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.

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 non-existence of gravitational radiation in a spherically pulsating star). In cosmological problems, symmetry finds a role to play 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:

These and other symmetries will be discussed below in more detail. This preservation feature can be used to motivate a useful definition of symmetries.

A rigorous definition of symmetries in general relativity has been given by Hall (2004). In this approach, the idea is to use (smooth) vector fields whose local flow diffeomorphisms preserve some property of the spacetime. (Note that one should emphasize in one's thinking this is a diffeomorphism--a transformation on a differential element. The implication is that the behavior of objects with extent may not be as manifestly symmetric.) This preserving property of the diffeomorphisms is made precise as follows. A smooth vector field X on a spacetime M is said to preserve a smooth tensor T on M (or T is invariant under X) if, for each smooth local flow diffeomorphism ϕ_t associated with X, the tensors T and ϕ_t*(T) are equal on the domain of ϕ_t. This statement is equivalent to the more usable condition that the Lie derivative of the tensor under the vector field vanishes:

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