Approximate Partial Noether Operators of the Schwarzschild Spacetime

Abstract

The objective of this paper is twofold: (a) to find a natural example of a perturbed Lagrangian that has different partial Noether operators with symmetries different from those of the underlying Lagrangian. First we regard the Schwarzschild spacetime as a perturbation of the Minkowski spacetime and investigate the approximate partial Noether operators for this perturbed spacetime. It is shown that the Minkowski spacetime has 12 partial Noether operators, 10 of which are different from the 17 Noether symmetries for this spacetime. It is found that for the perturbed Schwarzschild spacetime we recover the exact partial Noether operators as trivial first-order approximate partial Noether operators and there is no non-trivial approximate partial Noether operator as for the Noether case. As a consequence we state a conjecture. (b) Then we prove a conjecture that the approximate symmetries of a perturbed Lagrangian form a subalgebra of the approximate symmetries of the corresponding perturbed Euler-Lagrange equations and illustrate it by our examples. This is in contrast to approximate partial Noether operators.

Integration of the geodesic equations via Noether symmetries

“…has also five NSs [14] for constant potential, which are four KVs given in (26) for the Schwarzschild metric and the translation symmetry Y 0 = ∂ s .…”

“…It has been investigated the ANSs and conservation laws of the geodesic equations without the potential function for the Schwarzschild [13] and the RN [14] the spacetimes. Constructing the geometrical set of equations corresponding to the ANS equations with an arbitrary potential function, the ANSs of the geodesic Lagrangian for the Schwarzschild, the RN and the Bardeen spacetimes have been determined by Camci [15,16].…”

Fourth-Order Pattern Forming Pdes: Partial and Approximate Symmetries

“…The method not only gives rise to conserved quantities, but may also be used to obtain new closed form solutions [19]. Several other works in the literature have involved partial Lagrangians, namely [3,8,12].…”