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

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. A collection of charts which describes the whole manifold is an atlas.

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:

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

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:

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

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

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.

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.

Using general covariance, each observer describes physical quantities using vectors. Using Beth’s primed coordinates, denote the vector describing

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,

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.

An observer, Alf, defines spherical coordinates by the radar method, using time t, determined from a clock at an origin at A. Coordinate distance from A is the so called “tortoise” coordinate, r*, 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 g_{00} = k^{2}. |

Beth 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. |

In 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, g_{22} = −r^{2} and g_{33} = −r^{2}sin^{2}θ, and that, since Beth has increased the scale of local distances by a factor k, g_{11} = −k^{−2}. |

This result is usually found by solving 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. Using Cartesian space coordinates, Using Cartesian space coordinates,

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

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.

Concepts of General Relativity ↑ Riemann Curvature →

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