A projective vector field (projective) is a smooth vector field on a semi Riemannian manifold (p.ex. spacetime whose flow preserves the geodesic 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.

Decomposition

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

.

Equivalent conditions

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 (the projective algebra. (For non-connected manifolds you need to specify these 3 in one point per connected component.) Projectives also satisfy the properties:

Subalgebras

. These subalgebras are useful, for example, in classifying spacetimes in general relativity.

Affine algebra

Affine vector fields forms a Lie subalgebra (the affine algebra) and satisfies for connected M. Affines also preserve the Riemann, Ricci and Weyl tensors, i.e.

Homothetic algebra

Homothetic vector fields (the homothetic algebra) and satisfies for connected M

1. .

) at any point of the manifold.

Killing algebra

Killing vector fields (the Killing algebra) and satisfies for connected M

1. .

.

Applications

In general relativity, many spacetimes possess certain symmetries that can be characterised by vector fields on the spacetime. For example, Minkowski space.

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.

References