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 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).
In this approach, the idea is to use (smooth) vector fields whose local flow diffeomorphisms preserve some property of the spacetime.
This statement is equivalent to the more usable condition that the Lie derivative of the tensor under the vector field vanishes:
Killing vector fields find extensive applications (including in classical mechanics) and are related to conservation laws.
Homothetic vector fields find application in the study of singularities in general relativity.
The above three vector field types are special cases of projective vector fields which preserve geodesics without necessarily preserving the affine parameter.
where ϕ is a smooth real-valued function on M. A curvature collineation is a vector field which preserves the Riemann tensor:
A less well-known form of symmetry concerns vector fields that preserve the energy–momentum tensor.
When the energy–momentum tensor represents a perfect fluid, every Killing vector field preserves the energy density, pressure and the fluid flow vector field.
As mentioned at the start of this article, the main application of these symmetries occur in general relativity, where solutions of Einstein's equations may be classified by imposing some certain symmetries on the spacetime.
Classifying solutions of the EFE constitutes a large part of general relativity research.
Various approaches to classifying spacetimes, including using the Segre classification of the energy–momentum tensor or the Petrov classification of the Weyl tensor have been studied extensively by many researchers, most notably Stephani et al. (2003).
For example, the Schwarzschild solution has a Killing algebra of dimension four (three spatial rotational vector fields and a time translation), whereas the Friedmann–Lemaître–Robertson–Walker metric (excluding the Einstein static subcase) has a Killing algebra of dimension six (three translations and three rotations).
The Einstein static metric has a Killing algebra of dimension seven (the previous six plus a time translation).