III.1.4 Space-Time Symmetries and Diffeomorphisms
In General Relativity, space-time symmetries are used in studying exact solutions of Einstein s field equations, and they are different from internal symmetries. For example, the role of spherical symmetry is important in deriving the Schwarzschild solution and studying its physical consequences, such as the nonexistence of gravitational radiation in a spherically pulsating star. In the cosmological principle, symmetry restricts the type of universes that are consistent with large-scale observations, such as the Friedmann-Lema tre-Robertson-Walker metric. There various corresponding preserved properties, including: geodesics of the space-time, the metric tensor and the curvature tensor.
A rigorous definition of symmetries in General Relativity has been given by Hall in 2004, based on the idea of using smooth vector fields whose local flow diffeomorphisms preserve some property of the space-time. Because of this restricting diffeomorphism, the behavior of objects may not be as manifestly symmetric on large scales. A symmetry on the space-time is a smooth vector field whose local flow diffeomorphisms preserve some geometrical feature of the space-time, such as the metric, the energy-momentum tensor, or other aspects of the space-time, including its geodesic structure. The vector fields are called collineations or symmetries, and the set of all symmetry vector fields forms a Lie algebra under the Lie bracket operation, if the smoothness condition is fulfilled.
A Killing vector field is one of the most important types of symmetries and is defined to be a smooth vector field that preserves the metric tensor. Killing vector fields are related to conservation laws, and they have extensive applications, also in classical mechanics. An affine vector field preserves geodesics and preserves the affine parameter, while the homothetic vector fields are important in studying singularities. Other kinds of symmetry preserve the energy-momentum tensor, which are referred to as matter collineations or matter symmetries. It may be also shown that every Killing vector field is a matter collineation by the Einstein field equations. Therefore, a vector field that preserves the metric necessarily preserves the corresponding 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. When the energy-momentum tensor represents an electromagnetic field, a Killing vector field does not necessarily preserve the electric and magnetic fields.