Spacetime symmetries

Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact solutions of Einstein's field equations of general relativity. Spacetime symmetries are distinguished from internal symmetries.

Physical motivation

Physical problems are often investigated and solved by noticing features which have some form of symmetry. For example, in the Schwarzschild solution, the role of spherical symmetry is important in deriving the Schwarzschild solution and deducing the physical consequences of this symmetry (such as the nonexistence of gravitational radiation in a spherically pulsating star). In cosmological problems, symmetry plays a role in the cosmological principle, which restricts the type of universes that are consistent with large-scale observations (e.g. the Friedmann–Lemaître–Robertson–Walker (FLRW) metric). Symmetries usually require some form of preserving property, the most important of which in general relativity include the following:

preserving geodesics of the spacetime

preserving the metric tensor

preserving the curvature tensor

These and other symmetries will be discussed below in more detail. This preservation property which symmetries usually possess (alluded to above) can be used to motivate a useful definition of these symmetries themselves.