Motion of a hyperelastic sphere in Schwarzschild spacetime

We simulate the motion of a hyperelastic sphere in a background Schwarzschild spacetime using a finite element discretization and a Lagrangian formulation of the equations of motion. We set the initial spacetime coordinates and velocities of the nodes of the discretized sphere on a constant coordinate time hypersurface such that the tidal and elastic forces are balanced. We compute the coordinates and velocities of the center of mass in a Fermi normal frame (FNFnode) centered about a fiducial node. We then compute a geodesic that starts out with the same spacetime coordinates and velocities as the center of mass. Next, we construct a Fermi normal frame (FNFgeo) that is carried along the geodesic. The metric in each Fermi normal frame is nearly flat assuming that the size of the sphere is small compared to the curvature of spacetime. We simulate both a close encounter orbit as well as a radial plunge, and observe the sphere as it deforms, oscillates and rotates in the FNFgeo. The observed oscillation frequency agrees with the lowest-frequency ellipsoidal mode for small oscillations of a free solid elastic sphere. We integrate the stress-energy-momentum tensor in the FNFgeo to obtain the total momentum and total angular momentum about the geodesic and compute the spin. Finally, we observe how the spin and elastic energy change as the sphere interacts with the black hole.