Geodesic deviation in a nonlinear gravitational wave spacetime

TL;DR

This paper analyzes the tidal effects of a nonlinear gravitational wave using exact Einstein solutions, revealing nonlinear behavior in transverse separations and confirming results via tetrad formalism.

Contribution

It provides an exact solution for gravitational wave tidal effects in double-null coordinates, highlighting nonlinear transverse behaviors and validating findings with tetrad formalism.

Findings

Transverse separation components depend nonlinearly on coordinate v.

Behavior of separation vectors matches flat space in v and u components.

Results are consistent across different formalisms.

Abstract

The tidal effects generated by a nonlinear gravitational wave are investigated in double-null v - u coordinates, as an exact solution of Einstein's field equations. The components $\xi^{v}$ and $\xi^{u}$ of the separation vector behave as in flat space but the transversal components $\xi^{x}$ and $\xi^{y}$ depend nonlinearly on $v$ through the Bessel and Neumann functions, far from the null surface $v = 0$. We show that the same results are obtained by means of the tetrad formalism.