Metric" >metric</a>"", ""<span class="math"><i>g<sub>ij</sub></i>"", lowers the indices of ""<a href="http://www.rqgravity.net/IntroductionToTensors#hn_Index_Gymnastics" >contravariant</a>"" vectors in such a way that the inner product between vectors ""<span class="math"><i>x</i>"" and ""<span class="math"><i>y</i>"" is an ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_The_Dot_Product" >invariant</a>"". Using Minkowski spacetime, as described in [[http://rqgravity.net/FoundationsOfSpecialRelativity special relativity]] the metric is

""<img alt="GTR-2" title="The dot product" src="images/gtr/GTR-2.gif" align="texttop" vspace="0">""

In particular, the magnitude ""<span class="math">|<i>x</i>|"" of the vector ""<span class="math"><i>x</i>"" is invariant,

""<img alt="GTR-3" title="vector magnitude" src="images/gtr/GTR-3.gif" align="texttop" vspace="0">""

so that the metric is a means of determining magnitude of a vector in any coordinate system.

The laws of physics local to Beth, as described by Beth, use proper times and distances determined in her local measurements. If Alf is to analyse physical processes close to Beth, he must also determine proper times and distances, using remote measurements. Since the difference between Alf’s and Beth’s measurements is just that Alf and Beth define coordinates differently, their measurements are related by ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation>coordinate transformation</a>"". The definition of the metric ensures that, when Alf applies it to his own measurements, the magnitudes returned will be the proper times and distances of quantities local to Beth. The ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_The_Metric>metric</a>"" determines that the quantity ""<span class="math">|<i>x</i>|<sup>2</sup> = <i>g<sub>ij</sub>x<sup>i</sup>y<sup>j</sup></span>"" is the same in any coordinates. Beth could be anywhere in Alf’s coordinate space. For each point, ""<span class="math"><i>x</i>"", where Beth could be, there is a different metric, ""<span class="math"><i>g<sub>ij</sub></i>(<i>x</i>)</i>"".

<<""<span class="math"><b>Definition:</b> The <i>spacetime metric</i> is defined, on a given coordinate system, by""

""<img alt="GTR-4g" title="The spacetime metric is a function on spacetime coordinates" src="images/gtr/GTR-4g.gif" align="texttop" vspace="0">""

The spacetime metric is often simply called the metric. One should avoid this abuse of language, because the spacetime metric is a ""<a href=http://www.rqgravity.net/IntroductionToTensors#hn_Tensor_Fields>tensor field</a>"", that is to say it is a function having a different metric value at each point in spacetime. Calling it simply “the metric” confuses a ""<a href=http://www.rqgravity.net/IntroductionToTensors#hn_Tensor_Fields>metric field</a>"", which is a function of coordinate space, with the ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_The_Metric" >metric</a>"" at a given position.

##### Additions:

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesic_Motion geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle, the gravitational force we experience at the surface of the Earth is simply a [[http://rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, ""<a href="http://www.rqgravity.net/GeneralRelativity#hn_The_Principle_Of_General_Covariance" >general covariance</a>"" automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in [[FoundationsOfSpecialRelativity special relativity]], using the radar method. In this case the chart is made on Minkowski spacetime, which has constant ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_3_Dimensions_plus_Time>Minkowski metric</a>"", ""<span class="math"><i>h</i>"". ""<span class="math"><i>h</i>"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""<span class="math"><i>h</i>"" does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Tangent_Space>tangent chart</a>"", an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.

In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, ""<a href="http://www.rqgravity.net/GeneralRelativity#hn_The_Principle_Of_General_Covariance" >general covariance</a>"" automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in [[FoundationsOfSpecialRelativity special relativity]], using the radar method. In this case the chart is made on Minkowski spacetime, which has constant ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_3_Dimensions_plus_Time>Minkowski metric</a>"", ""<span class="math"><i>h</i>"". ""<span class="math"><i>h</i>"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""<span class="math"><i>h</i>"" does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Tangent_Space>tangent chart</a>"", an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.

##### Deletions:

In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, ""<a href="http://www.rqgravity.net/GeneralRelativity#hn_The_Principle_Of_General_Covariance" >general covariance</a>"" automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in [[FoundationsOfSpecialRelativity Special relativity]], using the radar method. In this case the chart is made on Minkowski spacetime, which has constant ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_3_Dimensions_plus_Time>Minkowski metric</a>"", ""<span class="math"><i>h</i>"". ""<span class="math"><i>h</i>"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""<span class="math"><i>h</i>"" does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Tangent_Space>tangent chart</a>"", an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.

##### Additions:

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesic_Motion geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without ever mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle the gravitational force we experience at the surface of the Earth is simply a [[http://rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

##### Deletions:

##### Additions:

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesic_Motion geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without every mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle the gravitational force we experience at the surface of the Earth is simply a [[http://rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

##### Deletions:

##### Additions:

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesic_Motion geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without every mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity. According to the equivalence principle the gravitational force we experience at the surface of the Earth is just the same as a [[http://rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces g-force]] experienced by an observer in an accelerated frame of reference.

##### Deletions:

##### Additions:

""<a href="GeneralRelativity#hn_The_Principle_of_General_Covariance">The Principle of General Covariance </a>""

""<a href="GeneralRelativity#hn_The_Special_Principle_of_Relativity">The Special Principle of Relativity</a>""

""<a href="GeneralRelativity#hn_The_Equivalence_Principle">The Equivalence Principle</a>""

""<a href="GeneralRelativity#hn_The_Special_Principle_of_Relativity">The Special Principle of Relativity</a>""

""<a href="GeneralRelativity#hn_The_Equivalence_Principle">The Equivalence Principle</a>""

##### Deletions:

##### Additions:

<<""<span class="math"><b>Answer 3.</b> I observe that I can, in principle, choose <a href="PhilosophicalBackground#hn_Reference_Frames">reference matter</a> anywhere I wish, and that I can define <a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski coordinates</a> relative to that matter. I now define spacetime by <i>imagining</i> all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.</span>""<<

##### Deletions:

##### Additions:

<<""<span class="math"><b>Answer 3.</b> I observe that I can, in principle, choose <a href="PhilosophicalBackground#hn_Reference_Frames">reference matter</a> anywhere I wish, and that I can define <a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski coordinates</a> relative to that matter. I now define spacetime by //imagining// all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.</span>""<<

##### Deletions:

##### Additions:

""<a href="GeneralRelativity#hn_What_is_Spacetime">What is Spacetime?</a>""

<<""<span class="math"><b>Answer 1.</b> There exists a <a href=http://plato.stanford.edu/entries/spacetime-theories/#5.2>substantive spacetime</a>, which is modelled (i.e. described) by the mathematical structure of a manifold.</span>""<<

<<""<span class="math"><b>Answer 2.</b> The mathematical structures of physics do not model anything. They are just algorithms whose validity rests only on correspondence between prediction and experiment.</span>""<<

This position can be regarded as the current orthodoxy. It is adopted by many theoretical physicists, especially quantum field theorists. There is no arguing with it. It is both as solid, and as absurd, as schoolboy [[http://en.wikipedia.org/wiki/Solipsism solipsism]]. It is clear to me that realism is a prerequisite for science as a meaningful activity, and that when I say "there is a tree in the park" I am describing reality. Our ability to describe a tree in the park refutes, at a very obvious level, the proposition that reality cannot be described. I always find it surprising when physicists advocate the idea that reality cannot be described, because it undermines the very purpose of physics. In fact, Einstein has already refuted the idea that science requires us to infer theory from agreement between prediction and experiment. [[FoundationsOfSpecialRelativity Special relativity]] is based on based on empirically verifiable //postulates//, the operational definitions of measurement. Einstein’s argument is the archetype according to which we should formulate modern scientific theory, starting with how we //define// the numerical quantities which we use in the scientific study of nature. Special relativity is imported, as local theory, into general relativity, and provides the basis for understanding what spacetime actually is:

<<""<span class="math"><b>Answer 3.</b> I observe that I can, in principle, choose ""<a href="PhilosophicalBackground#hn_Reference_Frames">reference matter</a>"" anywhere I wish, and that I can define ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski coordinates</a>"" relative to that matter. I now define spacetime by //imagining// all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.<<

>>""<span class="math"><i>“A physical quantity is defined by the series of operations and calculations of which it is the result”</i> — <a href=http://en.wikipedia.org/wiki/Arthur_Eddington>Sir Arthur Stanley Eddington</a>, The Mathematical Theory of Relativity, 2nd ed., p. 3, 1923</span."">>Only measurements which are actually carried out have physical reality, and generate coordinates for physical events, whereas spacetime consists of all the ways in which this can conceivably be done. So, spacetime does not model reality — only the small subset of spacetime for which there are actual measurements corresponds to reality. Spacetime models our //conception// of reality, not reality itself. At the same time, spacetime does contain the real observational results required for comparison between theory and experiments.

<<""<span class="math"><b>Answer 1.</b> There exists a <a href=http://plato.stanford.edu/entries/spacetime-theories/#5.2>substantive spacetime</a>, which is modelled (i.e. described) by the mathematical structure of a manifold.</span>""<<

<<""<span class="math"><b>Answer 2.</b> The mathematical structures of physics do not model anything. They are just algorithms whose validity rests only on correspondence between prediction and experiment.</span>""<<

This position can be regarded as the current orthodoxy. It is adopted by many theoretical physicists, especially quantum field theorists. There is no arguing with it. It is both as solid, and as absurd, as schoolboy [[http://en.wikipedia.org/wiki/Solipsism solipsism]]. It is clear to me that realism is a prerequisite for science as a meaningful activity, and that when I say "there is a tree in the park" I am describing reality. Our ability to describe a tree in the park refutes, at a very obvious level, the proposition that reality cannot be described. I always find it surprising when physicists advocate the idea that reality cannot be described, because it undermines the very purpose of physics. In fact, Einstein has already refuted the idea that science requires us to infer theory from agreement between prediction and experiment. [[FoundationsOfSpecialRelativity Special relativity]] is based on based on empirically verifiable //postulates//, the operational definitions of measurement. Einstein’s argument is the archetype according to which we should formulate modern scientific theory, starting with how we //define// the numerical quantities which we use in the scientific study of nature. Special relativity is imported, as local theory, into general relativity, and provides the basis for understanding what spacetime actually is:

<<""<span class="math"><b>Answer 3.</b> I observe that I can, in principle, choose ""<a href="PhilosophicalBackground#hn_Reference_Frames">reference matter</a>"" anywhere I wish, and that I can define ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski coordinates</a>"" relative to that matter. I now define spacetime by //imagining// all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.<<

>>""<span class="math"><i>“A physical quantity is defined by the series of operations and calculations of which it is the result”</i> — <a href=http://en.wikipedia.org/wiki/Arthur_Eddington>Sir Arthur Stanley Eddington</a>, The Mathematical Theory of Relativity, 2nd ed., p. 3, 1923</span."">>Only measurements which are actually carried out have physical reality, and generate coordinates for physical events, whereas spacetime consists of all the ways in which this can conceivably be done. So, spacetime does not model reality — only the small subset of spacetime for which there are actual measurements corresponds to reality. Spacetime models our //conception// of reality, not reality itself. At the same time, spacetime does contain the real observational results required for comparison between theory and experiments.

##### Deletions:

<<**Answer 1.** There exists a [[http://plato.stanford.edu/entries/spacetime-theories/#5.2 substantive spacetime]], which is modelled (i.e. described) by the mathematical structure of a manifold.<<

<<**Answer 2.** The mathematical structures of physics do not model anything. They are just algorithms whose validity rests only on correspondence between prediction and experiment.<<

This position can be regarded as the current orthodoxy. It is adopted by many theoretical physicists, especially quantum field theorists. There is no arguing with it. It is both as solid, and as absurd, as schoolboy [[http://en.wikipedia.org/wiki/Solipsism solipsism]]. It is clear to me that realism is a prerequisite for science as a meaningful activity, and that when I say "there is a tree in the park" I am describing reality. Our ability to describe a tree in the park refutes, at a very obvious level, the proposition that reality cannot be described.

I always find it surprising when physicists advocate the idea that reality cannot be described, because it undermines the very purpose of physics. In fact, Einstein has already refuted the idea that science requires us to infer theory from agreement between prediction and experiment. [[FoundationsOfSpecialRelativity Special relativity]] is based on based on empirically verifiable //postulates//, the operational definitions of measurement. Einstein’s argument is the archetype according to which we should formulate modern scientific theory, starting with how we //define// the numerical quantities which we use in the scientific study of nature. Special relativity is imported, as local theory, into general relativity, and provides the basis for understanding what spacetime actually is:

<<**Answer 3.** I observe that I can, in principle, choose ""<a href="PhilosophicalBackground#hn_Reference_Frames">reference matter</a>"" anywhere I wish, and that I can define ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski coordinates</a>"" relative to that matter. I now define spacetime by //imagining// all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.<<

>>""<span class="math">A physical quantity is defined by the series of operations and calculations of which it is the result — Eddington, The mathematical theory of relativity, p3, 1923)."">>Only measurements which are actually carried out have physical reality, and generate coordinates for physical events, whereas spacetime consists of all the ways in which this can conceivably be done. So, spacetime does not model reality — only the small subset of spacetime for which there are actual measurements corresponds to reality. Spacetime models our //conception// of reality, not reality itself. At the same time, spacetime does contain the real observational results required for comparison between theory and experiments.

##### Additions:

>>""<span class="math">A physical quantity is defined by the series of operations and calculations of which it is the result — Eddington, The mathematical theory of relativity, p3, 1923)."">>Only measurements which are actually carried out have physical reality, and generate coordinates for physical events, whereas spacetime consists of all the ways in which this can conceivably be done. So, spacetime does not model reality — only the small subset of spacetime for which there are actual measurements corresponds to reality. Spacetime models our //conception// of reality, not reality itself. At the same time, spacetime does contain the real observational results required for comparison between theory and experiments.

##### Deletions:

No Differences

##### Additions:

Despite the extreme reasonableness of this assumption, it is extraordinary in its strength. It appears that the mathematical constraints it places on physical laws are so strong that the only possible theory of gravity or of the structure of space and time is the geometrical theory described by general relativity. As will be seen in the [[Gravity mathematical development]], general covariance determines the whole of the theory of general relativity, while the special principle and the equivalence principle relate mathematical structure to physical observation.

The principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the principle in two ways, using it to establish the ""<a href="http://rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and the constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information">speed of light</a>"".

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesic_Motion geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without every mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity.

The principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the principle in two ways, using it to establish the ""<a href="http://rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and the constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information">speed of light</a>"".

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter</span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesic_Motion geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Equivalence_Principle_1 equivalence principle]] was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, with all its predictions, without every mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Its value is to relate the mathematical theory to our intuitive understanding and experience of gravity.

##### Deletions:

The special principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the special principle in two ways, using it to establish the ""<a href="http://rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information">speed of light</a>"".

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter"<./span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesic_Motion geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the equivalence principle was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, together with all its predictions, without every mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Nevertheless it is used to relate the mathematical theory to our intuitive understanding and experience of gravity.

##### Additions:

The special principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the special principle in two ways, using it to establish the ""<a href="http://rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information">speed of light</a>"".

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter"<./span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesic_Motion geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the equivalence principle was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, together with all its predictions, without every mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Nevertheless it is used to relate the mathematical theory to our intuitive understanding and experience of gravity.

>>""<span class="math"><b>N1*:</b> An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter"<./span>"">>In a geometrical theory of spacetime, [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter Newton’s first law]] determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is [[http://www.rqgravity.net/BasicsOfCurvature#hn_Geodesic_Motion geodesic motion]], meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the equivalence principle was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, together with all its predictions, without every mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Nevertheless it is used to relate the mathematical theory to our intuitive understanding and experience of gravity.

##### Deletions:

The special principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the special principle in two ways, using it to establish the ""<a href="http://rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information>speed of light</a>"".

In a geometrical theory of spacetime, Newton’s first law determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is geodesic motion, meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the equivalence principle was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, together with all its predictions, without every mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Nevertheless it is required to relate the mathematical theory our intuitive understanding and experience of gravity.

##### Additions:

======[[TheEquivalencePrinciple ←]] The General Theory of Relativity [[PhysicalPrinciples ↑]] [[MathematicalMethods →]]======

General relativity is based on three fundamental principles, the general principle , the special principle, and the principle of equivalence, together with the empirical fact that we establish coordinate systems locally through the physical measurement of time and position. The proper application of these principles requires a detailed understanding of mathematics. This page will discuss these ideas and their implication, and will motivate the mathematical treatments which follow.

====The Principle of General Covariance ====

together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy) and the empirical fact that we do not have a universe obeying [[https://en.wikipedia.org/wiki/Galilean_invariance Newtonian relativity]]. These notions are encapsulated in the ""<a href="PhilosophicalBackground#hn_The_General_Principle_of_Relativity">general principle of relativity</a>"",

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Coordinates>coordinate representation</a>"" changes with the coordinate system, but relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Contravariant_and_Covariant_Vectors>covariant</a>"". [[http://rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

Despite the extreme reasonableness of this assumption, it is extraordinary in its strength. It appears that the mathematical constraints it places on physical laws are so restrictive that the only possible theory of gravity or of the structure of space and time is the geometrical theory described by general relativity. As far as mathematical structure is concerned, general covariance determines the whole of the theory of general relativity, while the special principle and the equivalence principle are required to relate mathematical structure to physical observation.

====The Special Principle of Relativity ====

<<""<span class="math"><b>Special principle of relativity:</b> If a system of coordinates <i>K</i> is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates <i>K'</i> moving in uniform translation relatively to <i>K</i>. —Albert Einstein: The foundation of the general theory of relativity</span>""<<

The special principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the special principle in two ways, using it to establish the ""<a href="http://rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information>speed of light</a>"".

With [[http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity general relavity]], Einstein further extended the special principle, formulating the general principle and the principle of general covariance, but the special principle remains significant, as the means by which we use Newton’s laws to identify [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial frames]] as defining a special class of coordinate systems in which there is a natural correspondence between mathematical structure and physical behaviour.

====The Equivalence Principle====

In a geometrical theory of spacetime, Newton’s first law determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is geodesic motion, meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the equivalence principle was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, together with all its predictions, without every mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Nevertheless it is required to relate the mathematical theory our intuitive understanding and experience of gravity.

<<""<span class="math"><b>The Equivalence Principle:</b>"" ""<span class="math">We ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system" (Einstein 1907).""<<

====The Spacetime Manifold====

[[GeneralRelativity The General Theory of Relativity ↑]] [[MathematicalMethods Mathematical Methods →]]

General relativity is based on three fundamental principles, the general principle , the special principle, and the principle of equivalence, together with the empirical fact that we establish coordinate systems locally through the physical measurement of time and position. The proper application of these principles requires a detailed understanding of mathematics. This page will discuss these ideas and their implication, and will motivate the mathematical treatments which follow.

====The Principle of General Covariance ====

together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy) and the empirical fact that we do not have a universe obeying [[https://en.wikipedia.org/wiki/Galilean_invariance Newtonian relativity]]. These notions are encapsulated in the ""<a href="PhilosophicalBackground#hn_The_General_Principle_of_Relativity">general principle of relativity</a>"",

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Coordinates>coordinate representation</a>"" changes with the coordinate system, but relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Contravariant_and_Covariant_Vectors>covariant</a>"". [[http://rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

Despite the extreme reasonableness of this assumption, it is extraordinary in its strength. It appears that the mathematical constraints it places on physical laws are so restrictive that the only possible theory of gravity or of the structure of space and time is the geometrical theory described by general relativity. As far as mathematical structure is concerned, general covariance determines the whole of the theory of general relativity, while the special principle and the equivalence principle are required to relate mathematical structure to physical observation.

====The Special Principle of Relativity ====

<<""<span class="math"><b>Special principle of relativity:</b> If a system of coordinates <i>K</i> is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates <i>K'</i> moving in uniform translation relatively to <i>K</i>. —Albert Einstein: The foundation of the general theory of relativity</span>""<<

The special principle of relativity was first explicitly stated by [[http://en.wikipedia.org/wiki/Galileo_Galilei Galileo]], using an argument known as [[http://en.wikipedia.org/wiki/Galileo%27s_ship Galileo’s ship]], which he also tested by dropping objects from the mast of a moving ship. In [[FoundationsOfSpecialRelativity special relativity]], Einstein extended the application of the special principle in two ways, using it to establish the ""<a href="http://rqgravity.net/FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">coordinate system</a>"" and constancy of the ""<a href="http:// www.rqgravity.net/FoundationsOfSpecialRelativity#hn_Maximal_Speed_of_Information>speed of light</a>"".

With [[http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity general relavity]], Einstein further extended the special principle, formulating the general principle and the principle of general covariance, but the special principle remains significant, as the means by which we use Newton’s laws to identify [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial frames]] as defining a special class of coordinate systems in which there is a natural correspondence between mathematical structure and physical behaviour.

====The Equivalence Principle====

In a geometrical theory of spacetime, Newton’s first law determines uniform motion of inertial matter locally within an [[http://rqgravity.net/TheEquivalencePrinciple#hn_Inertial_Matter inertial reference frame]]. The consequence is geodesic motion, meaning that gravity is not an [[http://www.rqgravity.net/TheEquivalencePrinciple#hn_Active_and_Inertial_Forces active force]], in the sense of Newton. Although the equivalence principle was important in guiding Einstein toward the general theory of relativity, it is perfectly possible to develop the whole of general relativity, together with all its predictions, without every mentioning the force of gravity. Consequently, in a strict sense, we do not need the equivalence principle either. Nevertheless it is required to relate the mathematical theory our intuitive understanding and experience of gravity.

<<""<span class="math"><b>The Equivalence Principle:</b>"" ""<span class="math">We ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system" (Einstein 1907).""<<

====The Spacetime Manifold====

[[GeneralRelativity The General Theory of Relativity ↑]] [[MathematicalMethods Mathematical Methods →]]

##### Deletions:

In general relativity, Einstein put his physical ideas on the nature of time and space, into the mathematical language of [[IntroductionToTensors tensors]] and [[BasicsofCurvature Riemannian geometry]].

====""<a name="ThePrincipleOfGeneralCovariance"></a>""The Principle of General Covariance ====

together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy), encapsulated in the ""<a href="PhilosophicalBackground#hn_The_General_Principle_of_Relativity">general principle of relativity</a>"",

and the empirical fact that we do not have a universe obeying [[https://en.wikipedia.org/wiki/Galilean_invariance Newtonian relativity]].

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Coordinates>coordinate representation</a>"" changes with the coordinate system. Relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Contravariant_and_Covariant_Vectors>covariant</a>"". [[http://rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

====""<a name="TheSpacetimeManifold"></a>""The Spacetime Manifold====

[[GeneralRelativity The General Theory of Relativity ↑]] [[GTRTensors Riemann Curvature →]]

##### Additions:

This result is usually found by solving ""<a href=rqgravity.net/Gravitation#hn_Einsteins_Law_of_Gravitation>Einstein’s field equation</a>"". Here I have established it purely geometrically, because I think this gives greater insight and because it will simplify (slightly) the [[Schwarzschild calculation]] of the ""<a href=rqgravity.net/Gravitation#hn_The_Schwarzschild_Solution>Schwarzschild solution</a>"".

##### Deletions:

##### Additions:

======[[Gravity ←]] The General Theory of Relativity [[Gravity ↑]] [[GTRTensors →]]======

""<a href="GeneralRelativity#hn_The_Principle_Of_General_Covariance">The Principle of General Covariance </a>""

""<a href="GeneralRelativity#hn_The_Spacetime_Manifold">The Spacetime Manifold</a>""

""<a href="GeneralRelativity#hn_Differentiability">Differentiability</a>""

""<a href="GeneralRelativity#hn_Tangent_Charts">Tangent Charts</a>""

""<a href="GeneralRelativity#hn_Coordinate_Time_and_Proper_Time">Coordinate Time and Proper Time</a>""

""<a href="GeneralRelativity#hn_The_Spacetime_Metric">The Spacetime Metric</a>""

""<a href="GeneralRelativity#hn_Schwarzschild_Coordinates">Schwarzschild Coordinates</a>""

""<a href="GeneralRelativity#hn_The_Levi-Civita_Connection">The Levi-Civita Connection</a>""

General relativity is based on ""<a href="PhilosophicalBackground#hn_The_General_Principle_of_Relativity">the principle of uniformity in nature</a>"",

together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy), encapsulated in the ""<a href="PhilosophicalBackground#hn_The_General_Principle_of_Relativity">general principle of relativity</a>"",

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Coordinates>coordinate representation</a>"" changes with the coordinate system. Relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Contravariant_and_Covariant_Vectors>covariant</a>"". [[http://rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

According to the general principle, an observer anywhere can use the radar method to define locally ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski Coordinates</a>"", but there is no guarantee that a mapping of distant points to these coordinates can be made without distortion of the map. The situation is analogous to mapping the surface of the Earth. At any point of the Earth’s surface, a cartographer can make a locally flat map. He cannot extend the map without distortion, but this does not mean that geometry at other points of the Earth surface is different from the geometry seen by the cartographer.

Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called [[http://en.wikipedia.org/wiki/Manifold manifolds]]. Spacetime is described as a [[http://en.wikipedia.org/wiki/Lorentzian_manifold#Lorentzian_manifold Lorentzian manifold]]. By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment"> Pound-Rebka experiment</a>"". In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys [[FoundationsOfSpecialRelativity special relativity]]. Together with ""<a href=http://www.rqgravity.net/Gravitation#hn_Einsteins_Law_of_Gravitation>Einstein’s field equation</a>"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.

<<""<span class="math"><b>Definition:</b> A <i>manifold</i> is a structure in which any point has a neighbourhood which can be described by a <a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>coordinate system</a> or <a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>chart</a>.""<<

Typically a single coordinate system cannot be used to give a full description of a manifold. A collection of charts which describes the whole manifold is an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>atlas</a>"".

We can describe spacetime as a manifold, a geometrical structure which can be mapped onto an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>atlas</a>"", or collection of ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>charts</a>"". In common with many definitions of mathematical structures, this does not tell us what the manifold actually is. Instead it tells us what properties a manifold has, how a manifold behaves.

Here the manifold simply replaces Newton’s conception of ""<a href=http://rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute space</a>"" and ""<a href=http://rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

<<**Answer 3.** I observe that I can, in principle, choose ""<a href="PhilosophicalBackground#hn_Reference_Frames">reference matter</a>"" anywhere I wish, and that I can define ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski coordinates</a>"" relative to that matter. I now define spacetime by //imagining// all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.<<

A ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>chart</a>"" of spacetime need not be a physical map, like the maps in a world atlas. A mathematical idealisation suffices just as well — that is to say, the map may consist of tables of data and/or formulae. We may imagine, for example, the numbers, or coordinates, describing the times and positions of physical events mapped into a bank of computer memory. In principle, using a large enough bank of computer memory, this could be done to any precision, for as many points as one requires, and a map of a region of spacetime could be produced with any required level of detail, up to the limit of accuracy of measurement and the size of available computer memory.

In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, ""<a href="http://www.rqgravity.net/GeneralRelativity#hn_The_Principle_Of_General_Covariance" >general covariance</a>"" automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in [[FoundationsOfSpecialRelativity Special relativity]], using the radar method. In this case the chart is made on Minkowski spacetime, which has constant ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_3_Dimensions_plus_Time>Minkowski metric</a>"", ""<span class="math"><i>h</i>"". ""<span class="math"><i>h</i>"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""<span class="math"><i>h</i>"" does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Tangent_Space>tangent chart</a>"", an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.

A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation">transformed</a>"" into coordinates with Minkowski metric at the position of the observer.

In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer at a distance, and the same quantities as they would be described by a observer who determines them locally. Let Alf be an observer, with a clock at some point, ""<span class="math">A</span>"", the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at ""<span class="math">B</span>"", the origin of Beth’s coordinates. Alf and Beth both determine locally ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski Coordinates</a>"". Alf’s time axis is denoted the 0-axis, and his space axes are labelled ""<span class="math">1"", ""<span class="math">2"", ""<span class="math">3"". Beth’s coordinates are denoted with primes, ""<span class="math">0'"", ""<span class="math">1'"", ""<span class="math">2'"", ""<span class="math">3'"".

Using ""<a href="http://www.rqgravity.net/GeneralRelativity#hn_The_Principle_Of_General_Covariance" >general covariance</a>"", each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing ""<span class="math"><i>t</i>'"" seconds for a stationary object at the origin of Beth’s coordinates ""<span class="math">τ = τ<sup><i>i'</i></sup> = (<i>t'</i>, 0, 0, 0)"". ""<span class="math"><i>t'</i>"" is the actual amount of time measured by Beth using her own clock, and is known as //proper time//. In Alf’s coordinates, ""<img alt="GTR-1" title="coordinate transformation" src="images/gtr/GTR-1.gif" align="texttop" vspace="0">"" is found by ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation">coordinate transformation</a>"". If Beth is stationary in Alf’s coordinates, ""<span class="math">τ = τ<sup><i>i</i>'</sup> = (<i>kt'</i>, 0, 0, 0)"", where ""<span class="math"><i>k</i>"" is the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#hn_Gravitational_Red_Shift">gravitational redshift</a>"" factor, ""<span class="math"><i>k</i> = 1 + <i>z</i>"".

<< <<""<span class="math"><b>Definition:</b> <i>Proper length</i> is the length an object as it would be measured by an observer moving with that object.""<<

The ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_The_Metric" >metric</a>"", ""<span class="math"><i>g<sub>ij</sub></i>"", lowers the indices of ""<a href="http://www.rqgravity.net/IntroductionToTensors#hn_Index_Gymnastics" >contravariant</a>"" vectors in such a way that the inner product between vectors ""<span class="math"><i>x</i>"" and ""<span class="math"><i>y</i>"" is an ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_The_Dot_Product" >invariant</a>"". Using Minkowski spacetime, as described in [[http://rqgravity.net/FoundationsOfSpecialRelativity special relativity]] the metric is

The laws of physics local to Beth, as described by Beth, use proper times and distances determined in her local measurements. If Alf is to analyse physical processes close to Beth, he must also determine proper times and distances, using remote measurements. Since the difference between Alf’s and Beth’s measurements is just that Alf and Beth define coordinates differently, their measurements are related by ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation>coordinate transformation</a>"". The definition of the metric ensures that, when Alf applies it to his own measurements, the magnitudes returned will be the proper times and distances of quantities local to Beth. The ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_The_Metric>metric</a>"" determines that the quantity ""<span class="math">|<i>x</i>|<sup>2</sup> = <i>g<sub>ij</sub>x<sup>i</sup>y<sup>j</sup></span>"" is the same in any coordinates. Beth could be anywhere in Alf’s coordinate space. For each point, ""<span class="math"><i>x</i>"", where Beth could be, there is a different metric, ""<span class="math"><i>g<sub>ij</sub></i>(<i>x</i>)</i>"".

The spacetime metric is often simply called the metric. One should avoid this abuse of language, because the spacetime metric is a ""<a href=http://www.rqgravity.net/IntroductionToTensors#hn_Tensor_Fields>tensor field</a>"", that is to say it is a function having a different metric value at each point in spacetime. Calling it simply “the metric” confuses a ""<a href=http://www.rqgravity.net/IntroductionToTensors#hn_Tensor_Fields>metric field</a>"", which is a function of coordinate space, with the ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_The_Metric" >metric</a>"" at a given position.

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-5" title="rate of clocks and gravitational redshift" src="images/gtr/GTR-5.gif">In a static geometry, an observer, Alf, defines <a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_Spherical_Coordinates>spherical coordinates</a> by the radar method, using time <span class="math"><i>t</i></span>, determined from a clock at an origin at <span class="math">A</span>. Coordinate distance, <span class="math"><i>r*</i></span>, from <span class="math">A</span> is defined by setting the radial speed of light to unity. Spherical coordinates are <a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Orthonormal_Bases>orthogonal</a> so that the <a href=http://rqgravity.net/GeneralRelativity#hn_The_Spacetime_Metric>spacetime metric</a> is diagonal. A second observer, Beth, is at a constant position, <span class="math">B</span>, at radial coordinate <span class="math"><i>r</i></span> in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor <span class="math"><i>k</i> = 1 + <i>z</i> > 1</span> (the argument below also holds for <span class="math"><i>k</i> < 1</span>). Then light transmitted from Alf to Beth is redshifted by factor <span class="math"><i>k</i></span> and an interval <span class="math"><i>t</i></span> of Alf’s coordinate time at <span class="math">B</span> is measured by Beth as proper time interval <span class="math"><i>t' = kt</i></span>. Then the <a href=http://www.rqgravity.net/GeneralRelativity#hn_The_Spacetime_Metric>spacetime metric</a> has <span class="math"><i>g</i><sub>00</sub> = <i>k</i><sup>2</sup></i></span>.</td></table>""

This result is usually found by solving [[rqgravity.net/Gravitation#hn_Einsteins_Law_of_Gravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it will simplify (slightly) the [[Schwarzschild calculation]] of the [[rqgravity.net/Gravitation#hn_The_Schwarzschild_Solution Schwarzschild solution]].

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Orthonormal_Bases>orthogonal</a>"". In this case the metric is diagonal (as seen in ""<a href="GeneralRelativity#hn_Schwarzschild_Coordinates">Schwarzschild coordinates</a>"") and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the ""<a href= BasicsOfCurvature#Lens>lensed and mirrored </a>"" geometries, which are actually flat. To describe curvature requires a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement>connection</a>"" in addition to the metric field. Given the metric field, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""<span class="math"><i>x</i>"", say, and another set, at ""<span class="math"><i>x</i> +<i>dx</i>"", where ""<span class="math"><i>dx</i>"" is a small displacement, such that we can meaningfully describe a vector at ""<span class="math"><i>x</i>"" as being parallel to one at ""<span class="math"><i>x</i> +<i>dx</i>"" (other types of [[http://en.wikipedia.org/wiki/Connection_(mathematics) connection]] are used to transport other types of data).

In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between vectors at nearby points using ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement">parallel displacement</a>"" in tangent space, and projecting back into the curved surface. This is the [[http://en.wikipedia.org/wiki/Levi-Civita_connection Levi-Civita connection]], defined in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Transport">parallel transport</a>"", and is path dependent. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.

""<a href="GeneralRelativity#hn_The_Principle_Of_General_Covariance">The Principle of General Covariance </a>""

""<a href="GeneralRelativity#hn_The_Spacetime_Manifold">The Spacetime Manifold</a>""

""<a href="GeneralRelativity#hn_Differentiability">Differentiability</a>""

""<a href="GeneralRelativity#hn_Tangent_Charts">Tangent Charts</a>""

""<a href="GeneralRelativity#hn_Coordinate_Time_and_Proper_Time">Coordinate Time and Proper Time</a>""

""<a href="GeneralRelativity#hn_The_Spacetime_Metric">The Spacetime Metric</a>""

""<a href="GeneralRelativity#hn_Schwarzschild_Coordinates">Schwarzschild Coordinates</a>""

""<a href="GeneralRelativity#hn_The_Levi-Civita_Connection">The Levi-Civita Connection</a>""

General relativity is based on ""<a href="PhilosophicalBackground#hn_The_General_Principle_of_Relativity">the principle of uniformity in nature</a>"",

together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy), encapsulated in the ""<a href="PhilosophicalBackground#hn_The_General_Principle_of_Relativity">general principle of relativity</a>"",

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Coordinates>coordinate representation</a>"" changes with the coordinate system. Relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Contravariant_and_Covariant_Vectors>covariant</a>"". [[http://rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

According to the general principle, an observer anywhere can use the radar method to define locally ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski Coordinates</a>"", but there is no guarantee that a mapping of distant points to these coordinates can be made without distortion of the map. The situation is analogous to mapping the surface of the Earth. At any point of the Earth’s surface, a cartographer can make a locally flat map. He cannot extend the map without distortion, but this does not mean that geometry at other points of the Earth surface is different from the geometry seen by the cartographer.

Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called [[http://en.wikipedia.org/wiki/Manifold manifolds]]. Spacetime is described as a [[http://en.wikipedia.org/wiki/Lorentzian_manifold#Lorentzian_manifold Lorentzian manifold]]. By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#hn_The_Pound-Rebka_Experiment"> Pound-Rebka experiment</a>"". In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys [[FoundationsOfSpecialRelativity special relativity]]. Together with ""<a href=http://www.rqgravity.net/Gravitation#hn_Einsteins_Law_of_Gravitation>Einstein’s field equation</a>"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.

<<""<span class="math"><b>Definition:</b> A <i>manifold</i> is a structure in which any point has a neighbourhood which can be described by a <a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>coordinate system</a> or <a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>chart</a>.""<<

Typically a single coordinate system cannot be used to give a full description of a manifold. A collection of charts which describes the whole manifold is an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>atlas</a>"".

We can describe spacetime as a manifold, a geometrical structure which can be mapped onto an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>atlas</a>"", or collection of ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>charts</a>"". In common with many definitions of mathematical structures, this does not tell us what the manifold actually is. Instead it tells us what properties a manifold has, how a manifold behaves.

Here the manifold simply replaces Newton’s conception of ""<a href=http://rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute space</a>"" and ""<a href=http://rqgravity.net/PhilosophicalBackground#hn_Relationism> absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

<<**Answer 3.** I observe that I can, in principle, choose ""<a href="PhilosophicalBackground#hn_Reference_Frames">reference matter</a>"" anywhere I wish, and that I can define ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski coordinates</a>"" relative to that matter. I now define spacetime by //imagining// all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.<<

A ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Charts_or_Coordinate_Systems>chart</a>"" of spacetime need not be a physical map, like the maps in a world atlas. A mathematical idealisation suffices just as well — that is to say, the map may consist of tables of data and/or formulae. We may imagine, for example, the numbers, or coordinates, describing the times and positions of physical events mapped into a bank of computer memory. In principle, using a large enough bank of computer memory, this could be done to any precision, for as many points as one requires, and a map of a region of spacetime could be produced with any required level of detail, up to the limit of accuracy of measurement and the size of available computer memory.

In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, ""<a href="http://www.rqgravity.net/GeneralRelativity#hn_The_Principle_Of_General_Covariance" >general covariance</a>"" automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in [[FoundationsOfSpecialRelativity Special relativity]], using the radar method. In this case the chart is made on Minkowski spacetime, which has constant ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_3_Dimensions_plus_Time>Minkowski metric</a>"", ""<span class="math"><i>h</i>"". ""<span class="math"><i>h</i>"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""<span class="math"><i>h</i>"" does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Tangent_Space>tangent chart</a>"", an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.

A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation">transformed</a>"" into coordinates with Minkowski metric at the position of the observer.

In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer at a distance, and the same quantities as they would be described by a observer who determines them locally. Let Alf be an observer, with a clock at some point, ""<span class="math">A</span>"", the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at ""<span class="math">B</span>"", the origin of Beth’s coordinates. Alf and Beth both determine locally ""<a href="FoundationsOfSpecialRelativity#hn_Minkowski_Coordinates">Minkowski Coordinates</a>"". Alf’s time axis is denoted the 0-axis, and his space axes are labelled ""<span class="math">1"", ""<span class="math">2"", ""<span class="math">3"". Beth’s coordinates are denoted with primes, ""<span class="math">0'"", ""<span class="math">1'"", ""<span class="math">2'"", ""<span class="math">3'"".

Using ""<a href="http://www.rqgravity.net/GeneralRelativity#hn_The_Principle_Of_General_Covariance" >general covariance</a>"", each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing ""<span class="math"><i>t</i>'"" seconds for a stationary object at the origin of Beth’s coordinates ""<span class="math">τ = τ<sup><i>i'</i></sup> = (<i>t'</i>, 0, 0, 0)"". ""<span class="math"><i>t'</i>"" is the actual amount of time measured by Beth using her own clock, and is known as //proper time//. In Alf’s coordinates, ""<img alt="GTR-1" title="coordinate transformation" src="images/gtr/GTR-1.gif" align="texttop" vspace="0">"" is found by ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation">coordinate transformation</a>"". If Beth is stationary in Alf’s coordinates, ""<span class="math">τ = τ<sup><i>i</i>'</sup> = (<i>kt'</i>, 0, 0, 0)"", where ""<span class="math"><i>k</i>"" is the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#hn_Gravitational_Red_Shift">gravitational redshift</a>"" factor, ""<span class="math"><i>k</i> = 1 + <i>z</i>"".

<< <<""<span class="math"><b>Definition:</b> <i>Proper length</i> is the length an object as it would be measured by an observer moving with that object.""<<

The ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_The_Metric" >metric</a>"", ""<span class="math"><i>g<sub>ij</sub></i>"", lowers the indices of ""<a href="http://www.rqgravity.net/IntroductionToTensors#hn_Index_Gymnastics" >contravariant</a>"" vectors in such a way that the inner product between vectors ""<span class="math"><i>x</i>"" and ""<span class="math"><i>y</i>"" is an ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_The_Dot_Product" >invariant</a>"". Using Minkowski spacetime, as described in [[http://rqgravity.net/FoundationsOfSpecialRelativity special relativity]] the metric is

The laws of physics local to Beth, as described by Beth, use proper times and distances determined in her local measurements. If Alf is to analyse physical processes close to Beth, he must also determine proper times and distances, using remote measurements. Since the difference between Alf’s and Beth’s measurements is just that Alf and Beth define coordinates differently, their measurements are related by ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Coordinate_Transformation>coordinate transformation</a>"". The definition of the metric ensures that, when Alf applies it to his own measurements, the magnitudes returned will be the proper times and distances of quantities local to Beth. The ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_The_Metric>metric</a>"" determines that the quantity ""<span class="math">|<i>x</i>|<sup>2</sup> = <i>g<sub>ij</sub>x<sup>i</sup>y<sup>j</sup></span>"" is the same in any coordinates. Beth could be anywhere in Alf’s coordinate space. For each point, ""<span class="math"><i>x</i>"", where Beth could be, there is a different metric, ""<span class="math"><i>g<sub>ij</sub></i>(<i>x</i>)</i>"".

The spacetime metric is often simply called the metric. One should avoid this abuse of language, because the spacetime metric is a ""<a href=http://www.rqgravity.net/IntroductionToTensors#hn_Tensor_Fields>tensor field</a>"", that is to say it is a function having a different metric value at each point in spacetime. Calling it simply “the metric” confuses a ""<a href=http://www.rqgravity.net/IntroductionToTensors#hn_Tensor_Fields>metric field</a>"", which is a function of coordinate space, with the ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#hn_The_Metric" >metric</a>"" at a given position.

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-5" title="rate of clocks and gravitational redshift" src="images/gtr/GTR-5.gif">In a static geometry, an observer, Alf, defines <a href=http://www.rqgravity.net/IntroductionToVectorSpace#hn_Spherical_Coordinates>spherical coordinates</a> by the radar method, using time <span class="math"><i>t</i></span>, determined from a clock at an origin at <span class="math">A</span>. Coordinate distance, <span class="math"><i>r*</i></span>, from <span class="math">A</span> is defined by setting the radial speed of light to unity. Spherical coordinates are <a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Orthonormal_Bases>orthogonal</a> so that the <a href=http://rqgravity.net/GeneralRelativity#hn_The_Spacetime_Metric>spacetime metric</a> is diagonal. A second observer, Beth, is at a constant position, <span class="math">B</span>, at radial coordinate <span class="math"><i>r</i></span> in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor <span class="math"><i>k</i> = 1 + <i>z</i> > 1</span> (the argument below also holds for <span class="math"><i>k</i> < 1</span>). Then light transmitted from Alf to Beth is redshifted by factor <span class="math"><i>k</i></span> and an interval <span class="math"><i>t</i></span> of Alf’s coordinate time at <span class="math">B</span> is measured by Beth as proper time interval <span class="math"><i>t' = kt</i></span>. Then the <a href=http://www.rqgravity.net/GeneralRelativity#hn_The_Spacetime_Metric>spacetime metric</a> has <span class="math"><i>g</i><sub>00</sub> = <i>k</i><sup>2</sup></i></span>.</td></table>""

This result is usually found by solving [[rqgravity.net/Gravitation#hn_Einsteins_Law_of_Gravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it will simplify (slightly) the [[Schwarzschild calculation]] of the [[rqgravity.net/Gravitation#hn_The_Schwarzschild_Solution Schwarzschild solution]].

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""<a href=http://rqgravity.net/IntroductionToVectorSpace#hn_Orthonormal_Bases>orthogonal</a>"". In this case the metric is diagonal (as seen in ""<a href="GeneralRelativity#hn_Schwarzschild_Coordinates">Schwarzschild coordinates</a>"") and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the ""<a href= BasicsOfCurvature#Lens>lensed and mirrored </a>"" geometries, which are actually flat. To describe curvature requires a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement>connection</a>"" in addition to the metric field. Given the metric field, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""<span class="math"><i>x</i>"", say, and another set, at ""<span class="math"><i>x</i> +<i>dx</i>"", where ""<span class="math"><i>dx</i>"" is a small displacement, such that we can meaningfully describe a vector at ""<span class="math"><i>x</i>"" as being parallel to one at ""<span class="math"><i>x</i> +<i>dx</i>"" (other types of [[http://en.wikipedia.org/wiki/Connection_(mathematics) connection]] are used to transport other types of data).

In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between vectors at nearby points using ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Displacement">parallel displacement</a>"" in tangent space, and projecting back into the curved surface. This is the [[http://en.wikipedia.org/wiki/Levi-Civita_connection Levi-Civita connection]], defined in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through ""<a href="http://www.rqgravity.net/BasicsOfCurvature#hn_Parallel_Transport">parallel transport</a>"", and is path dependent. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.

##### Deletions:

""<a href="GeneralRelativity#ThePrincipleOfGeneralCovariance">The Principle of General Covariance </a>""

""<a href="GeneralRelativity#TheSpacetimeManifold">The Spacetime Manifold</a>""

""<a href="GeneralRelativity#Differentiability">Differentiability</a>""

""<a href="GeneralRelativity#TangentCharts">Tangent Charts</a>""

""<a href="GeneralRelativity#CoordinateTimeAndProperTime">Coordinate Time and Proper Time</a>""

""<a href="GeneralRelativity#TheSpacetimeMetric">The Spacetime Metric</a>""

""<a href="GeneralRelativity#SchwarzschildCoordinates">Schwarzschild Coordinates</a>""

""<a href="GeneralRelativity#TheLevi-CivitaConnection">The Levi-Civita Connection</a>""

General relativity is based on ""<a href="PhilosophicalBackground#TheGeneralPrincipleOfRelativity">the principle of uniformity in nature</a>"",

together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy), encapsulated in the ""<a href="PhilosophicalBackground#TheGeneralPrincipleOfRelativity">general principle of relativity</a>"",

The principle of [[https://en.wikipedia.org/wiki/General_covariance general covariance]] is the mathematical implementation of the general principle of relativity. In non-mathematical language it says //“local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”//. [[http://rqgravity.net/IntroductionToVectorSpace Vectors]] are not invariant, as their ""<a href=http://rqgravity.net/IntroductionToVectorSpace#Coordinates>coordinate representation</a>"" changes with the coordinate system. Relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be ""<a href=http://rqgravity.net/IntroductionToVectorSpace#ContravariantAndCovariantVectors>covariant</a>"". [[http://rqgravity.net/IntroductionToTensors Tensors]] are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

According to the general principle, an observer anywhere can use the radar method to define locally ""<a href="FoundationsOfSpecialRelativity#MinkowskiCoordinates">Minkowski Coordinates</a>"", but there is no guarantee that a mapping of distant points to these coordinates can be made without distortion of the map. The situation is analogous to mapping the surface of the Earth. At any point of the Earth’s surface, a cartographer can make a locally flat map. He cannot extend the map without distortion, but this does not mean that geometry at other points of the Earth surface is different from the geometry seen by the cartographer.

Mathematical structures which generalise the mapping properties of two dimensional surfaces to an arbitrary number of dimensions are called [[http://en.wikipedia.org/wiki/Manifold manifolds]]. Spacetime is described as a [[http://en.wikipedia.org/wiki/Lorentzian_manifold#Lorentzian_manifold Lorentzian manifold]]. By this we mean that, at each point in spacetime, it is possible to set up locally Minkowski coordinates. The observed laws of physics are the same near the origin of every set of locally defined coordinates, but there is no guarantee that processes can be viewed from a distance without distortion. In practice, we have seen that distortion, in the form of redshift, was detected in the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#ThePound-RebkaExperiment"> Pound-Rebka experiment</a>"". In general, identical clocks at distant points are not observed to run at the same speed at a clock at the origin. The relationship between clock time and measured distance is determined locally and obeys [[FoundationsOfSpecialRelativity special relativity]]. Together with ""<a href=http://www.rqgravity.net/Gravitation#Einstein’sLawOfGravitation>Einstein’s field equation</a>"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.

<<""<span class="math"><b>Definition:</b> A <i>manifold</i> is a structure in which any point has a neighbourhood which can be described by a <a href=http://www.rqgravity.net/BasicsOfCurvature#Charts>coordinate system</a> or <a href=http://www.rqgravity.net/BasicsOfCurvature#Charts>chart</a>.""<<

Typically a single coordinate system cannot be used to give a full description of a manifold. A collection of charts which describes the whole manifold is an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts>atlas</a>"".

We can describe spacetime as a manifold, a geometrical structure which can be mapped onto an ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts>atlas</a>"", or collection of ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts>charts</a>"". In common with many definitions of mathematical structures, this does not tell us what the manifold actually is. Instead it tells us what properties a manifold has, how a manifold behaves.

Here the manifold simply replaces Newton’s conception of ""<a href=http://rqgravity.net/PhilosophicalBackground#Relationism> absolute space</a>"" and ""<a href=http://rqgravity.net/PhilosophicalBackground#Relationism> absolute time</a>"". This is what I call a metaphysical manifold, because there is no observation of a substantive spacetime, and nor can there be one. We //observe// the behaviour of matter, and //infer// the existence of space-time structure, but cannot actually observe it. Since substantive spacetime is scientifically unverifiable, at best it lies outside the realms of science. At worst (as I maintain), it is in conflict with observations in quantum theory.

<<**Answer 3.** I observe that I can, in principle, choose ""<a href="PhilosophicalBackground#ReferenceFrames">reference matter</a>"" anywhere I wish, and that I can define ""<a href="FoundationsOfSpecialRelativity#MinkowskiCoordinates">Minkowski coordinates</a>"" relative to that matter. I now define spacetime by //imagining// all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.<<

A ""<a href=http://www.rqgravity.net/BasicsOfCurvature#Charts>chart</a>"" of spacetime need not be a physical map, like the maps in a world atlas. A mathematical idealisation suffices just as well — that is to say, the map may consist of tables of data and/or formulae. We may imagine, for example, the numbers, or coordinates, describing the times and positions of physical events mapped into a bank of computer memory. In principle, using a large enough bank of computer memory, this could be done to any precision, for as many points as one requires, and a map of a region of spacetime could be produced with any required level of detail, up to the limit of accuracy of measurement and the size of available computer memory.

In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, ""<a href="http://www.rqgravity.net/GeneralRelativity#ThePrincipleOfGeneralCovariance" >general covariance</a>"" automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in [[FoundationsOfSpecialRelativity Special relativity]], using the radar method. In this case the chart is made on Minkowski spacetime, which has constant ""<a href=http://www.rqgravity.net/IntroductionToVectorSpace#3DimensionsPlusTime>Minkowski metric</a>"", ""<span class="math"><i>h</i>"". ""<span class="math"><i>h</i>"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""<span class="math"><i>h</i>"" does not give physical magnitudes of vector quantities except at the position of the observer, i.e. the point of contact between spacetime and tangent space. Using a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#TangentSpace>tangent chart</a>"", an observer can define vectors at the origin, and he can translate them through small distances in his immediate neighbourhood, so long as differences between physical measurement and corresponding calculations in tangent space are negligible.

A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#CoordinateTransformation">transformed</a>"" into coordinates with Minkowski metric at the position of the observer.

In order to describe geometrical effects we distinguish between physical quantities described in a given coordinate system, by an observer at a distance, and the same quantities as they would be described by a observer who determines them locally. Let Alf be an observer, with a clock at some point, ""<span class="math">A</span>"", the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at ""<span class="math">B</span>"", the origin of Beth’s coordinates. Alf and Beth both determine locally ""<a href="FoundationsOfSpecialRelativity#MinkowskiCoordinates">Minkowski Coordinates</a>"". Alf’s time axis is denoted the 0-axis, and his space axes are labelled ""<span class="math">1"", ""<span class="math">2"", ""<span class="math">3"". Beth’s coordinates are denoted with primes, ""<span class="math">0'"", ""<span class="math">1'"", ""<span class="math">2'"", ""<span class="math">3'"".

Using ""<a href="http://www.rqgravity.net/GeneralRelativity#ThePrincipleOfGeneralCovariance" >general covariance</a>"", each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing ""<span class="math"><i>t</i>'"" seconds for a stationary object at the origin of Beth’s coordinates ""<span class="math">τ = τ<sup><i>i'</i></sup> = (<i>t'</i>, 0, 0, 0)"". ""<span class="math"><i>t'</i>"" is the actual amount of time measured by Beth using her own clock, and is known as //proper time//. In Alf’s coordinates, ""<img alt="GTR-1" title="coordinate transformation" src="images/gtr/GTR-1.gif" align="texttop" vspace="0">"" is found by ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#CoordinateTransformation">coordinate transformation</a>"". If Beth is stationary in Alf’s coordinates, ""<span class="math">τ = τ<sup><i>i</i>'</sup> = (<i>kt'</i>, 0, 0, 0)"", where ""<span class="math"><i>k</i>"" is the ""<a href="http://www.rqgravity.net/TheEquivalencePrinciple#GravitationalRedshift">gravitational redshift</a>"" factor, ""<span class="math"><i>k</i> = 1 + <i>z</i>"".

<< <<""<span class="math"><b>Definition:</b> <i>Proper length</i> is the length an object as it would be measured by an observer moving with that object."">>

The ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#TheMetric" >metric</a>"", ""<span class="math"><i>g<sub>ij</sub></i>"", lowers the indices of ""<a href="http://www.rqgravity.net/IntroductionToTensors#IndexGymnastics" >contravariant</a>"" vectors in such a way that the inner product between vectors ""<span class="math"><i>x</i>"" and ""<span class="math"><i>y</i>"" is an ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#TheDotProduct" >invariant</a>"". Using Minkowski spacetime, as described in [[http://rqgravity.net/FoundationsOfSpecialRelativity special relativity]] the metric is

The laws of physics local to Beth, as described by Beth, use proper times and distances determined in her local measurements. If Alf is to analyse physical processes close to Beth, he must also determine proper times and distances, using remote measurements. Since the difference between Alf’s and Beth’s measurements is just that Alf and Beth define coordinates differently, their measurements are related by ""<a href=http://rqgravity.net/IntroductionToVectorSpace#CoordinateTransformation>coordinate transformation</a>"". The definition of the metric ensures that, when Alf applies it to his own measurements, the magnitudes returned will be the proper times and distances of quantities local to Beth. The ""<a href=http://rqgravity.net/IntroductionToVectorSpace#TheMetric>metric</a>"" determines that the quantity ""<span class="math">|<i>x</i>|<sup>2</sup> = <i>g<sub>ij</sub>x<sup>i</sup>y<sup>j</sup></span>"" is the same in any coordinates. Beth could be anywhere in Alf’s coordinate space. For each point, ""<span class="math"><i>x</i>"", where Beth could be, there is a different metric, ""<span class="math"><i>g<sub>ij</sub></i>(<i>x</i>)</i>"".

The spacetime metric is often simply called the metric. One should avoid this abuse of language, because the spacetime metric is a ""<a href=http://www.rqgravity.net/IntroductionToTensors#Fields>tensor field</a>"", that is to say it is a function having a different metric value at each point in spacetime. Calling it simply “the metric” confuses a ""<a href=http://www.rqgravity.net/IntroductionToTensors#Fields>metric field</a>"", which is a function of coordinate space, with the ""<a href="http://www.rqgravity.net/IntroductionToVectorSpace#TheMetric" >metric</a>"" at a given position.

""<table width="100%" border="0" cellpadding=0 cellspacing=0><td><img class="right" alt="GTR-5" title="rate of clocks and gravitational redshift" src="images/gtr/GTR-5.gif">In a static geometry, an observer, Alf, defines <a href=http://www.rqgravity.net/IntroductionToVectorSpace#SphericalCoordinates>spherical coordinates</a> by the radar method, using time <span class="math"><i>t</i></span>, determined from a clock at an origin at <span class="math">A</span>. Coordinate distance, <span class="math"><i>r*</i></span>, from <span class="math">A</span> is defined by setting the radial speed of light to unity. Spherical coordinates are <a href=http://rqgravity.net/IntroductionToVectorSpace#OrthonormalBases>orthogonal</a> so that the <a href=http://rqgravity.net/GeneralRelativity#TheSpacetimeMetric>spacetime metric</a> is diagonal. A second observer, Beth, is at a constant position, <span class="math">B</span>, at radial coordinate <span class="math"><i>r</i></span> in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor <span class="math"><i>k</i> = 1 + <i>z</i> > 1</span> (the argument below also holds for <span class="math"><i>k</i> < 1</span>). Then light transmitted from Alf to Beth is redshifted by factor <span class="math"><i>k</i></span> and an interval <span class="math"><i>t</i></span> of Alf’s coordinate time at <span class="math">B</span> is measured by Beth as proper time interval <span class="math"><i>t' = kt</i></span>. Then the <a href=http://www.rqgravity.net/GeneralRelativity#TheSpacetimeMetric>spacetime metric</a> has <span class="math"><i>g</i><sub>00</sub> = <i>k</i><sup>2</sup></i></span>.</td></table>""

This result is usually found by solving [[rqgravity.net/Gravitation#Einstein’sLawOfGravitation Einstein’s field equation]]. Here I have established it purely geometrically, because I think this gives greater insight and because it will simplify (slightly) the [[Schwarzschild calculation]] of the [[rqgravity.net/Gravitation#TheSchwarzschildSolution Schwarzschild solution]].

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are ""<a href=http://rqgravity.net/IntroductionToVectorSpace#OrthonormalBases>orthogonal</a>"". In this case the metric is diagonal (as seen in ""<a href="GeneralRelativity#SchwarzschildCoordinates">Schwarzschild coordinates</a>"") and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the ""<a href= BasicsOfCurvature#Lens>lensed and mirrored </a>"" geometries, which are actually flat. To describe curvature requires a ""<a href=http://www.rqgravity.net/BasicsOfCurvature#ParallelDisplacement>connection</a>"" in addition to the metric field. Given the metric field, an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]] describes a relationship between a set of coordinate axes at ""<span class="math"><i>x</i>"", say, and another set, at ""<span class="math"><i>x</i> +<i>dx</i>"", where ""<span class="math"><i>dx</i>"" is a small displacement, such that we can meaningfully describe a vector at ""<span class="math"><i>x</i>"" as being parallel to one at ""<span class="math"><i>x</i> +<i>dx</i>"" (other types of [[http://en.wikipedia.org/wiki/Connection_(mathematics) connection]] are used to transport other types of data).

In general relativity the connection is defined in the same way as in the surface of the Earth. That is to say, it is defined between vectors at nearby points using ""<a href="http://www.rqgravity.net/BasicsOfCurvature#ParallelDisplacement">parallel displacement</a>"" in tangent space, and projecting back into the curved surface. This is the [[http://en.wikipedia.org/wiki/Levi-Civita_connection Levi-Civita connection]], defined in accordance with physical experience, that it makes sense to translate objects in space through small distances. As with Earth geometry, a relationship between coordinates at distant points can only be determined through ""<a href="http://www.rqgravity.net/BasicsOfCurvature#ParallelTransport">parallel transport</a>"", and is path dependent. Other affine connections are mathematically possible. For example, if rectangular coordinates were superimposed on the image in a convex mirror or a lens, the geometry would have an affine connection such that the apparently curved space is actually flat. Such connections do not appear to be physically interesting.