Encounter between an extended hyperelastic body and a Schwarzschild black hole with quadrupole-order effects

TL;DR

This paper models the relativistic interaction of a hyperelastic body with a Schwarzschild black hole, revealing effects of quadrupole order through finite element simulations and analyzing the resulting orbital dynamics.

Contribution

It introduces a finite element scheme for simulating hyperelastic bodies in general relativity and applies it to a black hole encounter, incorporating quadrupole effects.

Findings

The small body is captured into a highly eccentric orbit.

Quadrupole effects influence the body's internal and orbital dynamics.

Energy is deposited into the body's elastic deformation during interaction.

Abstract

We model the general relativistic interaction of a small hyperelastic sphere with a Schwarzschild black hole as it follows an initially marginally-bound orbit through a close encounter. While the interaction reveals effects that are encoded by the Mathisson-Papapetrou-Dixon (MPD) multipolar equations through quadrupole order, the calculation is made using an independent general relativistic finite element scheme that we described earlier (Phys.~Rev.~D 108(8):084020, October 2023). The finite element calculation is done in Schwarzschild coordinates, following a large and scalable number of mass elements in interaction with each other through elastic forces derived from a potential energy function and with the spacetime geometry. After the fact, we analyze the dynamics using a local Fermi coordinate system, computing (1) the deviation of the center of mass of the body relative to the initial marginally-bound orbit, (2) changes in orbital and spin angular momenta, and (3) the decrease in orbital energy and accompanying deposition of energy into internal elastic dynamics. The interaction leads to the capture of the small body into a highly eccentric orbit ($e \simeq 0.99998$ in a sample calculation).