Symmetry generators for null geodesic motion (Q794278)

Let \(L(t,x^ a,x^ a)\) be the Lagrange function modeling an arbitrary dynamical system and \(N=R\times TM\) the associated extended tangent space. The local natural coordinates on N are \((s,x^ a,\dot x^ a)\), s being an affine parameter and \(\dot x{}^ a=dx/ds\). A vector field Y on N with local representation \(Y=\tau(s,x^ b,\dot x^ b)\partial /\partial s+K^ a(s,x^ b,\dot x^ b)\partial /\partial x^ a+\eta^ a(s,x^ b,\dot x^ b)\partial /\partial \dot x^ a\) is said to be a conformal dynamical symmetry (CDS) iff \(L_ Y\Gamma =[Y,\Gamma]=g\Gamma\), where g is a scalar function, whenever \(\Gamma\) is the lift of the tangent vector to a null geodesic. The author shows that CDSs may be regarded as generators of Jacobi fields along null geodesics of M and gives methods for the construction of CDSs. The connections between CDSs and the conformal Killing tensors are also considered and a procedure relying on the conformal invariance of null geodesic paths in order to construct CDSs is described. Based on Noether's theorem, five invariants along null geodesics are found.