 Imagine n observers, Beth1, Beth2, …, Bethn, with rods of proper lengths l'1, l'2, …, l'n, positioned end to end, such that they form an unbroken circle of radius r in Alf’s coordinates. The angles, θ1, θ2, ... θn, subtended by the rods at Alf need not be equal. Let the redshift for Bethi be ki. The diagram is scaled such that coordinate distances determined by radar by Bethi are reduced by factor ki, so that Bethi places Alf at the centre of the circle. Since the proper length l'i is equal to the coordinate length determined locally by Bethi, we have g11 = −1 / k. Thus the metric is
The Schwarzschild solution to Einstein’s field equations for static coordinates, outside of an isolated spherically symmetric gravitating body is an example of a metric with this form, in which
Deletions:
"" An observer, Alf, defines spherical coordinates (t, r, θ, φ) with an origin at A. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the metric has g00 = k2. |
""
"" Alf and Beth each measure the coordinate distance of the other, using light speed equal to unity, as in special relativity. Using unit light speed Beth calculates coordinate distance r' = kr/i> to Alf (figure 1a). |
""
"" Imagine n observers, Beth1, Beth2, …, Bethn, with rods of proper lengths l'1, l'2, …, l'n, positioned end to end, such that they form an unbroken circle of radius r in Alf’s coordinates. The angles, θ1, θ2, ... θn, subtended by the rods at Alf need not be equal. Let the redshift for Bethi be ki. The diagram is scaled such that coordinate distances determined by radar by Bethi are reduced by factor ki, so that Bethi, places Alf at the centre of the circle. Since the proper length li is equal to the coordinate length determined locally by Bethi, we require that g11 = −k−2. Thus the metric is
The [[http://en.wikipedia.org/wiki/Schwarzschild_metric Schwarzschild]] solution to Einstein’s field equations for static coordinates, outside of an isolated spherically symmetric gravitating body is an example of a metric with this form, in which
""""
Coordinates defined such that the speed of light is unity allow no description of the region inside an event horizon. For stationary observers, if the redshift from A to B is kAB, and the redshift from B to C is kBC, then the redshift from A to C is kAC = kABkBC. Thus, for mutually stationary observers, the metric is a group.
For stationary observers, this form of the metric determines a mathematical [[http://en.wikipedia.org/wiki/Group_(mathematics) group]]. That is to say that if the redshift from ""A"" to ""B"" is ""kAB"", and the redshift from ""B"" to ""C"" is ""kBC"", then the redshift from ""A"" to ""C"" is ""kAC = kABkBC"", and the metric at ""C"" obeys
"" ""
Additions:
""| An observer, Alf, defines spherical coordinates (t, r, θ, φ) with an origin at A. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the metric has g00 = k2. |
""
"" Beth holds a small rod of proper length l' at B, perpendicular to AB. The defining condition for Schwarzschild coordinates is that the angle subtended by the rod at A is l' / r. Thus, in spherical coordinates g22 = −r2 and g33 = −r2sin2θ |
"".
Deletions:
"" An observer, Alf, defines spherical coordinates (t, r, θ, φ) with an origin at A. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the metric has g00 = k2. |
""
"" Beth holds a small rod of proper length l' at B, perpendicular to AB. The defining condition for Schwarzschild coordinates is that the angle subtended by the rod at A is l' / r. Thus, in spherical coordinates g22 = −r2 and g33 = −r2sin2θ |
"".
Additions:
""| An observer, Alf, defines spherical coordinates (t, r, θ, φ) with an origin at A. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the metric has g00 = k2. |
""
""| Beth holds a small rod of proper length l' at B, perpendicular to AB. The defining condition for Schwarzschild coordinates is that the angle subtended by the rod at A is l' / r. Thus, in spherical coordinates g22 = −r2 and g33 = −r2sin2θ |
"".
Deletions:
"" An observer, Alf, defines spherical coordinates (t, r, θ, φ) with an origin at A. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the metric has g00 = k2. |
""
"" Beth holds a small rod of proper length l' at B, perpendicular to AB. The defining condition for Schwarzschild coordinates is that the angle subtended by the rod at A is l' / r. Thus, in spherical coordinates g22 = −r2 and g33 = −r2sin2θ |
"".
Additions:
""| An observer, Alf, defines spherical coordinates (t, r, θ, φ) with an origin at A. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor k = 1 + z > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the metric has g00 = k2. |
""
""| Beth holds a small rod of proper length l' at B, perpendicular to AB. The defining condition for Schwarzschild coordinates is that the angle subtended by the rod at A is l' / r. Thus, in spherical coordinates g22 = −r2 and g33 = −r2sin2θ |
"".
<<""Definition: Schwarzschild coordinates have the metric""
Deletions:
"" An observer, Alf, defines spherical coordinates (t, r, θ, φ)with an origin at A. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor 1 + z = k > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the metric has g00 = k2. |
""
"""".
<<""Definition: Schwarzschild coordinates have the metric""
Additions:
"""".
""where k is the redshift of light from the origin to x, and f is an undetermined function."" <<
""Imagine n observers, Beth1, Beth2, …, Bethn, with rods of proper lengths l'1, l'2, …, l'n, positioned end to end, such that they form an unbroken circle of radius r in Alf’s coordinates. The angles, θ1, θ2, ... θn, subtended by the rods at Alf need not be equal. Let the redshift for Bethi be ki. The diagram is scaled such that coordinate distances determined by radar by Bethi are reduced by factor ki, so that Bethi, places Alf at the centre of the circle. Since the proper length li is equal to the coordinate length determined locally by Bethi, we require that g11 = −k−2. Thus the metric is
Deletions:
"" Beth holds a small rod of proper length l' at B, perpendicular to AB. The defining condition for [[http://en.wikipedia.org/wiki/Schwarzschild_coordinates Schwarzschild coordinates]] is that the angle subtended by the rod at A is l' / r. Thus, in spherical coordinates g22 = −r2 and g33 = −r2sin2θ |
"".
""where k is the redshift of light from the origin to x, and f is an undetermined function."" <<
"" Imagine n observers, Beth1, Beth2, …, Bethn, with rods of proper lengths l'1, l'2, …, l'n, positioned end to end, such that they form an unbroken circle of radius r in Alf’s coordinates. The angles, θ1, θ2, ... θn, subtended by the rods at Alf need not be equal. Let the redshift for Bethi be ki. The diagram is scaled such that coordinate distances determined by radar by Bethi are reduced by factor ki, so that Bethi, places Alf at the centre of the circle. Since the proper length li is equal to the coordinate length determined locally by Bethi, we require that g11 = −k−2. Thus the metric is
Additions:
""| Beth holds a small rod of proper length l' at B, perpendicular to AB. The defining condition for [[http://en.wikipedia.org/wiki/Schwarzschild_coordinates Schwarzschild coordinates]] is that the angle subtended by the rod at A is l' / r. Thus, in spherical coordinates g22 = −r2 and g33 = −r2sin2θ |
"".
Deletions:
""| Beth holds a small rod of proper length l'"" at B, perpendicular to AB. The defining condition for [[http://en.wikipedia.org/wiki/Schwarzschild_coordinates Schwarzschild coordinates]] is that the angle subtended by the rod at A is l' / r. Thus, in spherical coordinates g22 = −r2 and g33 = −r2sin2θ |
"".
Additions:
The metric field is often simply called the metric. One should avoid this abuse of language, because it confuses the metric field as a function of coordinate space with the metric at a given position.
""| An observer, Alf, defines spherical coordinates (t, r, θ, φ)with an origin at A. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor 1 + z = k > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the metric has g00 = k2. |
""
Deletions:
The metric field is often simply called the metric. One should avoid this abuse of language, because it confuses the metric field as a function of coordinate spacevalue with the metric at a given position.
""| An observer, Alf, defines spherical coordinates (r, θ, φ)with an origin at A. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor 1 + z = k > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the metric has g00 = k2. |
""
Additions:
""Schwarzschild Coordinates""
According to the general principle, an observer anywhere can use the radar method to define locally ""Minkowski Coordinates"", 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 "" Pound-Rebka experiment"". 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 ""Einstein’s field equation"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.
A ""chart"" 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, ""general covariance"" 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 ""Minkowski metric"", ""h"". ""h"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""h"" 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 ""tangent chart"", 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 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, ""A"", the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at ""B"", the origin of Beth’s coordinates. Alf and Beth both determine locally ""Minkowski Coordinates"". Alf’s time axis is denoted the 0-axis, and his space axes are labelled ""1"", ""2"", ""3"". Beth’s coordinates are denoted with primes, ""0'"", ""1'"", ""2'"", ""3'"".
<<""Definition: The metric field is defined, on a given coordinate system, by""
The metric field is often simply called the metric. One should avoid this abuse of language, because it confuses the metric field as a function of coordinate spacevalue with the metric at a given position.
====""""Schwarzschild Coordinates====
""| An observer, Alf, defines spherical coordinates (r, θ, φ)with an origin at A. A second observer, Beth, is at a constant position, B, at radial coordinate r in Alf’s coordinates. Suppose that Beth’s clock runs faster than Alf’s by a factor 1 + z = k > 1 (the argument below also holds for k < 1). Then light transmitted from Alf to Beth is redshifted by factor k and an interval t of Alf’s coordinate time at B is measured by Beth as proper time interval t' = kt. Then the metric has g00 = k2. |
""
""| Beth holds a small rod of proper length l'"" at B, perpendicular to AB. The defining condition for [[http://en.wikipedia.org/wiki/Schwarzschild_coordinates Schwarzschild coordinates]] is that the angle subtended by the rod at A is l' / r. Thus, in spherical coordinates g22 = −r2 and g33 = −r2sin2θ |
"".
<<""Definition: Schwarzschild coordinates have the metric""
"" ""
""where k is the redshift of light from the origin to x, and f is an undetermined function."" <<
""| Alf and Beth each measure the coordinate distance of the other, using light speed equal to unity, as in special relativity. Using unit light speed Beth calculates coordinate distance r' = kr/i> to Alf (figure 1a). |
""
""Imagine n observers, Beth1, Beth2, …, Bethn, with rods of proper lengths l'1, l'2, …, l'n, positioned end to end, such that they form an unbroken circle of radius r in Alf’s coordinates. The angles, θ1, θ2, ... θn, subtended by the rods at Alf need not be equal. Let the redshift for Bethi be ki. The diagram is scaled such that coordinate distances determined by radar by Bethi are reduced by factor ki, so that Bethi, places Alf at the centre of the circle. Since the proper length li is equal to the coordinate length determined locally by Bethi, we require that g11 = −k−2. Thus the metric is
The [[http://en.wikipedia.org/wiki/Schwarzschild_metric Schwarzschild]] solution to Einstein’s field equations for static coordinates, outside of an isolated spherically symmetric gravitating body is an example of a metric with this form, in which
Using cartesian space coordinates,
""""
|
""
Coordinates defined such that the speed of light is unity allow no description of the region inside an event horizon. For stationary observers, if the redshift from A to B is kAB, and the redshift from B to C is kBC, then the redshift from A to C is kAC = kABkBC. Thus, for mutually stationary observers, the metric is a group.
Deletions:
According to the general principle, an observer anywhere can use the radar method to define locally Minkowski coordinates, 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 "" Pound-Rebka experiment"". 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 special relativity. Together with ""Einstein’s field equation"", this determines a curved geometry which precisely accounts for Newton’s law of gravity.
A ""chart"" of spacetime is not 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 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, ""general covariance"" automatically allows claim that the same tensor equations hold in any coordinates. When possible, I will define coordinates as in special relativity, using the radar method. In this case the chart is made on Minkowski spacetime, which has constant ""Minkowski metric"", ""h"". ""h"" is a //non-physical metric//, analogous to the metric of the paper on which a map is drawn. ""h"" 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 ""tangent chart"", 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 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, ""A"", the origin of Alf’s coordinates. Let Beth be a distant observer, with a clock at ""B"", the origin of Beth’s coordinates. Alf and Beth both determine locally Minkowski coordinates. Alf’s time axis is denoted the 0-axis, and his space axes are labelled ""1"", ""2"", ""3"". Beth’s coordinates are denoted with primes, ""0'"", ""1'"", ""2'"", ""3'"".
<<""Definition: The metric field is defined, on a given coordinate system, by""
The metric field is often simply called the metric. Care should be taken. Failure to distinguish the metric at a given coordinate from the metric field as a function of coordinate space can lead to confusion.
"" Alf and Beth are stationary with respect to each other and Beth is at a coordinate distance r from Alf as measured in Alf’s coordinates. Alf and Beth each measure the coordinate length of a short rod located at Beth’s origin, and aligned on an axis with Alf and Beth. Using radar, Alf determines that the rod has a coordinate length d, while Beth determines a length d' in her coordinates. Because the rod is at Beth’s origin, d' is the true length, or proper length, of the rod. |
""
"" When Alf and Beth try to align their spacetime diagrams, maintaining synchronisation (lines of equal time are horizontal for both) and the constancy of the speed of light (light at 45°), they find a mismatch, because their clocks do not run at the same rate. In the diagrams, Beth’s clock runs faster than Alf’s by a factor k > 1; the arguments also hold when k < 1 and Beth’s clock is slower than Alf’s. Synchronisation requires that, for k > 1, Beth’s coordinates are enlarged by the factor k, so that if Beth’s coordinates are superimposed on Alf’s, and aligned at Beth, Alf appears at a coordinate distance r' = kr, displaced from his position in his own coordinates. |
""
"" The diagrams for measurement of the coordinate length of the rod are superimposed. The proper time interval 2d' in Beth’s coordinates corresponds to a time 2kd' in Alf’s coordinates. The proper length d' in Beth’s coordinates appears with coordinate distance d = d' ⁄ k in Alf’s coordinates. | |
""
"" Beth now turns the rod perpendicular to the axis from Alf. Beth calculates that the angle subtended in tangent space by the rod at Alf is θ' = d' ⁄ r'.
|
Imagine n observers, Beth1, Beth2, …, Bethn, with rods of proper lengths d'1, d'2, …, d'n, positioned end to end, such that they form an unbroken circle of radius r in Alf’s coordinates. The angles, θ1, θ2, ... θn, subtended by the rods at Alf are not all equal. Let the redshift for Bethi be ki. At each position i, Bethi applies a radial stretch with factor 1 ⁄ ki. The coordinate systems are then superposed with Alf at the centre of the circle in each observer’s coordinates. It is seen that θi = kiθi'. It is already established that ri' = kir. Then Alf’s coordinate length for Bethi’s rod is
equal to its proper length, as measured locally by Bethi (this is the defining condition for Schwarzschild coordinates).
|
""We have just seen that for quantities local to Beth, measured, or coordinate, time ""2kd'"" in Alf’s coordinates corresponds to a proper time ""2d'"", that coordinate distance ""d' ⁄ k"" corresponds to a proper distance ""d'"", and that angular distances are unchanged. Thus, a simple form for ""g"" can be given in ""spherical coordinates"",
""""
Using Cartesian space coordinates,
""""
The [[http://en.wikipedia.org/wiki/Schwarzschild_metric Schwarzschild]] solution to Einstein’s field equations for static coordinates, outside of an isolated spherically symmetric gravitating body is an example of a metric with this form, in which
""""
Additions:
====""""What is Spacetime?====
Deletions:
====""""What is spacetime?====
Additions:
<<**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.<<
Deletions:
<<**Answer 1.** There exists a [[http://plato.stanford.edu/entries/spacetime-theories/#5.2 substantive spacetime]], which is modelled (=described) by the mathematical structure of a manifold.<<
Additions:
Here the manifold simply replaces Newton’s conception of "" absolute space"" and "" absolute time"". 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.
The logical error lies in thinking that if we have a set of actual observations, B, and a theory, A with A=>B, then A must be true. In fact, there may be some other theory, C, which we may not know about, which also has C=>B, and such that C contradicts A. Modern physicists usually avoid the issue by denying that it is possible to describe nature:
Deletions:
In this conception, the manifold simply replaces Newton’s conception of "" absolute space"" and "" absolute time"". 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.
The logical error is in thinking that if we have a set of actual observations, B, and a theory A with A=>B, then A must be true. In fact, there may be some other theory, C, which we may not know about, which also has C=>B, and such that C contradicts A. Modern physicists usually avoid this issue by denying that it is possible to describe nature:
Additions:
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:
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 ""&emdash;"" 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:
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:
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 ""&emdash;"" 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:
I always find it surprising when physicists advance 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:
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 &emdash; 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:
If we have some object, and we can show that that object has the properties of a manifold, then we can say that the object is a manifold. We do this, for example, when we say that the surface of a sphere is a manifold. However, to do that we must first have the object. We know how the manifold behaves in general relativity, but it is natural to pose the question, "What is spacetime?". I know of three possible answers:
I always find it surprising when physicists advance 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:
Deletions:
If we have some object, and we can show that that object has the properties of a manifold, then we can say that the object is a manifold. We do this, for example, when we say that the surface of a sphere is a manifold. However, to do that we must first have the object. We know how the manifold behaves in general relativity, but it is natural to pose the question, "what is spacetime?". I know of three possible answers:
I always find it surprising when physicists advance 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. Special relativity is based on based on empirically verifiable //postulates//, the operational definitions of measurement. Einstein’s argument in special relativity 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:
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""The inconsistency between general relativity and classical electromagnetism is seen in the path of a signal is emitted from point ""C"", in line with ""AB"" and with small coordinate distance ""BC = l"". The coordinate time for the signal to reach ""B"" is ""l"", which corresponds to proper time ""kl"" on Beth’s clock. However, since ""BC"" is small, the proper distance ""BC"" as measured by Beth is ""k−1l"", so a signal emitted from ""C"" should be detected by Beth after a proper time interval ""k−1l"". Apparently ""k−1 = k"", so that ""k = 1"". There does not appear to be any classical resolution of this paradox, but a photon detected by Beth cannot also be detected by Alf. It is not meaningful to discuss the position of a quantum particle except in measurement, and it is therefore not meaningful to discuss the time at which a signal consisting of photons transmitted from ""C"" to ""A"" passes ""B"". Thus, we may think that the resolution should lie in the unification of quantum mechanics and general relativity.
Alf and Beth each measure the coordinate distance of the other, using light speed equal to unity, as in special relativity. Using unit light speed Beth calculates coordinate distance r' = kr to Alf.