Projective vector field

A projective vector field (projective) 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. More intuitively, the local flows of the projective map geodesics smoothly into geodesics without preserving the affine parameter.

Decomposition

In dealing with vector fields in general relativity, it is often useful to decompose the covariant derivative of $X$ into its 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 satisfies the condition: $\dim P(M)\leq n(n+2)$. 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 $M$. Projectives also satisfy the properties:

${\mathcal {L}}_{X}R^{a}{}_{bcd}=\delta ^{a}{}_{d}\psi _{b;c}-\delta ^{a}{}_{c}\psi _{b;d}$

${\mathcal {L}}_{X}R_{ab}=-3\psi _{a;b}$

Subalgebras of the projective algebra

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, $\psi =0$) and hence every affine is a projective. 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 satisfies $\dim A(M)\leq n(n+1)$. An affine vector is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying $X$, $h$ and $F$) at any point of $M$. Affines also preserve the Riemann, Ricci and Weyl tensors, i.e.

${\mathcal {L}}_{X}R^{a}{}_{bcd}=0$, ${\mathcal {L}}_{X}R_{ab}=0$, ${\mathcal {L}}_{X}C^{a}{}_{bcd}=0$

Homotheties

Homothetic vector fields (homotheties) preserve the metric up to a constant factor, i.e. $h={\mathcal {L}}_{X}g=2cg$. As $\nabla h=0$, every homothety is an affine and the set of all homotheties on $M$ forms a Lie subalgebra of $A(M)$ denoted by $H(M)$ (the homothetic algebra) and satisfies

$\dim H(M)\leq {\frac {1}{2}}n(n+1)+1$.

A homothetic vector field is uniquely determined by specifying the values of the vector field and its 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. $h={\mathcal {L}}_{X}g=0$. Taking $c=0$ 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 $M$ forms a Lie subalgebra of $H(M)$ denoted by $K(M)$ (the Killing algebra) and satisfies

$\dim K(M)\leq {\frac {1}{2}}n(n+1)$.

A Killing vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative (equivalently, specifying $X$ and $F$) at any point of $M$.

Applications

In general relativity, many spacetimes possess certain symmetries that can be characterised by vector fields on the spacetime. For example, Minkowski space admits the maximal projective algebra, i.e., if $M$ denotes Minkowski space, then $\dim P(M)=24$.

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

• Poor, W. (1981). Differential Geometric Structures. New York: McGraw Hill. ISBN 0-070-50435-0.
• Yano, K. (1970). Integral Formulas in Riemannian Geometry. New York: Marcel Dekker. ISBN ???.
• Hall, Graham (2004). Symmetries and Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics). Singapore: World Scientific Pub. ISBN 9-810-21051-5.