Projective vector field
Encyclopedia
A projective vector field (projective) is a smooth vector field
on a semi Riemannian manifold
(p.ex. spacetime
)
whose flow preserves the geodesic
structure of
without necessarily preserving the affine parameter
of any geodesic. More intuitively, the flow of the projective maps geodesics smoothly into geodesics without preserving the affine parameter.
on a semi Riemannian manifold
(p.ex. in general relativity
), it is often useful to decompose the covariant derivative
into its symmetric and skew-symmetric parts:
where
and
Note that
are the covariant components of 
.
to be projective is equivalent to the existence of a one-form
satisfying
which is equivalent to
The set of all global projective vector fields over a connected or compact manifold forms a finite-dimensional Lie algebra
denoted by
(the projective algebra) and satisfies for connected manifolds the condition: 
. Here a projective vector field is uniquely determined by specifying the values of 
, 
and 
(equivalently, specifying 
, 
, 
and 
) at any point of 
. (For non-connected manifolds you need to specify these 3 in one point per connected component.) Projectives also satisfy the properties:
. These subalgebras are useful, for example, in classifying spacetimes in general relativity.
s (affines) satisfy
(equivalently, 
) and hence every affine is a projective. Affines preserve the geodesic structure of the semi Riem. manifold (read spacetime) whilst also preserving the affine parameter. The set of all affines on 
forms a Lie subalgebra of 
denoted by 
(the affine algebra) and satisfies for connected M, 
. An affine vector is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying 
, 
and 
) at any point of 
. Affines also preserve the Riemann, Ricci and Weyl tensors, i.e.
, 
, 
. As 
, every homothety is an affine and the set of all homotheties on 
forms a Lie subalgebra of 
denoted by 
(the homothetic algebra) and satisfies for connected M
.
A homothetic vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying
, 
and 
) at any point of the manifold.
. Taking 
in the defining property of a homothety, it is seen that every Killing is a homothety (and hence an affine) and the set of all Killing vector fields on 
forms a Lie subalgebra of 
denoted by 
(the Killing algebra) and satisfies for connected M
.
A Killing vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying
and 
) at any point (for every connected component) of 
.
admits the maximal projective algebra, i.e. 
.
Many other applications of symmetry vector fields in general relativity may be found in Hall (2004) which also contains an extensive bibliography including many research papers in the field of symmetries in general relativity
.
Vector 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...
on a semi Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
(p.ex. 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...
)
whose flow preserves the geodesicGeodesic
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...
structure of
without necessarily preserving the affine parameterGeodesic (general relativity)
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational, force is a particular type of geodesic...
of any geodesic. More intuitively, the flow of the projective maps geodesics smoothly into geodesics without preserving the affine parameter.
Decomposition
In dealing with a vector field on a semi Riemannian manifoldRiemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
(p.ex. in 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...
), it is often useful to decompose the covariant derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
into its symmetric and skew-symmetric parts:
where
and
Note that
are the covariant components of .Equivalent conditions
Mathematically, the condition for a vector field to be projective is equivalent to the existence of a one-formOne-form
In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
satisfyingwhich is equivalent to
The set of all global projective vector fields over a connected or compact manifold forms a finite-dimensional 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...
denoted by
(the projective algebra) and satisfies for connected manifolds the condition: . Here a projective vector field is uniquely determined by specifying the values of , and (equivalently, specifying , , and ) at any point of . (For non-connected manifolds you need to specify these 3 in one point per connected component.) Projectives also satisfy the properties:Subalgebras
Several important special cases of projective vector fields can occur and they form Lie subalgebras of. These subalgebras are useful, for example, in classifying spacetimes in general relativity.Affine algebra
Affine vector fieldAffine vector field
An affine vector field is a projective vector field preserving geodesics and preserving the affine parameter...
s (affines) satisfy
(equivalently, ) and hence every affine is a projective. Affines preserve the geodesic structure of the semi Riem. manifold (read spacetime) whilst also preserving the affine parameter. The set of all affines on forms a Lie subalgebra of denoted by (the affine algebra) and satisfies for connected M, . An affine vector is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying , and ) at any point of . Affines also preserve the Riemann, Ricci and Weyl tensors, i.e., , Homothetic algebra
Homothetic vector fields (homotheties) preserve the metric up to a constant factor, i.e.. As , every homothety is an affine and the set of all homotheties on forms a Lie subalgebra of denoted by (the homothetic algebra) and satisfies for connected M.A homothetic vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying
, and ) at any point of the manifold.Killing algebra
Killing vector fields (Killings) preserve the metric, i.e.. Taking in the defining property of a homothety, it is seen that every Killing is a homothety (and hence an affine) and the set of all Killing vector fields on forms a Lie subalgebra of denoted by (the Killing algebra) and satisfies for connected M.A Killing vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying
and ) at any point (for every connected component) of .Applications
In general relativity, many spacetimes possess certain symmetries that can be characterised by vector fields on the spacetime. For example, Minkowski spaceMinkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
admits the maximal projective algebra, i.e. .Many other applications of symmetry vector fields in general relativity may be found in Hall (2004) which also contains an extensive bibliography including many research papers in the field of symmetries in general relativity
Spacetime symmetries
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...
.