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Just as the x, y, z coordinates of a point backup now on the axes one is using, so distances and time intervals, invariant in Newtonian physics, may depend on the reference frame of an observer, in relativistic physics. See length contraction and time dilation. This is the central lesson of special relativity.
The central lesson of general relativity is that spacetime cannot be a fixed background, but is rather a network of evolving relationships or "curved" spacetimes under shared acceleration forces.
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 timeProper time is time measured when the clock is at rest relative to the observer. The distinction between proper time and the measured time is necessary because of the effects of time dilation, which is outlined in Einstein's theory of special relativity. measured by an observer travelling between them. Spacetime together with this pseudo-metric makes up a pseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, (0,2) tensor which is nondegenerate at each point on the manifold. This tensor is called a pseudo-Riemannian metric or, simply, a pseudo-) metri.
One of the simplest interesting examples of a spacetime is R4 with the spacetime interval defined above. This is known as Minkowski spaceIn physics and mathematics, Minkowski space (or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a si, and is the usual geometric setting for special relativity. In contrast, General Relativity says that the underlying manifold will not be flat, if gravityThis article covers the physics of gravitation. See also gravity (disambiguation). Gravitation is the tendency of masses to move toward each other. The first mathematical formulation of the theory of gravitation was made by Sir Isaac Newton and proved ast 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.
A compact manifold can be turned into a spacetime if and only if its Euler characteristic is 0.
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.
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