Abstract
We study horizon shells and soldering freedom for extreme black holes and how supertranslation-like Bondi-Metzner-Sachs (BMS) symmetries appear as soldering transformations. Further, for a null shell placed infinitesimally close to the horizon of an extreme Reissner-Nordstr$\ddot{o}$m (RN) black hole, we show superrotation-like symmetries also arise as soldering freedom. Next, considering the interaction of impulsive gravitational waves supported at the horizon shell with test particles, we study how the "memory" (or the imprints) of BMS-like symmetries gets encoded in the geodesics (test particles) crossing the shell. Our study shows, timelike test particles get displaced from their initial plane when they cross the horizon shell. For a null geodesic congruence crossing the horizon shell, the optical tensors corresponding to the congruence suffer jumps. In both the cases, the changes are induced by BMS parameters that constitute the gravity wave and matter degrees of freedom of the shell.
Highlights
Gravitational memory effect has been an active area of research, and it has attracted researchers across the disciplines from classical gravity and gravitational wave astronomy [1,2,3,4,5,6] to researchers in the area of quantum gravity [7,8,9,10,11,12]
For a horizon shell in an Extreme Reissner–Nordström (ERN) black hole, we find that the effect of the passing of timelike geodesics across the horizon shell is to deflect the test particles off the initial surface, where they were placed before the interaction of impulsive gravitational waves (IGW)
The motivation of this work is to find the memory effect of the IGW supported at a horizon shell of extreme black holes for timelike and null geodesics crossing the null shell
Summary
Gravitational memory effect has been an active area of research, and it has attracted researchers across the disciplines from classical gravity and gravitational wave astronomy [1,2,3,4,5,6] to researchers in the area of quantum gravity [7,8,9,10,11,12]. In earlier studies [23,24], in the context of stitching two spacetimes across a null hypersurface, it has been shown that BMS-like [25] asymptotic symmetries can be recovered at the event horizon of black holes when one demands the induced metric on the horizon remains invariant under arbitrary coordinate transformations owing to satisfy the junction conditions. For a nonrotating and neutral black hole, it has been shown in [31], how BMS-like symmetries are encoded in the deviation vectors for timelike geodesics crossing the horizon shells. BMS symmetries arise as the freedom of allowing general coordinate transformations (diffeomorphisms) that preserve the induced metric on the event horizon, when we patch two spacetimes across it These possible transformations can be calculated by solving the Killing equation on the hypersurface [23]. If we set LZna 1⁄4 0, one gets a restriction on ZV:
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