Projective vector field

Definition

A projective vector field is a smooth vector field on a spacetime $M$ whose local flow diffeomorphisms preserve the geodesic structure of $M$ without necessarily preserving the affine parameter of any geodesic. The vector field is sometimes just referred to as projective. More intuitively, the local flows of the projective map geodesics smoothly into geodesics without preserving the affine parameter. In dealing with vector fields in general relativity, it is often useful to decompose the covariant derivative of $X$ into it's symmetric and skew-symmetric parts:

$X_{a;b}={\frac {1}{2}}h_{ab}+F_{ab}$

where $h_{ab}=({\mathcal {L}}_{X}g)_{ab}=X_{a;b}+X_{b;a}$ and $F_{ab}={\frac {1}{2}}(X_{a;b}-X_{b;a})$

Equivalent conditions

Mathematically, the condition for a vector field $X$ to be projective is equivalent to the existence of a one-form $\psi$ satisfying

$X_{ab;c}=R_{abcd}X^{d}+2g_{a(b}\psi _{c)}$

which is equivalent to

$h_{ab;c}=2g_{ab}\psi _{c}+g_{ac}\psi _{b}+g_{bc}\psi _{a}$

Projective vector fields may be defined on any n-dimensional manifold and the set of all global projective vector fields on such a manifold forms a finite-dimensional Lie algebra denoted by $P(M)$ (the projective algebra) and whose dimension cannot exceed $n(n+2)$. A projective vector field is uniquely determined by specifying the values of $X$, $\nabla X$ and $\nabla \nabla X$ (equivalently, specifying $X$, $h$, $F$ and $\psi$) at any point of $M$.

Subalgebras

Several important special cases of projective vector fields can occur and they form Lie subalgebras of $P(M)$. These subalgebras are useful, for example, in classifying spacetimes in general relativity.

Affines

Affine vector fields (affines) satisfy $\nabla h=0$ (equivalently, $\psi =0$). Affines preserve the geodesic structure of spacetime whilst also preserving the affine parameter. The set of all affines on $M$ forms a Lie subalgebra of $P(M)$ denoted by $A(M)$ (the affine algebra) and whose dimension cannot exceed $n(n+1)$. An affine vector is uniquely determined by specifying the values of the vector field and it's first covariant derivative (equivalently, specifying $X$, $h$ and $F$) at any point of $M$.

Homotheties

Homothetic vector fields (homotheties) preserve the metric up to a constant factor, i.e. $h={\mathcal {L}}_{X}g=cg$. The set of all homotheties on $M$ forms a subalgebra of $A(M)$ denoted by $H(M)$ (the homothetic algebra) whose maximum dimension is ${\frac {1}{2}}n(n+1)+1$. A homothetic vector field is uniquely determined by specifying the values of the vector field and it's first covariant derivative equivalently, specifying $X$, $F$ and $c$) at any point of the manifold.

Killings

Killing vector fields (Killings) preserve the metric, i.e. $h={\mathcal {L}}_{X}g=0$. The set of all Killing vector fields on $M$ forms a Lie subalgebra of $H(M)$ denoted by $K(M)$ (the Killing algebra) and has a maximum dimension of ${\frac {1}{2}}n(n+1)$. A Killing vector field is uniquely determined by specifying the values of the vector field and it's first covariant derivative (equivalently, specifying $X$ and $F$) at any point of $M$.