The role of symmetry in physics is important, for example, in simplifying solutions to many problems. The term spacetime symmetries refers to aspects of spacetime that can be described as exhibiting some form of symmetry. These symmetries are used to simplify problems and find ample application in the study of exact solutions of Einstein's field equations of general relativity. Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... Strictly speaking, any Lorentz metric is a solution of the Einstein field equation, as this amounts to nothing more than a mathematical definition of the energy-momentum tensor (by the field equations). ... For other topics related to Einstein see Einstein (disambig) Introduction In physics, the Einstein field equation or Einstein equation is a tensor equation in the Einsteins theory of general relativity. ... Two-dimensional visualization of space-time distortion. ...

Physical motivation

Quite often, physical problems may be investigated and solved by noticing features of the problem 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 (for example, 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-Robertson-Walker (FRW) metric). Symmetries usually require some form of preserving property, the most important of which in general relativity include the following: Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ... The Schwarzschild solution is one of the simplest and useful solutions of the Einstein field equations (see general relativity). ... The Cosmological Principle is a principle invoked in cosmology that severely restricts the large variety of possible cosmological theories: On large scales, the Universe is homogeneous and isotropic. ... The Friedmann-Lemaître-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ...

• preserving geodesics of the spacetime
• preserving the metric tensor
• preserving the curvature tensor

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

Mathematical definition

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: Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... In mathematics, a local diffeomorphism is a smooth map f : M → N between smooth manifolds such that for every point p of M there exists an open neighbourhood U of p such that f(U) is open in N and f|U : U → f(U) is a diffeomorphism. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... In mathematics, a Lie derivative is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented by vector fields, as...

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 energy-momentum 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: In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

the term on the right usually being written, with an abuse of notation, as .

Examples

Killing symmetry

One of the most important types of symmetry vector fields are Killing vector fields which are defined to be those smooth vector fields that preserve the metric tensor: In mathematics, a Killing vector field is a vector field on a Riemannian manifold that preserves the metric. ...

This is usually written in the expanded form as:

X_a;b + X_b;a = 0

Killing vector fields find extensive applications (including in classical mechanics) and are related to conservation laws. In physics, Classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies, and the forces that cause them. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

Homothetic symmetry

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. Singularity has several different meanings: mathematical singularity - a point where a mathematical function goes to infinity or is in certain other ways ill-behaved gravitational singularity - an infinity occurring in an astrophysical model, involving infinite curvature (a mathematical singularity) in the space/time continuum technological singularity - a predicted point in...

Affine symmetry

An affine vector field is one that satisfies:

An affine vector field preserves geodesics and preserves the affine parameter.

The above 3 types of vector fields are special cases of projective vector fields which preserve geodesics without necessarily preserving the affine parameter.

Conformal symmetry

A conformal vector field is one which satisfies:

where φ is a smooth real-valued function on M.

Curvature Symmetry

A curvature collineation is a vector field which preserves the Riemann tensor:

where R^a_bcd 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 infinite-dimensional. Every affine is a curvature collineation. In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... Curvature is the amount by which a geometric object deviates from being flat. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...

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 T_ab are the energy-momentum 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 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.

Applications of symmetry vector 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.

Classifications of spacetimes

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). 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 4-dimensional 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 Friedmann-Robertson-Walker (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 Segre classification is an algebraic classification of rank two symmetric tensors. ... The Petrov classsification provides a means of algebraically classifying the Weyl tensor. ... In differential geometry, the Weyl curvature tensor is the traceless component of the Riemann curvature tensor. ... The Friedmann-Lemaître-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ...

The assumption of a spacetime admitting a certain symmetry vector field can place severe restrictions on the spacetime.

See also

• Killing tensors
• Ricci decomposition
• Noether's theorem
• Lie groups

Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between the symmetries and the conservation laws. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

References

• Hall, Graham (2004). Symmetries and Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics). Singapore: World Scientific Pub. Co. Inc.. ISBN 9-810-21051-5. See Section 10.1 for a definition of symmetries.

• Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard (2003). Exact Solutions of Einstein's Field Equations. Cambridge: Cambridge University Press. ISBN 0-521-46136-7.

• Schutz, Bernard (1980). Geometrical Methods of Mathematical Physics. Cambridge: Cambridge University Press. ISBN 0-521-29887-3. See Chapter 3 for properties of the Lie derivative and Section 3.10 for a definition of invariance.