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

TL;DR

This work advances the modeling of extended bodies in general relativity by combining a general-relativistic hyperelasticity framework with a finite-element scheme to simulate a small elastic sphere on a marginally bound orbit around a Schwarzschild black hole. It goes beyond pole-dipole MPD theory by incorporating quadrupole and higher-order tidal couplings through a fully dynamical, self-consistent SEM tensor and internal dynamics, while neglecting the body's self-gravity in the spacetime exterior. The authors construct local Fermi frames to analyze energy, angular momentum, and center-of-mass deviations, and they decompose the body’s deformation into normal modes, finding dominant $n=0,\ell=2,m=±2$ modes influenced by rotation. The results show orbital energy can be transferred into internal elastic energy and spin, the center of mass deviates from a geodesic, and a highly eccentric bound orbit can result from a tidal encounter. This framework paves the way for implementing solid-body elasticity in future numerical relativity studies of neutron stars and compact-object binaries, including Kerr backgrounds and spin-curvature effects.

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).

Figures (15)

• Figure 1: The constant time slice $t = t_{\mathcal{O}}$ passes through the event ${\mathcal{O}}$ on the central worldline $X^\mu_{(cw)}(t)$. The dashed curve is a spacelike geodesic, orthogonal to the central worldline at ${\mathcal{O}}$. The spacelike geodesic passes through a nearby event ${\mathcal{P}}$. Events along the spacelike geodesic are assigned the Fermi coordinate time $\bar{t} = \bar{t}_{\mathcal{O}}$, which is the proper time at event ${\mathcal{O}}$ along the central worldline.
• Figure 2: The spacelike geodesic (dashed curve) is orthogonal to the central worldline $X^\mu_{(cw)}( t)$ at $\mathcal{O}$. The spacelike geodesic crosses the generic node's worldline $X^\mu_\zeta(t)$ at ${\mathcal{P}}$.
• Figure 3: The central worldline $X^\mu_{(cw)}(t)$ is attached to the fiducial node. The momentum and angular momentum of the body are computed on the slice of constant Fermi time $\bar{t} = \bar{t}_{\mathcal{O}}$. These quantities are used to compute the center of mass event which lies on the slice $t_{rest} = {\rm const}$.
• Figure 4: A $t = \mathrm{const}$ slice that crosses the worldlines of the generic nodes.
• Figure 5: Schwarzschild coordinates of the center of mass of the hyperelastic sphere for $\tilde{r}_p = 9.5, 10,$ and $10.5$. The orbits are counterclockwise in the equatorial plane.