A projective vector field (projective) is a smooth vector field on a semi Riemannian manifold (p.ex. spacetime) M {\displaystyle M} whose flow preserves the geodesic structure of M {\displaystyle M} 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 In dealing with a vector field X {\displaystyle X} 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:
X a ; b = 1 2 h a b + F a b {\displaystyle X_{a;b}={\frac {1}{2}}h_{ab}+F_{ab}}
where
h a b = ( L X g ) a b = X a ; b + X b ; a {\displaystyle h_{ab}=({\mathcal {L}}_{X}g)_{ab}=X_{a;b}+X_{b;a}}
and
F a b = 1 2 ( X a ; b − X b ; a ) {\displaystyle F_{ab}={\frac {1}{2}}(X_{a;b}-X_{b;a})}
Note that X a {\displaystyle X_{a}} are the covariant components of X {\displaystyle X} .
Equivalent conditions Mathematically, the condition for a vector field X {\displaystyle X} to be projective is equivalent to the existence of a one-form ψ {\displaystyle \psi } satisfying
X a ; b c = R a b c d X d + 2 g a ( b ψ c ) {\displaystyle X_{a;bc}\,=R_{abcd}X^{d}+2g_{a(b}\psi _{c)}}
which is equivalent to
h a b ; c = 2 g a b ψ c + g a c ψ b + g b c ψ a {\displaystyle h_{ab;c}\,=2g_{ab}\psi _{c}+g_{ac}\psi _{b}+g_{bc}\psi _{a}}
The set of all global projective vector fields over a connected or compact manifold forms a finite-dimensional Lie algebra denoted by P ( M ) {\displaystyle P(M)} (the projective algebra) and satisfies for connected manifolds the condition: dim P ( M ) ≤ n ( n + 2 ) {\displaystyle \dim P(M)\leq n(n+2)} . Here a projective vector field is uniquely determined by specifying the values of X {\displaystyle X} , ∇ X {\displaystyle \nabla X} and ∇ ∇ X {\displaystyle \nabla \nabla X} (equivalently, specifying X {\displaystyle X} , h {\displaystyle h} , F {\displaystyle F} and ψ {\displaystyle \psi } ) at any point of M {\displaystyle M} . (For non-connected manifolds you need to specify these 3 in one point per connected component.) Projectives also satisfy the properties:
L X R a
b c d = δ a
d ψ b ; c − δ a
c ψ b ; d {\displaystyle {\mathcal {L}}_{X}R^{a}{}_{bcd}=\delta ^{a}{}_{d}\psi _{b;c}-\delta ^{a}{}_{c}\psi _{b;d}}
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