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 finding ample application in the study of exact solutions of Einstein's field equations of general relativity.
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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 finds a role to play in the cosmological principle which restricts the type of universes that are consistent with largescale observations (e.g. the FriedmannRobertsonWalker (FRW) 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 in more detail later. 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. 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:
on M. This has the consequence that, given any two points p and q on M, the coordinates of T in a coordinate system around p are equal to the coordinates of T in a coordinate system around q. A symmetry on the spacetime is a smooth vector field whose local flow diffeomorphisms preserve some (usually geometrical) feature of the spacetime. The (geometrical) feature may refer to specific tensors (such as the metric, or the energymomentum tensor) or to other aspects of the spacetime such as it's geodesic structure. The vector fields are sometimes referred to as collineations, symmetry vector fields or just symmetries. The set of all symmetry vector fields on M forms a Lie algebra under the Lie bracket operation as can be seen from the identity:
the term on the right usually being written, with an abuse of notation, as .
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:
This is usually written in the expanded form as:
Killing vector fields find extensive applications (including in classical mechanics) and are related to conservation laws.
A homothetic vector field is one which satisfies:
where c is a real constant. Homothetic vector fields find application in the study of singularities in general relativity.
An affine vector field is one that satisfies:
An affine vector field preserves geodesics and preserves the affine parameter.
The above three vector field types are special cases of projective vector fields which preserve geodesics without necessarily preserving the affine parameter.
A conformal vector field is one which satisfies:
where φ is a smooth realvalued function on M.
A curvature collineation is a vector field which preserves the Riemann tensor:
where R^{a}_{bc d} are the components of the Riemann tensor. The set of all smooth curvature collineations forms a Lie algebra under the Lie bracket operation (if the smoothness condition is dropped, the set of all curvature collineations need not form a Lie algebra). The Lie algebra is denoted by CC(M) and may be infinitedimensional. Every affine vector field is a curvature collineation.
A less wellknown form of symmetry concerns vector fields that preserve the energymomentum tensor. These are variously referred to as matter collineations or matter symmetries and are defined by:
where T_{ab} are the energymomentum tensor components. The intimate relation between geometry and physics may be highlighted here, as the vector field X is regarded as preserving certain physical quantities along the flow lines of X, this being true for any two observers. In connection with this, it may be shown that every Killing vector field is a matter collineation (by the Einstein field equations, with or without cosmological constant). Thus, given a solution of the EFE, a vector field that preserves the metric necessarily preserves the corresponding energymomentum tensor. When the energymomentum tensor represents a perfect fluid, every Killing vector field preserves the energy density, pressure and the fluid flow vector field. When the energymomentum tensor represents an electromagnetic field, a Killing vector field does not necessarily preserve the electric and magnetic fields.
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 energymomentum tensor or the Petrov classification of the Weyl tensor have been studied extensively by many researchers, most notably Stephani et al. (2003). They also classify spacetimes using symmetry vector fields (especially Killing and homothetic symmetries). For example, Killing vector fields may be used to classify spacetimes, as there is a limit to the number of global, smooth Killing vector fields that a spacetime may possess (the maximum being 10 for 4dimensional spacetimes). Generally speaking, the higher the dimension of the algebra of symmetry vector fields on a spacetime, the more symmetry the spacetime admits. For example, the Schwarzschild solution has a Killing algebra of dimension 4 (3 spatial rotational vector fields and a time translation), whereas the FriedmannRobertsonWalker (FRW) metric (excluding the Einstein static subcase) has a Killing algebra of dimension 6 (3 translations and 3 rotations). The Einstein static metric has a Killing algebra of dimension 7 (the previous 6 plus a time translation).
The assumption of a spacetime admitting a certain symmetry vector field can place restrictions on the spacetime.
