Memory effect in anti–de Sitter spacetime

The geodesic deviation of a pair of test particles is a natural observable for the gravitational memory effect. Nevertheless in curved spacetime, this observable is plagued with various issues that need to be clarified before one can extract the essential part that is related to the gravitational radiation. In this paper we consider the anti–de Sitter (AdS) space as an example and analyze this observable carefully. We show that by employing the Fermi normal coordinates around the geodesic of one of the particles (i.e., the standard free falling frame attached to this particle), one can elegantly separate out the curvature contribution of the background spacetime to the geodesic deviation from the contribution of the gravitational wave. The gravitational wave memory obtained this way depends linearly and locally on the retarded metric perturbation caused by the gravitational wave, and, remarkably, it takes on exactly the same formula as in the flat case. To determine the memory, in addition to the standard tail contribution to the gravitational radiation, one needs to take into account the contribution from the reflected gravitational wave off the AdS boundary. For general curved spacetime, our analysis suggests that the use of a certain coordinate system adapted to the local geodesic (e.g., the Fermi normal coordinates system in the AdS case) would allow one to dissect the geodesic deviation of test particles and extract the relevant contribution to define the memory due to gravitational radiation.

Figure 1

A massless scattering in the $x^1-x^2$ plane.

Figure 2

Propagation of the gravitational waves and timelike geodesics.