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  The General Theory of 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 Principle of General Covariance

General relativity is based on the principle of uniformity in nature,

The principle of uniformity in nature: The fundamental behaviour of matter is always and everywhere the same

together with the empirical principle that we can only carry out measurements by comparing matter (& energy) relative to other matter (& energy), encapsulated in the general principle of relativity,

The general principle of relativity:  Local laws of physics are the same irrespective of the reference matter which a particular observer uses to quantify them.

and the empirical fact that we do not have a universe obeying Newtonian relativity.

The principle of general covariance is the mathematical implementation of the general principle of relativity. In non-mathematical language it says “local laws of physics are the same irrespective of the coordinate system which a particular observer uses to quantify them”. Vectors are not invariant, as their coordinate representation changes with the coordinate system. Relationships between vectors are unchanged by coordinate transformation. Such relationships are said to be covariant. Tensors are built from vectors. Relationships between tensors are also covariant. The general principle of relativity is then encapsulated in the principle of general covariance,

The principle of general covariance:  The equations describing local laws of physics have tensorial form.

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 coordinate system or chart.

Typically a single coordinate system cannot be used to give a full description of a manifold. A collection of charts which describes the whole manifold is an atlas.

What is Spacetime?

We can describe spacetime as a manifold, a geometrical structure which can be mapped onto an atlas, or collection of charts. In common with many definitions of mathematical structures, this does not tell us what the manifold actually is. Instead it tells us what properties a manifold has, how a manifold behaves.

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:

Answer 1. There exists a substantive spacetime, which is modelled (i.e. described) by the mathematical structure of a manifold.

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:

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

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

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

Answer 3. I observe that I can, in principle, choose reference matter anywhere I wish, and that I can define Minkowski coordinates relative to that matter. I now define spacetime by imagining all the conceivable ways in which coordinate systems can be set up in principle, dependent upon physical measurement.

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.

Position 3 avoids both the unjustified metaphysical assumption of answer 1 and the feeble abdication of the purpose of science inherent in answer 2. Instead it correctly bases science on observation. The proper use of mathematics will enable predictions to be rigorously deduced, not inferred by induction. Modern science can thus escape the problem that a premise cannot be deduced from a conclusion.


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 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.

Definition:  A tangent chart is defined to use coordinates as defined by an inertial observer using the radar method.

A tangent chart has a constant, non-physical, Minkowski metric, equal to to the physical metric at the observer’s origin of coordinates. Other coordinate choices are possible, but they can always be transformed into coordinates with Minkowski metric at the position of the observer.

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 = τi' = (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 = τi' = (kt', 0, 0, 0), where k is the gravitational redshift factor, k = 1 + z.

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.

The Spacetime Metric

The metric, gij, lowers the indices of contravariant vectors in such a way that the inner product between vectors x and y is an invariant. Using Minkowski spacetime, as described in special relativity the metric is
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. Since the difference between Alf’s and Beth’s measurements is just that Alf and Beth define coordinates differently, their measurements are related by coordinate transformation. The definition of the metric ensures that, when Alf applies it to his own measurements, the magnitudes returned will be the proper times and distances of quantities local to Beth. The metric determines that the quantity |x|2 = gijxiyj is the same in any coordinates. 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 spacetime metric is defined, on a given coordinate system, by

The spacetime metric is often simply called the metric. One should avoid this abuse of language, because the spacetime metric is a tensor field, that is to say it is a function having a different metric value at each point in spacetime. Calling it simply “the metric” confuses a metric field, which is a function of coordinate space, with the metric at a given position.

Schwarzschild Coordinates

GTR-5In a static geometry, an observer, Alf, defines spherical coordinates by the radar method, using time t, determined from a clock at an origin at A. Coordinate distance, r*, from A is defined by setting the radial speed of light to unity. Spherical coordinates are orthogonal so that the spacetime metric is diagonal. 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 spacetime metric has g00 = k2.

GTR-6Beth determines proper distances local to B using the radar method, with lightspeed equal to unity. Since Beth’s clock runs faster than Alf’s, proper distances local to Beth are greater than corresponding coordinate distances in Alf’s coordinates by a factor k. Using unit light speed Beth calculates coordinate distance r = kr* to Alf.
Using r as the radial coordinate, Beth increases the scale on a map of the (space) neighbourhood of B, so that the angle subtended at A by a small rod of proper length l at B, perpendicular to AB is l / r, as it would be in flat space. This is the defining condition for Schwarzschild coordinates.

Definition:  The Schwarzschild radial coordinate is given by the proper arc length of a small arc on a sphere at the origin divided by the angle subtended by that arc. :

GTR-7In Schwarzschild coordinates a ring of short rods at radial distance r from A can be drawn on Beth’s map to form a continuous circle, centre A, without overlaps. It follows that, in spherical coordinates with origin A and radial distance r, g22 = −r2 and g33 = −r2sin2θ, and that, since Beth has increased the scale of local distances by a factor k, g11 = −k−2.

Theorem:  Schwarzschild coordinates in vacuum have spacetime metric given by:
where k and k are functions of positon, x, and k(x) is the redshift of light from the origin to x.

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

The Levi-Civita Connection

A metric field is a measure of the distortion present in a chart. When the coordinate axes are perpendicular at each point, the coordinates are orthogonal. In this case the metric is diagonal (as seen in Schwarzschild coordinates) and the metric components are just the squares of the scale factors in each direction. More generally the metric will also have off-diagonal elements.

A metric field is not sufficient to describe curvature — we have seen examples of distorted spaces, like the 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.

The General Theory of Relativity ↑Riemann Curvature →
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