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. In cosmological problems, symmetry plays a role in the cosmological principle, which restricts the type of universes that are consistent with large-scale observations. 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 property which symmetries usually possess can be used to motivate a useful definition of these symmetries themselves.
Mathematical definition
A rigorous definition of symmetries in general relativity has been given by Hall. In this approach, the idea is to use 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 on a spacetime is said to preserve a smooth tensor on if, for each smooth local flow diffeomorphism associated with, the tensors and are equal on the domain of. This statement is equivalent to the more usable condition that the Lie derivative of the tensor under the vector field vanishes: on. This has the consequence that, given any two points and on, the coordinates of in a coordinate system around are equal to the coordinates of in a coordinate system around. A symmetry on the spacetime is a smooth vector field whose local flow diffeomorphisms preserve some feature of the spacetime. The feature may refer to specific tensors or to other aspects of the spacetime such as its 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 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
Killing symmetry
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 and are related to conservation laws.
Homothetic symmetry
A homothetic vector field is one which satisfies: where is a real constant. Homothetic vector fields find application in the study of singularities in general relativity.
Affine symmetry
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 curvature collineation is a vector field which preserves the Riemann tensor: where are the components of the Riemann tensor. The set of all smooth curvature collineations forms a Lie algebra under the Lie bracket operation. The Lie algebra is denoted by and may be infinite-dimensional. Every affine vector field is a curvature collineation.
Matter symmetry
A less well-known form of symmetry concerns vector fields that preserve the energy-momentum tensor. These are variously referred to as matter collineations or matter symmetries and are defined by: where are the energy-momentum tensor components. The intimate relation between geometry and physics may be highlighted here, as the vector field is regarded as preserving certain physical quantities along the flow lines of, this being true for any two observers. In connection with this, it may be shown that every Killing vector field is a matter collineation. Thus, given a solution of the EFE, 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.
Local and global symmetries
Applications
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.
Spacetime classifications
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.. They also classify spacetimes using symmetry vector fields. 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. 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, whereas the Friedmann-Lemaître-Robertson-Walker metric has a Killing algebra of dimension 6. The Einstein static metric has a Killing algebra of dimension 7. The assumption of a spacetime admitting a certain symmetry vector field can place restrictions on the spacetime.
