In special relativity and general relativity, time and three-dimensionalspace are treated together as a single four-dimensional manifold called spacetime (alternatively, space-time). A point in spacetime may be referred to as an event. Each event has four coordinates (t, x, y, z); or, in angular coordinates, t, r, θ, and φ.
The central lesson of general relativity is that spacetime cannot be a fixed background, but is rather a network of evolving relationships.
A spacetime interval between two events is the frame-invariant quantity analogous to distance in Euclidean space. The spacetime interval s along a curve is defined by:
where c is the speed of light (see sign convention). A basic assumption of relativity is that coordinate transformations have to leave intervals invariant. Intervals are invariant under Lorentz transformations.
The spacetime intervals on a manifold define a pseudo-metric called the Lorentz metric. This metric is very similar to distance in Euclidean space. However, note that whereas distances are always positive, intervals may be positive, zero, or negative. Events with a spacetime interval of zero are separated by the propagation of a light signal. Events with a positive spacetime interval are in each other's future or past, and the value of the interval defines the proper time measured by an observer travelling between them. Spacetime together with this pseudo-metric makes up a pseudo-Riemannian manifold.
One of the simplest interesting examples of a spacetime is R4 with the spacetime interval defined above. This is known as Minkowski space, and is the usual geometric setting for special relativity. In contrast, General Relativity says that the underlying manifold will not be flat, if gravity is present, and thus it calls for the use of spacetime rather than Minkowski space.
Strictly speaking one can also consider events in Newtonian physics as a single spacetime. This is Galilean-Newtonian relativity, and the coordinate systems are related by Galilean transformations. However, since these preserve spatial and temporal distances independently, such a spacetime can be decomposed unarbitrarily, which is not possible in the general case.
Any non-compact 4-manifold can be turned into a spacetime.
Many spacetimes have physical interpretations which most physicists would consider bizarre or unsettling. For example, a compact spacetime has closed timelike curves, which violate our usual ideas of causality. For this reason, mathematical physicists usually consider only restricted subsets of all the possible spacetimes. One way to do this is to study "realistic" solutions of the equations of General Relativity. Another way is add some additional "physically reasonable" but still fairly general geometric restrictions, and try to prove interesting things about the resulting spacetimes. The latter approach has led to some important results, most notably the Penrose-Hawking singularity theorems.
In mathematical physics it is also usual to restrict the manifold to be connected and Hausdorff. A Hausdorff spacetime is always paracompact.
