n""-dimensions was developed by generalising the geometry of the surface of the Earth to arbitrary curved surfaces, and to an arbitrary number of dimensions. This page introduces principle ideas in non-Euclidean geometry visually, with special reference to mapping the Earth’s surface. In [[NonEuclideanGeometry Non-Euclidean Geometry]], these ideas are applied more generally.
====""""Intrinsic and Extrinsic Curvature====
In many ways the term curvature is misleading when applied to spacetime. There are two types of curvature, known as intrinsic and extrinsic curvature. Extrinsic curvature is the familiar concept of curvature, visible curvature, or the shape of a two dimensional surface as it appears to us in three dimensional space. Intrinsic curvature refers to the geometrical properties of a space found from measurements taken within the space.
Intrinsic and extrinsic curvature are quite distinct ideas. A geometry can have intrinsic curvature but no extrinsic curvature, and one can have extrinsic curvature but no intrinsic curvature. The curvature of spacetime is intrinsic, because it refers to geometrical properties defined by measurement within spacetime. Extrinsic curvature has no meaning for spacetime, because there is no outside space to look at it from. To introduce the concept of intrinsic curvature we look at some simple examples.
"" Sphere: When you construct two equal lines of longitude, on a sphere AD and BC, you find that CD is less than AB, so the parallel postulate does not hold. By the same token, the circumference of a circle is less than 2πr, where r is radial distance measured in the surface of the sphere.
 Cylinder: A cylinder has extrinsic curvature, but no intrinsic, or Gaussian, curvature, because it can be made by rolling up a piece of paper without changing distance relationships between points. The circumference of a circle drawn on the paper is still 2πr when the paper is rolled into a cylinder. |
""
""Cone: Like the cylinder you can make a cone by rolling flat paper. A circle enclosing the apex has circumference less than 2πr, a circle anywhere else has circumference equal to 2πr. The cone is intrinsically flat except at the apex, where the geometry has a singularity.
 World map: A map of the world has intrinsic curvature, but no extrinsic curvature. It is drawn on flat paper, and does not look curved, but distances described by the map are geographical. It has exactly the same geometry as the world itself, and geometrically approximates a sphere. |
""
""Lenses and mirrors: The image under a magnifying glass is curved according to any normal definition; it has extrinsic curvature. But the distance between two points in the image is measured by a ruler, and is the same as the original. The geometry is intrinsically flat. The same is true of an image in a curved mirror.
 |  |
""
====""""Curvature of Spacetime====
There is no absolute distance scale. In practice coordinates are always defined with respect to ""reference matter"" within a neighbourhood of an observer. Distances are established locally by measuring matter relative to other matter. If all matter locally were to “shrink”, including ourselves and all the matter contained our measuring apparatus, we would have no means to detect the fact. Our fundamental length scales would “shrink” in direct proportion, and it would make no difference to the numerical results of local distance measurements. In the absence of such a difference, “shrinking” is meaningless. To talk about shrinking we need to make a comparison between the coordinates we define here and now, using here and now clocks and rulers, and the coordinates defined somewhere else.
The implication is that, if you hold up a stick, and a friend stands some distance away and measures its length by some triangulation technique based on Euclidean geometry, then he is not guaranteed to get the same answer as if you bring the stick to him and he measures its length directly. Of course, in this instance, we could not detect a difference between the two measurements, but the effects of non-Euclidean geometry can be measured, for example in the [[http://en.wikipedia.org/wiki/Pound-Rebka_experiment Pound-Rebka experiment]]. In [[FoundationsOfSpecialRelativity special relativity]], distance measurements are reduced to measurements of time. The statement that spacetime is not flat is precisely equivalent to the statement that identical clocks at different positions do not necessarily measure time at the same rate.
The parallel postulate breaks down precisely because distance is defined by local comparison. It holds within the geometry of a piece of paper, because we can take a ruler and slide it across the paper, and thereby compare the distance between parallel lines at one place and the corresponding distance at another. On the scale of astronomical distances, no such sliding of rulers is possible — or if it is, we cannot say that the ruler itself does not “expand” or “shrink” in the process. Special relativity describes measurements of time and position, using physical measurement to set up a reference frame and coordinates within a neighbourhod of an observer. According to the ""general principle of relativity"" another observer at a distant point can carry out exactly the same kind of measurements, and set up coordinates in exactly the same way, but special relativity does not provide us with a way to compare the coordinates set up by one observer with those of a second, remote observer. The absence of direct comparison between distant coordinate systems creates the possibility that the parallel postulate does not hold and that spacetime is non-Euclidean.
""  Special relativity assumed that the redshift factor, k, depends only on the magnitude of relative velocity. Local laws of physics would not be affected if k also depends on relative position. Because of the dependency of distance measurements on time, this will result in “shrinking” or “expansion” of one observer’s coordinates compared to the other’s, separate from, and in addition to, special relativistic, velocity dependent, time dilation. |
""
====""""Positive and Negative Curvature====
{{image class="right" alt="curvature-6" title="parallel postulate relaxed" url="images/curvature/Curvature-6N.gif"}}Intrinsic curvature is characterised by geometric properties. For example we can characterise curvature by comparison with the parallel postulate. If we measure the distance ""AB = d"" between any two points, and measure two equal distances ""AD = BC = h"", perpendicular to ""AB"", then we have no prior reason to assume that ""AB"" is equal to ""DC"", as measured by an observer at ""D"". So, ""DC = kAB = kd"" where ""k"" is a non-constant scale factor. This leads to a broad characterisation:
""
| Negative curvature: |
|
k increasing with h. |
|
| Zero curvature: |
|
k constant as h increases. |
| Positive curvature: |
|
k decreasing as h increases. |
""
{{image class="left" alt="curvature-7" title="characterisation of curvature" url="images/curvature/Curvature-7N.gif"}}The parallel postulate is a not an ideal criterion on which to base a categorisation of geometry, because curvature is better understood at a point, not on a line. A more natural, characterisation of a geometry is found by considering the length of an arc, ""CD"", of a circle of radius, ""r"", subtended by an angle, ""θ"", at the origin, ""O"". Use a very small angle, ""θ"", and drop perpendiculars of equal length from ""CD"" to a base line, ""AB"", through the origin ""O"". Then, in Euclidean geometry the length of ""CD"" is ""rθ"", almost equal to ""AB"". But in general the length of ""CD"" is ""krθ"", and the value of ""k"" characterises the geometry according to the relationships:
""
| Negative curvature: |
|
k increasing with r. |
|
| Zero curvature: |
|
k constant as r increases. |
| Positive curvature: |
|
k decreasing as r increases. |
""
""""{{image class="right" alt="curvature-8" title="Saddle Geometry" url="images/curvature/Curvature-8N.gif"}}**Saddle:** ""k"" increases with ""r"". So, curvature is negative.
{{image class="left" alt="curvature-9" title="Spherical Geometry" url="images/curvature/Curvature-9N.gif"}}
**Sphere:** ""k"" decreases as ""r"" increases. Curvature is positive.
"" | ""
====""""Singularities====
{{image class="right" alt="curvature-10" title="Trumpet Geometry" url="images/curvature/Curvature-10N.gif"}}If in a small region the value of the circumference of a circle approaches ""2πr"", the geometry locally approximates a flat geometry. It has a flat //tangent space// (red). If there is no unique tangent space the geometry has a ""singularity"", as at the apex. The practical implication is that, in general relativity, a singularity is a point where we do not know how to formulate the laws of physics.
[[http://en.wikipedia.org/wiki/Derivative Differentiation]] is used to determine whether ""k"" is increasing or decreasing, but in a small region of the surface, the geometry approaches that of the tangent space, in which ""k = 1"" is constant. ""k"" has a [[http://en.wikipedia.org/wiki/Stationary_point stationary point]]. At a stationary point, one cannot tell from the first derivative whether ""k"" is increasing or decreasing. It is necessary to look to the second derivative. Then the characterisation of curvature at a point is more accurately stated:
""
| Negative curvature: |
|
k accelerating with r. |
|
| Zero curvature: |
|
k constant as r increases. |
| Positive curvature: |
|
k decelerating as r increases. |
""
<<""Definition: A singularity is a point where the second derivative of k is not defined..""
<<
====""""Tensor Curvature====
"" The characterisation of curvature as positive or negative is simplistic in the general case. In general k may have dependencies on both position and direction. In two dimensions we need a quantity which can describe two vector directions to fully describe curvature. This will be a quantity with two vector indices, i.e. a tensor. In three dimensions, three indices are needed. The general description of curvature in spacetime requires a tensor with four vector indices. Curvature is most generally described by the Riemann curvature tensor. |
""
====""""Charts or Coordinate Systems====
"" We can produce flat maps of regions of the Earth’s surface. By a map, we mean that we define a function from the points of the surface to the points of the map. The function must have sensible properties. The smaller the region being mapped, the less distortion we find in the map.
A coordinate system, or chart» is a map such that the coordinates in the flat space are the same as the coordinates used to describe the geometry. For example, an equirectangular projection uses equal spacing between lines of longitude and latitude. It is a chart for geographic» (longitude-latitude) coordinates. An atlas» is a collection of charts covering a the surface of the Earth. In non-Euclidean geometry, we generalise this idea, and say that an atlas» is a collection of charts covering a manifold»
 |
""<<""Definition: The space described by a geometry is a manifold.""<< <<""Definition: A chart is a map on flat space of a region of a manifold.""<< << ""Definition: An atlas is a collection of charts covering a manifold.""<<
====""""The Metric====
Typically, charts are subject to scaling distortions; distance on the map is not proportional to the real distance between points. A //metric// is a function which undoes local scaling distortions and returns real distances as determined by measurement. This only works for distances short enough that the difference between a curved space and a flat space is not noticeable. In a two dimensional space, like the surface of the Earth, the metric is represented as a ""2 × 2"" matrix. In spacetime a ""4 × 4"" matrix will be needed. For reasons to do with Pythagoras’ theorem the components of the matrix are proportional to squared distances.
====""""Tangent Space====
"" The idea of a tangent space for two dimensional surfaces embedded in 3-space is clear. When we discuss the geometry of spacetime, we dispense with the idea of embedding in higher dimensonal space, but we retain the notion that a tangent space a flat space which meets the surface at a point, and which has geometrical properties identical to those of the surface within a small enough neighbourhood of the point. Thus a chart in which the scale is 1:1 at X is a tangent space at X. For example, an equirectangular projection, scaled such that cartographical distances on the projection are 1:1 with geographical distances at the equator, is a tangent space at any point on the equator. |
""
<<""Definition: A tangent chart is a chart which is also a tangent space.""
<<
====""""Parallel Displacement====
{{image class="right" alt="curvature2-4" title="Parallel Displacement" url="images/curvature/Curvature2-4N.gif"}}Dynamical properties like momentum are represented by vectors associated with moving objects. A vector is loosely described as an object with magnitude and direction, and may be represented as an arrowed line. In a curved surface, a vector must be defined at a particular position. Translating a vector on a curved surface does not generally make sense, but we can translate one in tangent space. In a small neighbourhood, the surface is indinstinguishable from the tangent space. So, for a small enough distance between ""X"" and ""Y"", we can translate a vector in tangent space at ""X"" and project it to an indistinguishable vector at ""Y"". This is //parallel displacement//.
In the ""lensed and mirrored "" spaces, scaling distortions do not imply that the space is curved. The ""metric "" is not suffient on its own to say whether a space is intrinsically curved. We also need to know what happens when a ruler (more generally a system of measurement) moves from one point to another. This requires a definition of parallel, and is given mathematically by an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]].
<< "" Definition: An affine connection is a rule which defines parallel vectors, when the vectors are defined at points separated by a small displacement.""
<<
On the geometry of the Earth’s surface, and in general relativity, the affine connection is given by parallel displacement, in accordance with physical experience that it is meaningful to move a ruler parallel to itself over short distances.
""""====""""Parallel Transport====
"" Parallel transport means repeating parallel displacement for small distances along all path. Observe that the result of parallel transport depends on the path taken. The red vector at W, at the equator, points due North. Under parallel transport along the equator(green) to E, at a longitude 90° east of W, it continues to point due North. But if we parallel transport it to the North pole, N, then turn right and parallel transport it back to the equator, we arrive at E with the vector pointing due east. |
""
====""""Geodesic Motion====
In a flat space, a straight line can be defined by translating a vector in the direction in which it is pointing. The same idea applies in curved space. A [[http://en.wikipedia.org/wiki/Geodesic geodesic]] is defined by parallel transport of a vector along its own axis. Since this is true in any number of dimensions and bodies always move in the direction of their velocity vector, straight line motion will be replaced by geodesic motion in curved spacetime.
It is easy to see that the shortest distance between two points is a geodesic, because it is indistinguishable from a straight line in any small neighbourhood, so that it is made up of shortest distances. The converse is not true. The long route round the equator from W to E is also a geodesic. There are an indefinite number of geodesics between two points at opposite ends of a diameter on a sphere.
[[BasicsOfCurvature Basics of Curvature ↑]] [[TheEquivalencePrinciple The Equivalence Principle →]]
Additions:
""
""====""""====
======[[FoundationsOfSpecialRelativity ←]] Basics of Curvature [[PhysicalPrinciples ↑]] [[TheEquivalencePrinciple →]]======
A gentle introduction to ideas encapsulated in the mathematics of Riemannian Geometry.
====""""The Development of Non-Euclidean Geometry====
Euclidean geometry incorporates the parallel postulate, that parallel lines can be extended indefinitely always the same distance apart, but this assumes that the measurements made in one place are relevant to measurements made elsewhere. For centuries people had tried to prove the parallel postulate but [[http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss Gauss]] was the first to realise that it might not actually be true. As part of his employment in cartography, mapping the surface of the Earth using measurements made on the surface of the Earth, Gauss developed mathematics to describe relationships between measurements within [[http://en.wikipedia.org/wiki/Gaussian_curvature curved surfaces]]. This was the beginning of the study of [[http://en.wikipedia.org/wiki/Non-Euclidean_Geometry non-Euclidean geometry]] and the origin of the term [[http://en.wikipedia.org/wiki/Curvature curvature]] as it is used to describe spacetime. We will need to recognise that this use of the word curvature is rather different from the common idea of a curved surface.
Gauss developed mathematics for two dimensional curved surfaces embedded in three-space, and he also recognised that it is not necessarily the case that 3-space is everywhere Euclidean. He even had surveyers under his direction take measurements between peaks in the Andes to establish whether the angles of a triangle add to 180° (they did).
Unknown to each other, [[http://en.wikipedia.org/wiki/J%C3%A1nos_Bolyai János Bolyai]] and [[http://en.wikipedia.org/wiki/Nikolai_Ivanovich_Lobachevsky Nikolai Lobachevski]] each developed non-Euclidean geometries, but the major step was made by [[http://en.wikipedia.org/wiki/Bernhard_Riemann Riemann]], who generalised two dimensional non-Euclidean geometry and found a mathematical structure for dealing with general geometries in any number of dimensions. Riemannian geometry is certainly one of the greatest achievements in the history of mathematics. For it, and his numerous other achievements, many of them also of a difficult nature, Riemann gets my vote as the greatest mathematician in history, ahead of both Gauss and [[http://en.wikipedia.org/wiki/Isaac_Newton Newton]].
Riemann considered himself physicist, more than mathematician — not many agreed, perhaps largely because so little of what he did was understood by others. He certainly hoped to gain insight into gravity from the study of non-Euclidean geometry. In this, of course, in the absence of relativity, he was not successful, but when Einstein found himself faced with the same problem, his friend [[http://en.wikipedia.org/wiki/Marcel_Grossman Grossman]] directed him to Riemann’s work. The main part of general relativity is the synthesis of Riemannian geomentry with special relativity.
Non-Euclidean geometry in ""n""-dimensions was developed by generalising the geometry of the surface of the Earth to arbitrary curved surfaces, and to an arbitrary number of dimensions. This page introduces principle ideas in non-Euclidean geometry visually, with special reference to mapping the Earth’s surface. In [[NonEuclideanGeometry Non-Euclidean Geometry]], these ideas are applied more generally.
====""""Intrinsic and Extrinsic Curvature====
In many ways the term curvature is misleading when applied to spacetime. There are two types of curvature, known as intrinsic and extrinsic curvature. Extrinsic curvature is the familiar concept of curvature, visible curvature, or the shape of a two dimensional surface as it appears to us in three dimensional space. Intrinsic curvature refers to the geometrical properties of a space found from measurements taken within the space.
Intrinsic and extrinsic curvature are quite distinct ideas. A geometry can have intrinsic curvature but no extrinsic curvature, and one can have extrinsic curvature but no intrinsic curvature. The curvature of spacetime is intrinsic, because it refers to geometrical properties defined by measurement within spacetime. Extrinsic curvature has no meaning for spacetime, because there is no outside space to look at it from. To introduce the concept of intrinsic curvature we look at some simple examples.
""Sphere: When you construct two equal lines of longitude, on a sphere AD and BC, you find that CD is less than AB, so the parallel postulate does not hold. By the same token, the circumference of a circle is less than 2πr, where r is radial distance measured in the surface of the sphere.
Cylinder: A cylinder has extrinsic curvature, but no intrinsic, or Gaussian, curvature, because it can be made by rolling up a piece of paper without changing distance relationships between points. The circumference of a circle drawn on the paper is still 2πr when the paper is rolled into a cylinder. |
""
""Cone: Like the cylinder you can make a cone by rolling flat paper. A circle enclosing the apex has circumference less than 2πr, a circle anywhere else has circumference equal to 2πr. The cone is intrinsically flat except at the apex, where the geometry has a singularity.
World map: A map of the world has intrinsic curvature, but no extrinsic curvature. It is drawn on flat paper, and does not look curved, but distances described by the map are geographical. It has exactly the same geometry as the world itself, and geometrically approximates a sphere. |
""
""Lenses and mirrors: The image under a magnifying glass is curved according to any normal definition; it has extrinsic curvature. But the distance between two points in the image is measured by a ruler, and is the same as the original. The geometry is intrinsically flat. The same is true of an image in a curved mirror.
| |
""
====""""Curvature of Spacetime====
There is no absolute distance scale. In practice coordinates are always defined with respect to ""reference matter"" within a neighbourhood of an observer. Distances are established locally by measuring matter relative to other matter. If all matter locally were to “shrink”, including ourselves and all the matter contained our measuring apparatus, we would have no means to detect the fact. Our fundamental length scales would “shrink” in direct proportion, and it would make no difference to the numerical results of local distance measurements. In the absence of such a difference, “shrinking” is meaningless. To talk about shrinking we need to make a comparison between the coordinates we define here and now, using here and now clocks and rulers, and the coordinates defined somewhere else.
The implication is that, if you hold up a stick, and a friend stands some distance away and measures its length by some triangulation technique based on Euclidean geometry, then he is not guaranteed to get the same answer as if you bring the stick to him and he measures its length directly. Of course, in this instance, we could not detect a difference between the two measurements, but the effects of non-Euclidean geometry can be measured, for example in the [[http://en.wikipedia.org/wiki/Pound-Rebka_experiment Pound-Rebka experiment]]. In [[FoundationsOfSpecialRelativity special relativity]], distance measurements are reduced to measurements of time. The statement that spacetime is not flat is precisely equivalent to the statement that identical clocks at different positions do not necessarily measure time at the same rate.
The parallel postulate breaks down precisely because distance is defined by local comparison. It holds within the geometry of a piece of paper, because we can take a ruler and slide it across the paper, and thereby compare the distance between parallel lines at one place and the corresponding distance at another. On the scale of astronomical distances, no such sliding of rulers is possible — or if it is, we cannot say that the ruler itself does not “expand” or “shrink” in the process. Special relativity describes measurements of time and position, using physical measurement to set up a reference frame and coordinates within a neighbourhod of an observer. According to the ""general principle of relativity"" another observer at a distant point can carry out exactly the same kind of measurements, and set up coordinates in exactly the same way, but special relativity does not provide us with a way to compare the coordinates set up by one observer with those of a second, remote observer. The absence of direct comparison between distant coordinate systems creates the possibility that the parallel postulate does not hold and that spacetime is non-Euclidean.
""| Special relativity assumed that the redshift factor, k, depends only on the magnitude of relative velocity. Local laws of physics would not be affected if k also depends on relative position. Because of the dependency of distance measurements on time, this will result in “shrinking” or “expansion” of one observer’s coordinates compared to the other’s, separate from, and in addition to, special relativistic, velocity dependent, time dilation. |
""
====""""Positive and Negative Curvature====
{{image class="right" alt="curvature-6" title="parallel postulate relaxed" url="images/curvature/Curvature-6N.gif"}}Intrinsic curvature is characterised by geometric properties. For example we can characterise curvature by comparison with the parallel postulate. If we measure the distance ""AB = d"" between any two points, and measure two equal distances ""AD = BC = h"", perpendicular to ""AB"", then we have no prior reason to assume that ""AB"" is equal to ""DC"", as measured by an observer at ""D"". So, ""DC = kAB = kd"" where ""k"" is a non-constant scale factor. This leads to a broad characterisation:
""
| Negative curvature: |
|
k increasing with h. |
|
| Zero curvature: |
|
k constant as h increases. |
| Positive curvature: |
|
k decreasing as h increases. |
""
{{image class="left" alt="curvature-7" title="characterisation of curvature" url="images/curvature/Curvature-7N.gif"}}The parallel postulate is a not an ideal criterion on which to base a categorisation of geometry, because curvature is better understood at a point, not on a line. A more natural, characterisation of a geometry is found by considering the length of an arc, ""CD"", of a circle of radius, ""r"", subtended by an angle, ""θ"", at the origin, ""O"". Use a very small angle, ""θ"", and drop perpendiculars of equal length from ""CD"" to a base line, ""AB"", through the origin ""O"". Then, in Euclidean geometry the length of ""CD"" is ""rθ"", almost equal to ""AB"". But in general the length of ""CD"" is ""krθ"", and the value of ""k"" characterises the geometry according to the relationships:
""
| Negative curvature: |
|
k increasing with r. |
|
| Zero curvature: |
|
k constant as r increases. |
| Positive curvature: |
|
k decreasing as r increases. |
""
""""{{image class="right" alt="curvature-8" title="Saddle Geometry" url="images/curvature/Curvature-8N.gif"}}**Saddle:** ""k"" increases with ""r"". So, curvature is negative.
{{image class="left" alt="curvature-9" title="Spherical Geometry" url="images/curvature/Curvature-9N.gif"}}
**Sphere:** ""k"" decreases as ""r"" increases. Curvature is positive.
"" | ""
====""""Singularities====
{{image class="right" alt="curvature-10" title="Trumpet Geometry" url="images/curvature/Curvature-10N.gif"}}If in a small region the value of the circumference of a circle approaches ""2πr"", the geometry locally approximates a flat geometry. It has a flat //tangent space// (red). If there is no unique tangent space the geometry has a ""singularity"", as at the apex. The practical implication is that, in general relativity, a singularity is a point where we do not know how to formulate the laws of physics.
[[http://en.wikipedia.org/wiki/Derivative Differentiation]] is used to determine whether ""k"" is increasing or decreasing, but in a small region of the surface, the geometry approaches that of the tangent space, in which ""k = 1"" is constant. ""k"" has a [[http://en.wikipedia.org/wiki/Stationary_point stationary point]]. At a stationary point, one cannot tell from the first derivative whether ""k"" is increasing or decreasing. It is necessary to look to the second derivative. Then the characterisation of curvature at a point is more accurately stated:
""
| Negative curvature: |
|
k accelerating with r. |
|
| Zero curvature: |
|
k constant as r increases. |
| Positive curvature: |
|
k decelerating as r increases. |
""
<<""Definition: A singularity is a point where the second derivative of k is not defined..""
<<
====""""Tensor Curvature====
""| The characterisation of curvature as positive or negative is simplistic in the general case. In general k may have dependencies on both position and direction. In two dimensions we need a quantity which can describe two vector directions to fully describe curvature. This will be a quantity with two vector indices, i.e. a tensor. In three dimensions, three indices are needed. The general description of curvature in spacetime requires a tensor with four vector indices. Curvature is most generally described by the Riemann curvature tensor. |
""
====""""Charts or Coordinate Systems====
""We can produce flat maps of regions of the Earth’s surface. By a map, we mean that we define a function from the points of the surface to the points of the map. The function must have sensible properties. The smaller the region being mapped, the less distortion we find in the map.
A coordinate system, or chart» is a map such that the coordinates in the flat space are the same as the coordinates used to describe the geometry. For example, an equirectangular projection uses equal spacing between lines of longitude and latitude. It is a chart for geographic» (longitude-latitude) coordinates. An atlas» is a collection of charts covering a the surface of the Earth. In non-Euclidean geometry, we generalise this idea, and say that an atlas» is a collection of charts covering a manifold»
|
""<<""Definition: The space described by a geometry is a manifold.""<< <<""Definition: A chart is a map on flat space of a region of a manifold.""<< << ""Definition: An atlas is a collection of charts covering a manifold.""<<
====""""The Metric====
Typically, charts are subject to scaling distortions; distance on the map is not proportional to the real distance between points. A //metric// is a function which undoes local scaling distortions and returns real distances as determined by measurement. This only works for distances short enough that the difference between a curved space and a flat space is not noticeable. In a two dimensional space, like the surface of the Earth, the metric is represented as a ""2 × 2"" matrix. In spacetime a ""4 × 4"" matrix will be needed. For reasons to do with Pythagoras’ theorem the components of the matrix are proportional to squared distances.
====""""Tangent Space====
""| The idea of a tangent space for two dimensional surfaces embedded in 3-space is clear. When we discuss the geometry of spacetime, we dispense with the idea of embedding in higher dimensonal space, but we retain the notion that a tangent space a flat space which meets the surface at a point, and which has geometrical properties identical to those of the surface within a small enough neighbourhood of the point. Thus a chart in which the scale is 1:1 at X is a tangent space at X. For example, an equirectangular projection, scaled such that cartographical distances on the projection are 1:1 with geographical distances at the equator, is a tangent space at any point on the equator. |
""
<<""Definition: A tangent chart is a chart which is also a tangent space.""
<<
====""""Parallel Displacement====
{{image class="right" alt="curvature2-4" title="Parallel Displacement" url="images/curvature/Curvature2-4N.gif"}}Dynamical properties like momentum are represented by vectors associated with moving objects. A vector is loosely described as an object with magnitude and direction, and may be represented as an arrowed line. In a curved surface, a vector must be defined at a particular position. Translating a vector on a curved surface does not generally make sense, but we can translate one in tangent space. In a small neighbourhood, the surface is indinstinguishable from the tangent space. So, for a small enough distance between ""X"" and ""Y"", we can translate a vector in tangent space at ""X"" and project it to an indistinguishable vector at ""Y"". This is //parallel displacement//.
In the ""lensed and mirrored "" spaces, scaling distortions do not imply that the space is curved. The ""metric "" is not suffient on its own to say whether a space is intrinsically curved. We also need to know what happens when a ruler (more generally a system of measurement) moves from one point to another. This requires a definition of parallel, and is given mathematically by an [[http://en.wikipedia.org/wiki/Affine_connection affine connection]].
<< "" Definition: An affine connection is a rule which defines parallel vectors, when the vectors are defined at points separated by a small displacement.""
<<
On the geometry of the Earth’s surface, and in general relativity, the affine connection is given by parallel displacement, in accordance with physical experience that it is meaningful to move a ruler parallel to itself over short distances.
""""====""""Parallel Transport====
""| Parallel transport means repeating parallel displacement for small distances along all path. Observe that the result of parallel transport depends on the path taken. The red vector at W, at the equator, points due North. Under parallel transport along the equator(green) to E, at a longitude 90° east of W, it continues to point due North. But if we parallel transport it to the North pole, N, then turn right and parallel transport it back to the equator, we arrive at E with the vector pointing due east. |
""
====""""Geodesic Motion====
In a flat space, a straight line can be defined by translating a vector in the direction in which it is pointing. The same idea applies in curved space. A [[http://en.wikipedia.org/wiki/Geodesic geodesic]] is defined by parallel transport of a vector along its own axis. Since this is true in any number of dimensions and bodies always move in the direction of their velocity vector, straight line motion will be replaced by geodesic motion in curved spacetime.
It is easy to see that the shortest distance between two points is a geodesic, because it is indistinguishable from a straight line in any small neighbourhood, so that it is made up of shortest distances. The converse is not true. The long route round the equator from W to E is also a geodesic. There are an indefinite number of geodesics between two points at opposite ends of a diameter on a sphere.
[[BasicsOfCurvature Basics of Curvature ↑]] [[TheEquivalencePrinciple The Equivalence Principle →]]
Deletions:
""
""====""""====
======[[ParticlesOrFields ←]] A Gravitating Particle [[RelationalQuantumGravity ↑]] [[SpacetimeStructure →]]======
Relational quantum gravity modifies the standard, continuum, form of qed and introduces an effective minimal interval of proper time between interactions of an elementary particle. The result of this modification is that instead of Minkowski spacetime, we find a curved spacetime obeying Einstein’s field equation.
====""""The k-Calculus====
Special relativity was described by Einstein as a principle theory, as distinct from a fundamental theory, because it derives its results from principles of measurement, and does not explain fundamental physical mechanisms. It enables us to derive the mathematical properties of Minkowski spacetime, but does not explain the underlying structure, or give the reason why light behaves as it does. In [[FoundationsOfSpecialRelativity Foundations Of Special Relativity]], the structure of Minkowski spacetime was found from the radar method, using two way transmission of light. As used in the radar method, light is the sensor for the properties of spacetime, not the cause of them. The radar method assumes a greater importance in relational quantum gravity. In [[QuantumElectrodynamics qed]], two-way transmission of photons is responsible for the electromagnetic force, and hence for the all the structures of matter we observe in our immediate environment. The radar method can use light as a sensor for the properties of spacetime because it uses the same physical process, two-way photon exchange, which is also the cause of the structure of spacetime.
The ""k-calculus»"" was introduced by [[http://en.wikipedia.org/wiki/Hermann_Bondi Hermann Bondi]] as a simple means of introducing [[FoundationsOfSpecialRelativity special relativity]]. In the ""k""-calculus, the metric is determined by the reflection of light (the radar method). Any other method of measurement may be used provided that it is calibrated to, and hence equivalent to, radar. In special relativity ""k"" may be identified with Doppler shift, ""k = 1 + z"" . The treatment of ""stationary observers"" generalises the ""k""-calculus to the case in which ""k"" may be identified with gravitational redshift. This approach is directly based on the [[TheEquivalencePrinciple equivalence principle]], has advantages in (relative) mathematical and conceptual simplicity, and leads to a formulation of general relativity which is mathematically equivalent to other formulations.
Here I generalise the radar method to allow that the minimum time for the return of a photon may depend not only on the speed of light, ""c"", but also on an effective minimum ""proper time"" interval between the absorption and re-emission of a reflected photon. In the idealised case of a single elementary particle, neglecting spin, the geometrical effect is calculated. I will show that an effective minimal time interval of reflection proportional to the mass of the reflecting elementary particle leads the replacement of Minkowski geometry by ""Schwarzschild"".
====""""A Modification to Radar====
Consistent with Einstein’s 1905 paper and the internationally agreed empirical definition of the metre, Bondi's ""k""-calculus for special relativity postulates instantaneous reflection of radar at the event whose position is to be determined. Although reflection clearly takes place on a very small timescale, there is no empirical basis on which we can say it is actually instantaneous. A natural generalisation is to hypothesise a small time delay between absorption and emission in proper time of a fundamental charged particle (electron or quark) reflecting electromagnetic radiation.
An intrinsic delay between the interactions of elementary particles affects the empirical definition of spacetime measurement (e.g. [[http://en.wikipedia.org/wiki/International_System_of_Units SI units]]). We seek to analyse the geometric implications. The metric is determined as in the ""k""-calculus for special relativity, from the minimum time for the return of information reflected at an event. But now this minimum net time depends not only on the maximum theoretical speed of information, ""c"", but also on an effective least proper time between absorption and emission in the reflection of a photon. Let the effective time delay be ""4GM"", where ""M"" is the mass of the reflecting particle and ""4G"" is a constant of proportionality. It will be seen that ""G"" may be identified with the gravitational constant. Special relativity can be recovered in the limit in which ""G"" goes to zero (allowing ""G"" to go to zero introduces the Landau pole, so this limit may not be strictly valid).
There are several reasons for introducing such a delay. Firstly, as shown here, the delay perturbs the metric, resulting in a curved spacetime obeying ""Einstein’s field equation"". If the reflection of a photon were instantaneous, the physical metric would be Minkowski. The intrinsic time delay in reflection, ""4GM"", causes a small amount of curvature, in accordance with Einstein's field equation. Secondly, it is well known that some small scale modification is needed to qed in order to remove the ""ultraviolet divergence"" and resolve the ""Landau pole"". The delay introduced here is an effective cut-off and allows the construction of a consistent qed. Finally, a minimum time between interactions proportional to mass may be related to the concept of inertia; if the interactions of muons and electrons with photons are discrete and identical, then it is natural that the acceleration due to the electromagnetic field will be proportional to the frequency of interaction. An intrinsic delay between interactions proportional to mass will result in accelerations inversely proportional to mass.
The calculation performed here applies to fundamental stable particles which can emit and absorb photons. In the real world these are charged particles with spin, electrons or quarks. It does not directly apply to macroscopic bodies. If radar is used to measure the position of the moon, for example, then an individual reflected photon can be said only to measure the position of a single electron in the surface of the moon. A classical radar pulse contains many such photons, the time delay at each reflection being dependant on the mass of an electron, not the mass of the moon. A more realistic calculation should also take into account charge and spin, and would be expected to yield Kerr-Newman geometry. It is not known how to carry out such a calculation within the ""k""-calculus, but it appears reasonable to separate off the contributions due to charge and spin, and to regard the calculation below of a Schwarzschild geometry surrounding a ""particle in a position eigenstate"" as a genuine indicator of an effective time interval between the interactions of an elementary particles.
====""""A Particle in a Position Eigenstate====
Since I am discussing measurement of position, I will describe ""eigenstates"" of position. There is no such thing as a perfect eigenstate of position of a massive point particle. Nonetheless such states ""span"" ""Hilbert space"" and will be sufficient for a description of geometry. For the purpose of analysis, I will consider a static system and calculate the metric at particular time. I consider only a system with a single gravitating particle, at ""O"". I will assume distance and time scales such that ""cosmological expansion"" is negligible.
"" An isolated elementary particle in an eigenstate of position has spherical symmetry and spacetime diagrams may be used to show a radial coordinate in n dimensions without loss of generality. A spacetime diagram is drawn, showing a tangent space with an origin at O, so that light is shown at 45°, lines of equal time are horizontal and time is proper time for the gravitating particle. In tangent space at O, the coordinate distance between Beth and the particle is r. |
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"" As shown in tangent space at O, Beth uses the radar method to determine a distance coordinate for the particle. Beth cannot resolve the points A and E where a photon is absorbed and emitted and places the particle at apparent position P. If the effective delay in the reflection is 4GM, then the coordinate coordinate distance of P from Beth is ρ = r + 2GM. |
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"" As calculated for stationary observers, the physical metric (with angular directions suppressed) is
 
So, Beth’s clock runs fast by a factor k compared to proper time, t, for the particle at O. Minkowski coordinates (t, r) for tangent space at O are stretched by a factor k−1 in the time direction and by k in the radial direction compared to Minkowski coordinates, (T, R) for tangent space at B, as determined by Beth using the radar method. Since the radar method determines that R = T, we have
So,
Using the apparent position ρ as the radial coordinate, and substituting ρ, we find the Schwarzschild metric:
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It will be observed that the event horizon, ""ρ = 2GM"", maps to ""r = 0"". Only space outside the event horizon is mapped in these coordinates. The space inside the event horizon, ""ρ < 2GM"", has no physical meaning. A particle which is point-like in a tangent space at the position of the particle, is mapped to the event horizon in a tangent space at the position of a distant observer. The effect of the discrete interval of proper time between interactions is a singularity at ""ρ = 0"" which “magnifies” a point-like particle to the size of the event horizon.
====""""The Path of Light====
"" With r as the radial coordinate, the metric is
Superficially, it appears that the speed of light is less than unity in these coordinates. Light should then be drawn at a greater than 45° to the horizontal. The inconsistency is resolved because this refers to the mean velocity of light, as determined from measurement using the radar method. The true path of light is plotted at 45°. The coordinate singularity, or event horizon in the Schwarzschild geometry, is actually at the position of the particle itself, while the singularity at ρ = 0 is an illusory point, which does not correspond to a position in space. |
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====""""Einstein’s Field Equation====
For a single particle in an eigenstate of position, stress energy is given by
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By considering the coordinate transformation ""r → r + δr"", we replace the delta function with a uniform density over a small region. We now take the limit ""δr → 0"" to see that Einstein’s field equation holds for the single particle case
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The derivation here applies only to the case where the gravitating source is a single particle with known position. The next section considers the issues in the general case.
[[OriginOfCurvature A Gravitating Particle ↑]] [[SpacetimeStructure The Emergence of Spacetime Structure →]]
Additions:
The ""k-calculus»"" was introduced by [[http://en.wikipedia.org/wiki/Hermann_Bondi Hermann Bondi]] as a simple means of introducing [[FoundationsOfSpecialRelativity special relativity]]. In the ""k""-calculus, the metric is determined by the reflection of light (the radar method). Any other method of measurement may be used provided that it is calibrated to, and hence equivalent to, radar. In special relativity ""k"" may be identified with Doppler shift, ""k = 1 + z"" . The treatment of ""stationary observers"" generalises the ""k""-calculus to the case in which ""k"" may be identified with gravitational redshift. This approach is directly based on the [[TheEquivalencePrinciple equivalence principle]], has advantages in (relative) mathematical and conceptual simplicity, and leads to a formulation of general relativity which is mathematically equivalent to other formulations.
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Deletions:
The ""k-calculus»"" was introduced by [[http://en.wikipedia.org/wiki/Hermann_Bondi Hermann Bondi]] as a simple means of introducing [[FoundationsOfSpecialRelativity special relativity]]. In the ""k""-calculus, the metric is determined by the reflection of light (the radar method). Any other method of measurement may be used provided that it is calibrated to, and hence equivalent to, radar. In special relativity ""k"" may be identified with Doppler shift, ""k = 1 + z"" . The treatment of ""stationary observers"" generalises the ""k""-calculus to the case in which ""k"" may be identified with gravitational redshift. This approach is directly based on the [[Equivalence equivalence principle]], has advantages in (relative) mathematical and conceptual simplicity, and leads to a formulation of general relativity which is mathematically equivalent to other formulations.
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