Influence of internal structure on the motion of test bodies in extreme mass ratio situations

TL;DR

This work extends the motion of small bodies in General Relativity beyond point-particle approximations by employing a multipolar expansion up to quadrupole order. It derives the full set of equations of motion, conserved quantities, and a realistic quadrupole model that includes spin-induced and tidal deformations in Kerr spacetime, and it analyzes equatorial, aligned-spin orbits through an effective potential whose minima yield circular orbits and binding energies. The authors systematically separate spin and quadrupole contributions, quantify tidal disruption limits, and compare gauge-invariant binding-energy–angular-momentum relations to conservative self-force results and post-Newtonian Hamiltonians, highlighting where structure-induced corrections are most significant. The results underscore the importance of incorporating finite-size effects in gravitational-wave modeling and provide a framework for future Kerr self-force comparisons and intermediate-mass-ratio extensions.

Abstract

We investigate the motion of test bodies with internal structure in General Relativity. With the help of a multipolar approximation method for extended test bodies we derive the equations of motion up to the quadrupolar order. The motion of pole-dipole and quadrupole test bodies is studied in the context of the Kerr geometry. For an explicit quadrupole model, which includes spin and tidal interactions, the motion in the equatorial plane is characterized by an effective potential and by the binding energy. We compare our findings to recent results for the conservative part of the self-force of bodies in extreme mass ratio situations. Possible implications for gravitational wave physics are outlined.

Figures (6)

• Figure 1: Energy $E=U_\pm$, in units of $\underline{m}$, as a function of the radius $r$, in units of $M$.
• Figure 2: Plots illustrating spin and self-force corrections to the binding energy of a multipolar object in Schwarzschild spacetime $\hat{a}_1 = 0$. In \ref{['spinPlot']} and \ref{['spinPlotZoom']} the corrections are normalized by subtracting their dependence on $q$, $\hat{a}_2$, and $C_{ES^2}$, see (\ref{['SFcurve']}), (\ref{['Scurve']}), (\ref{['S2curve']}), and (\ref{['CEScurve']}) [belonging to curves (*), (**), (***), and (****), respectively].
• Figure 3: Binding energy in terms of $\hat{J}$. This plot makes clear that one has to be careful when in comes to the parameterization of the strength of different corrections. In contrast to the $l_c$-parameterization, effects from the internal structure -- in this case the spin -- of the test body, appear more pronounced in the $\hat{J}$-parametrization. The quadrupole curve is not affected.
• Figure 4: Plots illustrating tidal and self-force corrections to the binding energy of a multipolar object in Schwarzschild spacetime $\hat{a}_1 = 0$. In \ref{['tidePlot']} and \ref{['tidePlotZoom']} the corrections are normalized by subtracting their dependence on $q$, $\hat{R}$, $k_2$, and $j_2$, see (\ref{['SFcurve']}), (\ref{['k2curve']}), and (\ref{['j2curve']}) [belonging to curves (*), (**), and (***), respectively].
• Figure 5: Pole-Dipole corrections in Kerr spacetime.