Spacetime symmetries
Encyclopedia
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
.
is important in deriving the Schwarzschild solution
and deducing the physical consequences of this symmetry (such as 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-Lemaitre-Robertson-Walker (FLRW) 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.
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 
(or 
is invariant under 
) 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 (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 
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
.
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.
where c is a real constant. Homothetic vector fields find application in the study of singularities
in general relativity.
An affine vector field preserves geodesic
s 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.
where
is a smooth real-valued function on 
.
where
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
and may be infinite
-dimension
al. Every affine vector field is a curvature collineation.
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 (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.
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-Lemaitre-Robertson-Walker (FLRW) 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.
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
that can be described as exhibiting some form of symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
. The role of symmetry in physics
Symmetry in physics
In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...
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
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
.
Physical motivation
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 symmetrySpherically symmetric spacetime
A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the group SO and the orbits of this group are 2-dimensional spheres . The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one...
is important in deriving the Schwarzschild solution
Deriving the Schwarzschild solution
The Schwarzschild solution is one of the simplest and most useful solutions of theEinstein field equations . It describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object...
and deducing the physical consequences of this symmetry (such as the non-existence of gravitational radiation in a spherically pulsating star). In cosmological problems, symmetry finds a role to play in the cosmological principle
Cosmological Principle
In modern physical cosmology, the cosmological principle is the working assumption that observers on Earth do not occupy an unusual or privileged location within the universe as a whole, judged as observers of the physical phenomena produced by uniform and universal laws of physics...
which restricts the type of universes that are consistent with large-scale observations (e.g. the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric). Symmetries usually require some form of preserving property, the most important of which in general relativity include the following:
- 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 diffeomorphismsLocal diffeomorphism
In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a function between smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below....
preserve some property of the spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. 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 (or is invariant under ) 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 derivativeLie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
of the tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
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 (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 forms a Lie algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
under the Lie bracket
Lie bracket
Lie bracket can refer to:*A bilinear binary operation defined on elements of a Lie algebra*Lie bracket of vector fields...
operation as can be seen from the identity:
the term on the right usually being written, with an abuse of notation
Abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...
, as
.Killing symmetry
A Killing vector field is one of the most important types of symmetries and is defined to be a smooth vector fieldVector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
that preserves the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
:
This is usually written in the expanded form as:
Killing vector fields find extensive applications (including in classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
) and are related to conservation laws.
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
Gravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...
in general relativity.
Affine symmetry
An affine vector field is one that satisfies:An affine vector field preserves geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
s 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.
Conformal symmetry
A conformal vector field is one which satisfies:where
is a smooth real-valued function on .Curvature symmetry
A curvature collineation is a vector field which preserves the Riemann tensor:where
are the components of the Riemann tensor. The set of all smoothSmooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
curvature collineations forms a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
under the Lie bracket
Lie bracket
Lie bracket can refer to:*A bilinear binary operation defined on elements of a Lie algebra*Lie bracket of vector fields...
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
and may be infiniteInfinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al. 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 (by the Einstein field equations, with or without cosmological constantCosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
). 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
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 classificationSegre classification
The Segre classification is an algebraic classification of rank two symmetric tensors. The resulting types are then known as Segre types. It is most commonly applied to the energy-momentum tensor and primarily finds application in the classification of exact solutions in general relativity....
of the energy-momentum tensor or the Petrov classification
Petrov classification
In differential geometry and theoretical physics, the Petrov classification describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold....
of the Weyl tensor
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...
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-Lemaitre-Robertson-Walker (FLRW) 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.
See also
- Field (physics)Field (physics)In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
- Killing tensor
- Lie groups
- Noether's theoremNoether's theoremNoether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
- Ricci decompositionRicci decompositionIn semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties...
- Symmetry in physicsSymmetry in physicsIn physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...