Most recent edit on 2008-09-11 09:39:17 by CharlesFrancis

Deletions:

Definition: Proper time for an object is time measured on a clock moving with that object.

Edited on 2008-09-11 09:37:56 by CharlesFrancis

Additions:
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: Proper time for an object is time measured on a clock moving with that object.

Deletions:

Definition: Proper time for an object is time measured on a clock moving with that object.

Edited on 2008-09-11 09:36:09 by CharlesFrancis

Additions:

Definition: Proper time for an object is time measured on a clock moving with that object.

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'.
Using general covariance, each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing t' seconds for a stationary object at the origin of Beth’s coordinates τ = τ^i' = (t', 0, 0, 0). t' is the actual amount of time measured by Beth using her own clock, and is known as proper time. In Alf’s coordinates, GTR-1

is found by coordinate transformation. If Beth is stationary in Alf’s coordinates, τ = τ^i' = (kt', 0, 0, 0), where k is the gravitational redshift factor, k = 1 + z.

Deletions:
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'.
Using general covariance, each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing t' seconds for a stationary object at the origin of Beth’s coordinates τ = τ^i' = (t', 0, 0, 0). t' is the actual amount of time measured by Beth using her own clock, and is known as proper time. In Alf’s coordinates, GTR-1

is found by coordinate transformation. If Beth is stationary in Alf’s coordinates, τ = τ^i' = (kt', 0, 0, 0), where k is the gravitational redshift factor, k = 1 + z.

Edited on 2008-08-13 03:08:39 by CharlesFrancis

Additions:

← Concepts of General Relativity ↑ →

Deletions:

← Concepts of General Relativity →

Edited on 2008-05-22 02:53:51 by CharlesFrancis

Additions:
Using general covariance, each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing t' seconds for a stationary object at the origin of Beth’s coordinates τ = τ^i' = (t', 0, 0, 0). t' is the actual amount of time measured by Beth using her own clock, and is known as proper time. In Alf’s coordinates, GTR-1

is found by coordinate transformation. If Beth is stationary in Alf’s coordinates, τ = τ^i' = (kt', 0, 0, 0), where k is the gravitational redshift factor, k = 1 + z.

Deletions:
Using general covariance, each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing t' seconds for a stationary object at the origin of Beth’s coordinates τ = τ^i' = (t', 0, 0, 0). t' is the actual amount of time measured by Beth using her own clock, and is known as proper time. In Alf’s coordinates, GTR-1

is found by coordinate transformation. If Beth is stationary in Alf’s coordinates, τ = τ^i' = (kt', 0, 0, 0), where k is the gravitational redshift factor.

Edited on 2008-05-04 02:06:04 by CharlesFrancis

Additions:

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.

""<table width="100%" border="0" cellspacing=0 cellpadding=0><td><img class="right" alt="GTR-8" title="The rod is now perpendicular to a line from Alf to Beth" src="images/gtr/GTR-8N.gif">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'. The angle seen by Alf is θ = kθ'. Alf’s coordinate length of the rod, <table border=0><td></td></table>

Deletions:

""<table width="100%" border="0" cellspacing=0 cellpadding=0><td><img class="right" alt="GTR-8" title="The rod is now perpendicular to a line from Alf to Beth" src="images/gtr/GTR-8.gif">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'. The angle seen by Alf is θ = kθ'. Alf’s coordinate length of the rod, <table border=0><td></td></table>

Edited on 2008-05-02 09:47:24 by CharlesFrancis

Additions:
The metric field is a measure of the distortion present in a chart. It is not sufficient to describe curvature — we have seen examples of distorted spaces, like the lensed and mirrored geometries, which are actually flat. To describe curvature requires a connection in addition to the metric field. Given the metric field, an affine connection» describes a relationship between a set of coordinate axes at x, say, and another set, at x +dx, where dx is a small displacement, such that we can meaningfully describe a vector at x as being parallel to one at x +dx (other types of connection» are used to transport other types of data).

Deletions:
The metric field is a measure of the distortion present in a chart. It is not sufficient to describe curvature — we have seen examples of distorted spaces, like the lensed and mirrored geometries, which are actually flat. To describe curvature requires a connection in addition to the metric field. Given the metric field, an affine connection» describes a relationship between a set of coordinate axes at x, say, and another set, at x +dx, where dx is a small displacement, such that we can meaningfully describe a vector at x as being parallel to one at x +dx (other types of connection» are used to transport other types of data).

Edited on 2008-05-01 05:43:18 by CharlesFrancis

Additions:
These forms of the metric, g, are coordinate dependent. The metric is a tensor quantity, and will be used in tensor equations. Tensor equations are covariant, so that if a tensor equation holds in one set of coordinates, then it will hold in any coordinates. In particular, a tensor equation will hold after a Lorentz transform of either Alf’s or Beth’s coordinates, so the choice of stationary observers places no limitation on generality.

Deletions:
GTR-13

These forms of the metric, g, are coordinate dependent. The metric is a tensor quantity, and will be used in tensor equations. Tensor equations are covariant, so that if a tensor equation holds in one set of coordinates, then it will hold in any coordinates. In particular, a tensor equation will hold after a Lorentz transform of either Alf’s or Beth’s coordinates, so the choice of stationary observers places no limitation on generality.

Edited on 2008-05-01 05:10:41 by CharlesFrancis

Additions:
</td></table>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 <a href=http://www.teleconnection.info/rqg/IntroductionToVectorSpace#SphericalCoordinates>spherical» coordinates</a>, 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

Deletions:
</td></table> 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 <a href=http://www.teleconnection.info/rqg/IntroductionToVectorSpace#SphericalCoordinates>spherical» coordinates</a>, 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

Edited on 2008-05-01 05:08:24 by CharlesFrancis

Additions:
Using Cartesian space coordinates,
For stationary observers, this form of the metric determines a mathematical group». That is to say that if the redshift from A to B is k_AB, and the redshift from B to C is kBC, then the redshift from A to C is k_AC = k_ABk_BC, and the metric at C obeys

Deletions:
Using cartesian space coordinates,
For stationary observers, this form of the metric determines a mathematical http://en.wikipedia.org/wiki/Group_(mathematics)». 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

Edited on 2008-05-01 05:02:53 by CharlesFrancis

Additions:
GTR-11

Deletions:
GTR-11

Edited on 2008-05-01 04:59:49 by CharlesFrancis

Additions:
GTR-2

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'. The angle seen by Alf is θ = kθ'. Alf’s coordinate length of the rod,

<img alt="GTR-10" title="Covariant components of the metric for mutually stationary observers in spherical coordinates" src="images/gtr/GTR-10.gif" align="texttop" vspace="0"> Using cartesian space coordinates, <img alt="GTR-11" title="Covariant components of the metric for mutually stationary observers in Cartesian coordinates" src="images/gtr/GTR-11.gif" align="texttop" vspace="0">
For stationary observers, this form of the metric determines a mathematical http://en.wikipedia.org/wiki/Group_(mathematics)». 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

<img alt="GTR-13" title="k in Schwarzschild coordinates" src="images/gtr/GTR-13.gif" align="texttop" vspace="0">""

Deletions:
GTR-8

<img alt="GTR-10" title="Covariant components of the metric for mutually stationary observers" src="images/gtr/GTR-10.gif" align="texttop" vspace="0"> <img alt="GTR-11" title="k in Schwarzschild coordinates" src="images/gtr/GTR-11.gif" align="texttop" vspace="0">""

Edited on 2008-05-01 02:32:40 by CharlesFrancis

Additions:
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 general relativity, spacetime is described as a differentiable manifold. This means that we assume that we may apply the mathematical operation of differentiation to functions defined on the mapping space and we claim that differentiation is physically meaningful when those functions correspond to measurable physical quantities. For example, in classical physics we regularly differentiate a velocity to find an acceleration, or a potential to find a force. In the strictest sense, a map which consists of tables of data, or a bank of computer memory, is not continuous and is not differentiable. In computer science derivatives are regularly approximated with difference relations. Similarly, in applications to physics, differentiability does not require taking a mathematical limit, but merely requires a step size small enough that any errors are smaller than measurement errors and taking it smaller makes no practical difference to calculation. General relativity is a theory of classical matter, and is used to study large scale phenomena, anything from the gravity of a planet to the large scale structure of the universe. On such scales the assumption of differentiability is entirely reasonable and leads to no practical issues or disagreement between theory and observation. It will be necessary to revisit questions of continuity and differentiability in the study of quantum phenomena.
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.
Using general covariance, each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing t' seconds for a stationary object at the origin of Beth’s coordinates τ = τ^i' = (t', 0, 0, 0). t' is the actual amount of time measured by Beth using her own clock, and is known as proper time. In Alf’s coordinates, GTR-1

is found by coordinate transformation. If Beth is stationary in Alf’s coordinates, τ = τ^i' = (kt', 0, 0, 0), where k is the gravitational redshift factor.

Deletions:
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, tables of data and formulae containing a numerical description. 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 general relativity, spacetime is described as a differentiable manifold. This means that we assume that we may apply the mathematical operation of differentiation to functions defined on the mapping space and we claim that differentiation is physically meaningful when those functions correspond to measurable physical quantities. For example, in classical physics we regularly differentiate a velocity to find an acceleration, or a potential to find a force. It may be said that, in the strictest sense, a map which consists of tables of data, or a bank of computer memory, is not continuous and is not differentiable. In computer science derivatives are regularly approximated with difference relations. Similarly, in applications to physics, differentiability does not require taking a mathematical limit, but merely requires a step size small enough that any errors are smaller than measurement errors and taking it smaller makes no practical difference to calculation.
General relativity is a theory of classical matter, and is used to study large scale phenomena, anything from the gravity of a planet to the large scale structure of the universe. On such scales the assumption of differentiability is entirely reasonable and leads to no practical issues or disagreement between theory and observation. It will be necessary to revisit questions of continuity and differentiability in the study of quantum phenomena.
In principle many forms of coordinates can be used in 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.
Using general covariance, each observer describes physical quantities using vectors. Using Beth’s primed coodinates, denote the vector describing t' seconds for a stationary object at the origin of Beth’s coordinates τ = τ^i' = (t', 0, 0, 0). t' is the actual amount of time measured by Beth using her own clock, and is known as proper time. In Alf’s coordinates, GTR-1

is found by coordinate transformation. If Beth is stationary in Alf’s coordinates, τ = τ^i' = (kt', 0, 0, 0), where k is the gravitational redshift factor.

Edited on 2008-04-20 20:48:17 by CharlesFrancis

Additions:

""<table width="100%" border="0" cellspacing=0 cellpadding=0><td><img class="right" alt="GTR-6" title="The rod is now perpendicular to a line from Alf to Beth" src="images/gtr/GTR-6.gif">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'. The angle seen by Alf is θ = kθ'. Alf’s coordinate length of the rod, <table border=0><td></td></table>

Deletions:

""<table width="100%" border="0" cellspacing=0 cellpadding=0><td><img class="right" alt="GTR-6" title="The rod is now perpendicular to a line from Alf to Beth" src="images/gtr/GTR-6.gif">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'. The angle seen by Alf is θ = kθ'. Alf’s coordinate length of the rod, <table border=0><td></td></table>

Edited on 2008-04-13 03:32:22 by CharlesFrancis

Additions:
large blue square

← Concepts of General Relativity →

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

The Spacetime Manifold

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 manifolds». Spacetime is described as a 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.

Definition: A manifold is a structure in which any point has a neighbourhood which can be described by a cordinate system.

Typically a single coordinate system cannot be used to give a full description of a manifold.

Differentiability

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, tables of data and formulae containing a numerical description. 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 general relativity, spacetime is described as a differentiable manifold. This means that we assume that we may apply the mathematical operation of differentiation to functions defined on the mapping space and we claim that differentiation is physically meaningful when those functions correspond to measurable physical quantities. For example, in classical physics we regularly differentiate a velocity to find an acceleration, or a potential to find a force. It may be said that, in the strictest sense, a map which consists of tables of data, or a bank of computer memory, is not continuous and is not differentiable. In computer science derivatives are regularly approximated with difference relations. Similarly, in applications to physics, differentiability does not require taking a mathematical limit, but merely requires a step size small enough that any errors are smaller than measurement errors and taking it smaller makes no practical difference to calculation.
General relativity is a theory of classical matter, and is used to study large scale phenomena, anything from the gravity of a planet to the large scale structure of the universe. On such scales the assumption of differentiability is entirely reasonable and leads to no practical issues or disagreement between theory and observation. It will be necessary to revisit questions of continuity and differentiability in the study of quantum phenomena.

Tangent Charts

In principle many forms of coordinates can be used in 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.

Definition: A tangent chart is defined by the radar method and has a constant, non-physical, Minkowski metric, equal to to the physical metric at an observer’s origin of coordinates.

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, 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'.
Using general covariance, each observer describes physical quantities using vectors. Using Beth’s primed coodinates, denote the vector describing t' seconds for a stationary object at the origin of Beth’s coordinates τ = τ^i' = (t', 0, 0, 0). t' is the actual amount of time measured by Beth using her own clock, and is known as proper time. In Alf’s coordinates, GTR-1

is found by coordinate transformation. If Beth is stationary in Alf’s coordinates, τ = τ^i' = (kt', 0, 0, 0), where k is the gravitational redshift factor.

Definition: Proper time is the amount of time for an object, as it would be measured on a local clock moving with that object.

Definition: Proper length is the length an object as it would be measured by an observer moving with that object.

Stationary Observers

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'. The angle seen by Alf is θ = kθ'. Alf’s coordinate length of the rod,

is equal to its proper length, measured locally by Beth. ====

<a name="TheSpacetimeMetric"></a>The Spacetime Metric==== The <a href="http://www.teleconnection.info/rqg/IntroductionToVectorSpace#TheMetric»" >metric</a>, gij, lowers the indices of <a href="http://www.teleconnection.info/rqg/IntroductionToTensors#IndexGymnastics»" >contravariant</a> vectors in such a way that the inner product between vectors x and y is an <a href="http://www.teleconnection.info/rqg/IntroductionToVectorSpace#TheDotProduct»" >invariant</a>, <img alt="GTR-8" title="The dot product" src="images/gtr/GTR-8.gif" align="texttop" vspace="0"> In particular, the magnitude |x| of the vector x is invariant, <img alt="GTR-9" title="vector magnitude" src="images/gtr/GTR-9.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. 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. General relativity allows that, in principle, Beth could be anywhere in Alf’s coordinate space. For each point, x, where Beth could be, there is a different metric, gij(x). <Definition: ; The metric field is defined, on a given coordinate system, by <img alt="GTR-12g" title="The metric field is a function on spacetime coordinates" src="images/gtr/GTR-12g.gif" align="texttop" vspace="0"> 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. 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 <a href=http://www.teleconnection.info/rqg/IntroductionToVectorSpace#SphericalCoordinates>spherical» coordinates</a>, <img alt="GTR-10" title="Covariant components of the metric for mutually stationary observers" src="images/gtr/GTR-10.gif" align="texttop" vspace="0"> 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 <img alt="GTR-11" title="k in Schwarzschild coordinates" src="images/gtr/GTR-11.gif" align="texttop" vspace="0"> These forms of the metric, g, are coordinate dependent. The metric is a [[introductionToTensors tensor]] quantity, and will be used in tensor equations. Tensor equations are <a href="http://www.teleconnection.info/rqg/IntroductionToTensors#ThePrincipleOfGeneralCovariance»" >covariant</a>, so that if a tensor equation holds in one set of coordinates, then it will hold in any coordinates. In particular, a tensor equation will hold after a <a href= IntroductionToVectorSpace#CoordinateTransformation">Lorentz transform</a> of either Alf’s or Beth’s coordinates, so the choice of stationary observers places no limitation on generality. ====<a name="TheLevi-CivitaConnection"></a>The Levi-Civita Connection==== The metric field is a measure of the distortion present in a chart. It 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.teleconnection.info/rqg/GeneralRelativity#http://www.teleconnection.info/rqg/BasicsOfCurvature#ParallelDisplacement>connection</a>"»" in addition to the metric field. Given the metric field, an affine connection» describes a relationship between a set of coordinate axes at x, say, and another set, at x +dx, where dx is a small displacement, such that we can meaningfully describe a vector at x as being parallel to one at x +dx (other types of 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 parallel displacement in tangent space, and projecting back into the curved surface. This is the 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 parallel transport, 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.
Einstein was not satisfied that the Levi-Civita connection is physically correct. He observed an inconsistency between general relativity and classical electromagnetism. Such an inconsistency can be seen in the mismatch of Alf’s and Beth’s coordinates. Light emitted from the end of the short rod and seen by Beth does not behave in the same way as light emitted from the same point, in the same direction, and seen by Alf — if it did, Alf’s coordinate length would be altered by the same factor as his coordinate time rather than by the inverse factor, and space would have to be flat. The inconsistency lead Einstein to investigate a different form of parallelism which he called distant parallelism, also known as teleparallelism». The possibility of such a connection had previously been pointed out by Cartan», known as the best geometer in Europe, and who was the first to recognise the distinct roles of the connection and the metric in the description of curvature.
Einstein’s attempts at unification of general relativity and classical electromagnetism using teleparallelism failed. The teleconnection in relational quantum gravity seeks to unify general relativity with quantum electrodynamics and uses a different form of distant parallelism, one rooted in a philosophical issue also raised by Einstein. The hole argument» gave him considerable trouble in his original formulation of general relativity. In general relativity, the metric is defined at all points in space. Empirically, spacetime coordinates can only be defined in the presence of matter. It is physically meaningless to define a metric when there is nothing to be measured, as in the vacuum of space. The teleconnection will describe a relationship between coordinate axes, at remote points, x and y, such that we can meaningfully translate momentum from x to y, and such that the Levi-Civita connection is restored in the limit of small displacement between x and y.
Concepts of General Relativity ↑ Riemann Curvature →

Deletions:
large blue square

← Large Scale Structure of the Universe →

“Einstein’s biggest blunder”, the cosmological constant, is introduced. Weyl’s postulate is described, which treats the motions of galaxies as a “cosmic fluid” and allows us to talk of “cosmic time” and the large scale structure of the universe. Spaces of constant curvature are treated and the meaning of cosmological expansion is described. The cosmological principle, which essentially states that the universe is everywhere the same at any cosmic time, is used to derive Friedmann’s equation for the expansion of the universe. The equation is solved and the Friedmann models are described.

The Cosmological Constant

“Much later, when I was discussing cosmological problems with Einstein, he remarked that the introduction of the cosmological term was the biggest blunder of his life” — George Gamow, My World Line (1970).
“It is just as well that Einstein made this remark to Gamow, otherwise Gamow would have been severely tempted to make it up.” — J.P. Leahy.

A universe governed by a universal attractive force cannot remain stable. Matter in it must either fly apart until stopped by the attractive force whereupon it must fall back in on itself, or it must fly apart indefinitely. This is as true of a Newtonian Cosmology as it is of a relativistic one. At the time general relativity was produced, the observable stars showed no evidence of either expansion or collapse. It was widely thought that the universe should endure from everlasting to everlasting. In order to make this possible Einstein made a modification to the field equation, by including a repulsive force to balance gravitational attraction. The only simple modification preserving tensor properties is to add a term proportional to the metric,

The new term must have a value of Λ sufficiently small not to modify gravitational effects observed in the solar system, but plays a role on cosmological scales. Unfortunately it does not lead, as Einstein had hoped, to a static universe, but to an unstable solution, in which the slightest local variation leads to the start of expansion.
In the 1920’s Edwin Hubble» made observations on distant galaxies showing that expansion was a fact, formulated as Hubble’s law» in 1929. Einstein removed the cosmological term, but in the 1990’s analysis of distant supernova showed a redshift-magnitude relation which cannot be explained by standard general relativity without a cosmological constant. Most cosmologists now accept the reality of the Cosmological constant.
Nonetheless, no theoretical reason for the existence of the cosmological constant has ever been presented. We should be suspicious of a physical law without a physical cause. Fudge factors to make equations fit data are often indicative of a deeper underlying fault in theory. The existence of such a fault in general relativity is known. As seen in the mismatch of Alf and Beth’s Spacetime diagrams, and Einstein himself pointed out, the affine connection is not consistent with classical electromagnetism. It seems unreasonable that one would be try to modify the general principle of relativity. We might suspect, as did Einstein, that the affine connection is not correct, and that a different redshift law might result from changing it. The teleconnection reconciles general relativity with quantum theory and with quantum electrodynamics, and gives a redshift law consistent with Supernova Redshifts without the requirement of a cosmological constant.

Weyl’s Postulate

The general principle of relativity enables us to define time from clock processes locally, and to claim that a similar definition of time is always possible anywhere in the universe, with the exception of possible singularities. We require a further assumption in order to discuss universal time, or concepts like the age of the universe. This issue was addressed by Herman Weyl» in 1923. Weyl argued that in order to discuss the distant we should base our ideas, in so far as is possible, on what we can observe in our own neighbourhood.

An observer can use the radar method to define synchronous surfaces in his neighbourhood with respect to his own proper time.

<table width=100% cellspacing=0 cellpadding=0><td><img class="right" alt="Cosmology-3" title="a synchronous slice in the Earth frame" src="images/cosmology/Cosmology-3N.gif">From the concept of proper time, or time as measured by an observer moving with a clock, we may define Earth time, i.e. proper time for the Earth. Time measured by atomic clocks on satellites is adjusted several times daily to remain synchronised with Earth time.</td></table> <table width=100% cellspacing=0 cellpadding=0><td><img class="right" alt="Cosmology-4" title="a synchronous slice in the solar system" src="images/cosmology/Cosmology-4N.gif">Although we cannot use radar on the Sun, It remains meaningful to define a synchronous time for the solar system, using adjustments for gravitational and Doppler shifts as required.</td></table> <table width=100% cellspacing=0 cellpadding=0><td><img class="right" alt="Cosmology-5" title="a synchronous slice in the Milky Way" src="images/cosmology/Cosmology-5N.gif">Similarly, given the motions of, and distances to, galactic stars, it is meaningful to discuss proper time for the galaxy, or for the local group.</td></table> <table width=100% cellspacing=0 cellpadding=0><td><img class="right" alt="Cosmology-6" title="a synchronous slice in the cosmic fluid" src="images/cosmology/Cosmology-6N.gif">Weyl’s postulate assumes that the overall motion of many galaxies may be likened to a “cosmic fluid”, with each galaxy moving on a geodesic from a point in the finite or infinite past, in such a way that synchronous slices of proper time for each galaxy can be combined to form synchronous slices in the cosmic fluid. These synchronous slices are labelled by cosmic time. Although cosmic time is defined globally, it is not a fundamental global property in the sense of Newtonian absolute time, but rather consists of the synchronisation of proper time for particles in the cosmic fluid.</td></table> <Definition: ; Cosmic time is determined from synchronous slices of proper time defined in the cosmic fluid. ====<a name="TheCosmicFluid"></a>The Cosmic Fluid==== A perfect fluid is characterised by three quantities, defined at each point in the fluid: 4-velocity, ua, and two scalars, density, ρ, and pressure, p. The most straightforward assumption for the <a href=http://www.teleconnection.info/rqg/Gravitation#Einstein’sLawOfGravitation>stress-energy» tensor</a> is that it is the sum of a part due to inertial motions, and a part due to pressure, <img alt="Gravitation-7" title="Stress-Energy for a perfect fluid" src="images/cosmology/Cosmology-7.gif" align="texttop" vspace="3"> for some symmetric tensor Sab. The only rank 2 tensors associated with the fluid are uaub and the metric, gab. The simplest assumption is that, for some constants λ and μ, <img alt="Gravitation-8" title="Part of stress-energy due to pressure" src="images/cosmology/Cosmology-8.gif" align="texttop" vspace="3"> The <a href=http://www.teleconnection.info/rqg/Gravitation#TheStress-EnergyTensor>law» of local energy-momentum conservation says that the stress-energy tensor</a> should satisfy <img alt="Gravitation-9" title="Law of energy-momentum conservation" src="images/cosmology/Cosmology-9.gif" align="texttop" vspace="3"> We require that it reduces to the [[http://en.wikipedia.org/wiki/Continuity_equation continuity equation]] and to the [[http://en.wikipedia.org/wiki/Navier-Stokes_equations#Derivation_and_description Navier-Stokes equations]] in the appropriate limit. From this we find λ =1 and μ = −1. Then the stress-energy tensor is <img alt="Gravitation-10" title="Stress-energy for a perfect fluid" src="images/cosmology/Cosmology-10.gif" align="texttop" vspace="3"> ====<a name="TheCosmologicalPrinciple"></a>The Cosmological Principle==== The [[http://en.wikipedia.org/wiki/Cosmological_principle cosmological principle]] states that in each era as defined by Weyl’s postulate, discounting local variations, the general behaviour of matter is similar in every part of the universe. More precisely, the cosmological principle incorporates assumptions of [[http://en.wikipedia.org/wiki/Homogeneity_%28physics%29 homogeneity]] and [[http://en.wikipedia.org/wiki/Isotropic isotropy]]. <Homogeneity states that the distribution of matter is even in each epoch. << <Isotropy states that there are no prefered directions in the distribution of matter in space. <a href="http://www.teleconnection.info/gfx/2dFzcone.gif"><img» class="right" alt="Cosmology-11b" title="Galaxies up to redshift ~ 0.2. Image courtesy of the 2dFGRS Team" src="images/cosmology/Cosmology-11b.gif"></a>The cosmological principle is not open to direct empirical test, since we can only observe a small part of the universe. Cosmological models have been developed which do not obey it. However, no effects contravening it have been observed and it gains some support from the degree of homogeneity and isotropy in the microwave background. Clearly the universe is not homogeneous on the scale of the Solar system, or of the Milky Way, and nor is it homogeneous on the scale of the local cluster or supercluster of galaxies. On much larger scales the distribution of matter does appear uniform. The scale on which the matter distribution becomes uniform may be seen in the <a href=http://teleconnection.info/rqg/images/cosmology/2df_rotslice.mpg>rotating» slice movie</a>, courtesy of the [[http://www.roe.ac.uk/~jap/2df 2dFGRS]] team. On the scale of the slice, galaxies appear as specks of dust, bearing in mind that the region plotted, up to redshifts, z ≈ 0.2, is small compared to the universe as a whole. ====<a name="SpacesOfConstantCurvature"></a>Spaces of Constant Curvature====<table width=100%> <td align=center><img alt="Cosmology1-12a" title="In a positive curvature universe, wrapping a plane in tangent space onto a sphere removes scaling distortions" src="images/cosmology/Cosmology1-12a.gif"></td></table> <table width=100%> <td align=center><img alt="Cosmology1-13a" title="In a positive curvature universe, wrapping a plane in tangent space onto a sphere removes scaling distortions" src="images/cosmology/Cosmology1-13aN.gif"></td></table>It follows from homogeneity and isotropy, together with <a href=http://www.teleconnection.info/rqg/Gravitation#Einstein’sLawOfGravitation>Einstein’s» field equation</a> that, on the large scale, the universe has constant curvature. This means that any plane in tangent space can be mapped onto a space of constant curvature in such a way that scaling distortions are removed. If the universe has positive curvature in the space coordinates, then any synchronous plane in tangent space can be mapped onto a sphere. This does not mean that space is really wrapped around a sphere, but that the metric field, is an identical function on tangent space to that of a spherical geometry. The point of contact, or “North pole” is completely arbitrary; tangent space can be drawn at any point and the map to a sphere is essentially unchanged. <table width=100%><td></td></table> <img class="right" alt="Cosmology-13" title="Polar coordinates in two space dimensions" src="images/cosmology/Cosmology-13N.gif">Using polar coordinates (r, φ) to describe any plane, with an origin at the observer, where <r> is determined from lightspeed, the metric locally is <img alt="Gravitation-14" title="metric in flat two dimensional space, using polar coordinates" src="images/cosmology/Cosmology-14.gif" align="texttop" vspace="3"> This simply states that the distance associated with a small coordinate change dr is dr, and the distance associated with a coordinate change dφ; is rdφ. A flat space in two dimensions has this metric globally, multiplied by a constant scale factor. <img class="right" alt="Cosmology-15" title="Polar coordinates on the surface of a sphere" src="images/cosmology/Cosmology-15N.gif">The extension of this metric on a space of uniform positive curvature is identical to the metric of an imagined sphere. Radial coordinate distance is mapped to an arc of angle ρ on a sphere of radius a, so that r = aρ. A small arc of angle dρ corresponds to radial distance adρ, and a coordinate change dφ corresponds to distance asinrdφ. Thus, in ρ-φ coordinates, the metric is <img alt="Gravitation-16" title="metric in two dimensional space of constant positive curvature, using polar coordinates" src="images/cosmology/Cosmology-16.gif" align="texttop" vspace="3"> a is called the //scale factor//, and gives a measure of the scale on which curvature is apparent. In some old accounts a was called the radius of the universe, but this terminology becomes meaningless for spaces of zero or negative curvature. Uniform negative curvature brings in √−1. Instead of sinρ we have sinhρ = isin(iρ). <img alt="Gravitation-17" title="metric in two dimensional space of constant negative curvature, using polar coordinates" src="images/cosmology/Cosmology-17.gif" align="texttop" vspace="3"> This can be checked by an explicit calculation of the <a href=http://www.teleconnection.info/rqg/GTRTensors#TheRiemannCurvatureTensor>Riemann» tensor</a>. Let f(ρ) = sinρ, ρ, or sinhρ for positive, zero and negative curvature, respectively. Then the three possibilities can be written in one, <img alt="Gravitation-19" title="metric in two dimensional space of constant positive, negative, or zero, curvature, using polar coordinates" src="images/cosmology/Cosmology-19.gif" align="texttop" vspace="3"> Now consider three dimensions, using <a href=http://www.teleconnection.info/rqg/IntroductionToVectorSpace#SphericalCoordinates>spherical» coordinates</a> (ρ, θ, φ). At a zenith angle of θ = 90°, small changes, dρ, dθ and dφ, in ρ, θ and φ lead to distance changes, dρ, af(ρ)dθ and af(ρ)dφ. So, the metric at θ = 90° is, <img alt="Gravitation-20" title="metric at θ =&90° in a 3-dimensional space of constant positive, negative, or zero, curvature, using spherical coordinates" src="images/cosmology/Cosmology-20.gif" align="texttop" vspace="3"> <img class="right" alt="Cosmology-21" title="Spherical coordinates for a space of constant curvature" src="images/cosmology/Cosmology-21N.gif">Rotate the plane through angle 90 − θ° about φ = 0 (blue to red; the position of φ = 0 is arbitrary, so there is no loss of generality). After rotation, there is no change to distances in the ρ or θ directions, but the distance associated with a change dφ is reduced by a factor sinθ. The metric is <img alt="Gravitation-22" title="metric in a 3-dimensional space of constant positive, negative, or zero, curvature, using spherical coordinates" src="images/cosmology/Cosmology-22.gif" align="texttop" vspace="3"> This is usually written in the form, <img alt="Gravitation-23" title="line element in a 3-dimensional space of constant positive, negative, or zero, curvature" src="images/cosmology/Cosmology-23.gif" align="texttop" vspace="3"> where ds is the proper distance due to small changes in coordinate position, (dρ, dθ, dφ). This form of the metric is known as the [[http://en.wikipedia.org/wiki/Line_element line element]], since it gives the length of a small displacement in any direction. ====<a name="CosmologicalExpansion"></a>Cosmological Expansion==== The introduction of a time coordinate, t, allows a to vary in time; a : t → a(t). With a (+, −, −, −) signature, the metric is <img alt="Gravitation-24" title="spacetime metric, with 3-space of constant positive, negative, or zero, curvature" src="images/cosmology/Cosmology-24.gif" align="texttop" vspace="3"> After the substitution, adτ = dt, the metric in τ-ρ coordinates is <img alt="Gravitation-25" title="spacetime metric in τ-ρ coordinates, with 3-space of constant positive, negative, or zero, curvature" src="images/cosmology/Cosmology-25.gif" align="texttop" vspace="3"> In these coordinates the radial speed of light is 1. Homogeneity and isotropy require that the particles of the cosmic fluid (galaxies) remain on lines of constant ρ (a change in ρ would define a prefered direction). A [[http://en.wikipedia.org/wiki/Penrose_diagram Penrose diagram]] plots the Universe in τ-ρ coordinates. <table width=100%> <td align=center><img alt="Cosmology1-26" title="A positive curvature universe in &tau-ρ coordinates" src="images/cosmology/Cosmology1-26N.gif"></td></table> In the Penrose diagram we can see the meaning of the scale factor more clearly. At each point the diagram scales by the factor, a(t), relative to locally Minkowski coordinates. Cosmological expansion is seen as the “shrinking” of matter in inverse proportion to a(τ), which, in turn, is due to the changing rate of cosmic time with respect to τ. In conventional distance units, r = a(t)ρ. The metric in t-r coordinates is <img alt="Cosmology-27" title="spacetime metric in t-r coordinates, with 3-space of constant positive, negative, or zero, curvature" src="images/cosmology/Cosmology-27.gif" align="texttop" vspace="3"> which reduces to Minkowski metric for <img alt="Cosmology-186" title="r much less than a" src="images/cosmology/Cosmology-186.gif" align="texttop" vspace="1"> <img alt="Cosmology-187" title="Minkowski metric for a small regions of spacetime" src="images/cosmology/Cosmology-187.gif" align="texttop" vspace="3">. Given the scale factor as a function of time, the universe may be plotted relative to locally defined distances (i.e. such that the size of galaxies does not change). <table width=100%> <td align=center><img alt="Cosmology1-28" title="A positive curvature universe in t-r coordinates" src="images/cosmology/Cosmology1-28N.gif"></td></table> In a positive curvature universe, each plane with the observer at the origin is the geometrical equivalent of a sphere of radius a(t). If a is increasing with t, the universe expands relative to locally defined coordinates based on cosmic time. Note that the central galaxy in the figure is quite arbitrary. Any galaxy could be chosen as central, and the overall picture would be unchanged. Expansion takes place for the universe as a whole, relative to local distances in which the size of galaxies remains fixed. It does not make sense to talk of expansion locally, because locally we can only talk about scale relative to local reference matter. Empty space cannot be measured. We can talk of the expansion of the universe, but the expansion of space is an empirically meaningless concept. It does not make sense to talk of expansion locally. Within general relativity, one coordinate system is as good as another. The Penrose diagram and the expanding universe diagram have equal status as descriptions of reality. It is as valid to say that the universe expands relative to the size of galaxies as it is to say that local distances shrink with respect to the universe. However, the Penrose diagram has the important feature of retaining the constancy of the speed of light, according to which local coordinate systems are defined in practice, and shows the paths of matter as straight lines (ignoring local gravity), respecting Newton’s first law. It will gain an increased significance when the teleconnection is defined in relational quantum gravity. We may be encouraged to think that the Penrose diagram, with its picture of a universe of constant size is in some sense the more fundamental view of the universe. In this view curvature and universal attraction are understood as effects of the changing speed of clocks with the scale factor. ====<a name="CosmologicalRedshift"></a>Cosmological Redshift==== Inertial radial motions are locally constant with respect to the inertial motions of galaxies, hence they are constant everywhere and are straight lines on a Penrose diagram. Because radial paths of inertial objects and of light are straight lines, parallel transport is given by translation. A vector representing the momentum of a signal from a distant galaxy to the origin is shown in red on the diagram. A local displacement vector, showing the size of a galaxy, is shown in cyan. The relative length of these vectors is changes proportional to the scale factor, a(t). Thus a signal from a galaxy sent at cosmic time t, and detected on Earth at t0 (now) is subject to a cosmological redshift proportional to expansion, <img alt="Gravitation-29" title="redshift is proportional to expansion" src="images/cosmology/Cosmology-29.gif" align="texttop" vspace="3"> At time t, the distance, r(t), to a galaxy at current distance coordinate r0 = r(t0) <img alt="Gravitation-30" title="Hubble’s law" src="images/cosmology/Cosmology-30.gif" align="texttop" vspace="3"> <Definition: ; Hubble’s constant is <img alt="Gravitation-31g" title="Hubble’s constant" src="images/cosmology/Cosmology-31g.gif" align="middle"> << <Definition: ; The current value of Hubble’s constant is H0 = H(t0). [[http://en.wikipedia.org/wiki/Hubble's_law Hubble’s constant]] is not a constant of nature, but varies over cosmological timescales. It may be taken as constant in our era. We have <img alt="Gravitation-32" title="expansion for nearby galaxies" src="images/cosmology/Cosmology-32.gif" align="texttop" vspace="3"> For nearby galaxies, the distance coordinate may be identified with distance (this is not a useful, or even very meaningful, measure of distance for a galaxy far enough away that the universe expands appreciably in the time taken for light to travel from that galaxy). Differentiating gives Hubble’s law, for the speed, v, of recession of nearby galaxies, <img alt="Gravitation-33" title="Hubble’s law" src="images/cosmology/Cosmology-33.gif" align="texttop" vspace="3"> << Hubble’s law states that the speed of recession of nearby galaxies (beyond the <a href=http://en.wikipedia.org/wiki/Virgo_Supercluster>local» supercluster</a>) is proportional to their distance.<< ====<a name="FriedmannEquation"></a>The Friedmann Equation==== In 1922 [[http://en.wikipedia.org/wiki/Alexander_Friedmann Alexander Friedmann]] solved Einstein’s field equation for a homogeneous isotropic cosmology ([[Friedmann calculation]]). <The Friedmann equation: <img alt="Gravitation-34g" title="The Friedmann Equation" src="images/cosmology/Cosmology-34g.gif" align="texttop" vspace="3"> where k = 1 for a space of positive curvature, k = −1 for a space of negative curvature, and k = 0 for flat space. << Density, ρ includes both matter density and energy density, including radiation. Observation shows that the universe is matter dominated, and has been since it was about 1⁄10 000 of its current size, 350 000 or so years after the big bang. In this case continuity of mass gives ρa3 is constant. <img alt="Gravitation-35" title="density" src="images/cosmology/Cosmology-35.gif" align="texttop" vspace="3"> It is convenient to define the cosmological parameters <img alt="Gravitation-36" title="cosmological parameter" src="images/cosmology/Cosmology-36.gif" align="texttop" vspace="3"> <img alt="Gravitation-37" title="cosmological parameter" src="images/cosmology/Cosmology-37.gif" align="texttop" vspace="3"> <img alt="Gravitation-38" title="cosmological parameter" src="images/cosmology/Cosmology-38.gif" align="texttop" vspace="3"> Then Friedmann’s equation is <img alt="Gravitation-39" title="The Friedmann equation" src="images/cosmology/Cosmology-39.gif" align="texttop" vspace="3"> Setting t = t0 gives the identity <img alt="Gravitation-40" title="Identity for cosmological parameters" src="images/cosmology/Cosmology-40.gif" align="texttop" vspace="3"> Thus, for no-Λ models, Ω = 1 is critical density, above which the universe has positive curvature and is finite, and below which it has negative curvature and is infinite. Experiments in observational cosmology are much concerned with determining the values of the cosmological parameters, H0, Ω, Ωk and ΩΛ. ====<a name="FriedmannModels"></a>Friedmann Models==== Cosmological models in which expansion is governed by the Friedmann equation are called [[//http://en.wikipedia.org/wiki/Friedmann-Lema%C3%AEtre-Robertson-Walker_metric Friedmann models]], or FRW models in honour of Robertson and Walker who worked on the problem some ten years after Friedmann. The Friedmann equation may be solved for different values of the cosmological parameters. (<a href=http://www.teleconnection.info/rqg/Friedmann#Solutions>Solution</a>"")». Near the big bang, all models are approximated by the Einstein-de Sitter, flat space, no-Λ model.

Using parallel transport of photon momentum under the Levi-Civita connection, there is strong evidence from supernova data and WMAP» for the “concordance model” (magenta), a flat space model with Ω ≈ 0.27 and Ω_Λ ≈ 0.73. Using the teleconnection, supernova data favours an “Einstein preferred” model (cyan) with positive curvature, Ω_Λ ≈ 0 and Ω ≈ 2.

Timeline for Processes from the Big Bang

High energy processes in elementary particle physics can be investigated empirically on earth using particle accelerators». Based on what is found we can calculate the processes taking place in the high energy densities near the Big Bang. Big bang models based on general relativity get a great deal of empirical support, not only from the the observation of the cosmic microwave background» (CMB), but also from the observed ratio of hydrogen to helium and other light elements. This is a prediction of Big Bang nucleosynthesis» (BBN), and is critically dependent on the age of the universe, which determines the rate of expansion near the big bang and the length time during which free neutrons were able to decay to protons.

Large Scale Structure of the Universe ↑ Relativistic Quantum Theory →

Edited on 2008-04-10 01:57:21 by CharlesFrancis

Additions:
large blue square

← Large Scale Structure of the Universe →

The Cosmological Constant

Weyl’s Postulate

An observer can use the radar method to define synchronous surfaces in his neighbourhood with respect to his own proper time.

RQG : GeneralRelativity

← Concepts of General Relativity ↑ →

← Concepts of General Relativity →

← Concepts of General Relativity →

The Spacetime Manifold

Differentiability

Tangent Charts

Coordinate Time and Proper Time

Stationary Observers

← Large Scale Structure of the Universe →

The Cosmological Constant

Weyl’s Postulate

Timeline for Processes from the Big Bang

← Large Scale Structure of the Universe →

The Cosmological Constant

Weyl’s Postulate