Symmetries and conserved quantities in geodesic motion

Abstract

Recently obtained results linking several constants of motion to one (non-Noetherian) symmetry to the problem of geodesic motion in Riemannian space-times are applied. The construction of conserved quantities in geodesic motion as well as the deduction of geometrical statements about Riemannian space-times are achieved.

Approximate spacetime symmetries and conservation laws

“…The same can also be said for more general collineations. Extensive discussions of exact symmetries and associated integrals of the geodesic equation may be found in [19]. Many of these symmetries are non-Noetherian in the sense that they preserve the equations of motion, but not the action.…”

“…The evolution of this quantity crucially depends on how the parameters A a (s) and B ab (s) in (19) are connected along Γ. It is well-known that P ξ is conserved if ξ a is Killing and no matter flows across the boundary of the worldtube.…”

“…Define a Jacobi field ψ a (x, γ) to be a solution of (21) throughout this region. It is explicitly given by (17) for some initial data with the form (19). The set of all Jacobi fields about a fixed γ forms a 20-dimensional group in four spacetime dimensions.…”

“…This is done by first applying the KT equations to the given parameters along Γ from γ 0 to γ (τ (x)). The geodesic deviation equation (21) is then integrated between this latter point and x using the initial conditions (19). Both of these operations simply require finding the solutions to well-behaved ordinary differential equations.…”

Noetherian first integrals

“…In [5] the above procedure, which is known in the relevant literature [9,10,11,12,13] as the method of s-and g-symmetries, has been applied to the two-dimensional autonomous isotropic harmonic oscillator with equations of motion In the case of the isotropic oscillator the formalism of the s-and g-symmetries [9,10,11,12,13] has been used [5] to obtain the off-diagonal component of the conserved Jauch-Hill-Fradkin tensor [4,14] A 12 =ẋẏ + xy (1.5) and in the case of the anisotropic oscillator the corresponding conserved quantities, 6) and its complex conjugate. In both cases the first integrals were described [5] as nonnoetherian.…”