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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.


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, 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.

Tangent Charts

In principle many forms of coordinates can be used for mapping spacetime, but it is useful to use charts which make the description as simple as possible. If we can find a simple description using tensor equations in a particular set of coordinates, 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.

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 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, 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.

The Spacetime Metric

The metric, g_ij, lowers the indices of contravariant vectors in such a way that the inner product between vectors x and y is an invariant,
In particular, the magnitude |x| of the vector x is invariant,
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, g_ij(x).


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.

Stationary Observers

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'. 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.

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,

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.

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 k_BC, then the redshift from A to C is k_AC = k_ABk_BC, and the metric at C obeys

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

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 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).

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.

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