A Maximum Entropy Conjecture for Black Hole Mergers

Abstract

The final state of a binary black hole merger is predicted with high precision by numerical relativity, but could there be a simple thermodynamic principle within general relativity that governs the selection of the remnant? Using post-Newtonian relations between the mass M (including the binding energy) and angular momentum J of quasi-circular, nonspinning binaries, we uncover a puzzling result: When the binary's instantaneous M and J are mapped to those of a hypothetical Kerr black hole, the corresponding entropy exhibits a maximum during the evolution. This maximum occurs at values of M and J strikingly close to those of the final remnant predicted by numerical relativity. Consistent behavior is observed when using the relation between M and J obtained from numerical relativity evolution. Although this procedure is somewhat ad hoc, the agreement between the masses and spins of the final state obtained from numerical relativity and the results of this maximum entropy procedure is remarkable, with agreement to within a few percent when using either post-Newtonian or numerical relativity results for M and J. These findings allow us to propose an entropy maximization conjecture for binary black hole mergers, hinting that thermodynamic principles may govern the selection of the final black hole state.

Figures (4)

• Figure 1: Entropy $S_\mathrm{Kerr}(J)$ of the remnant, normalized to $4\pi M_\mathrm{tot}^2/\hbar,$ as a function of the normalized angular momentum $J/M_\mathrm{tot}^2 = j\nu,$ for an equal-mass binary. At each instant of PN evolution, one may consider a hypothetical remnant inheriting the angular momentum and energy of the binary. While the PN formalism does not, by itself, determine the ultimate outcome of the evolution, the maximum-entropy conjecture provides a natural criterion for its termination: For a given PN order, the conjecture dictates that we should end the sequence of circular orbits at the value of angular momentum that maximizes the entropy of the remnant, $J_* = \mathop{\mathrm{argmax}}\limits_J S_\mathrm{Kerr}(J)$, marked with a star for each PN curve. Reference values of $J/M_\mathrm{tot}^2$ are displayed as vertical lines, corresponding to (black) the value of final angular momentum predicted by numerical relativity $J_\mathrm{NR}=\chi_{\mathrm{NR}}M_{\mathrm{NR}}^2$Scheel:2008rj, and (green) the value of angular momentum for which the specific heat $C_J$ diverges, $J_\mathrm{NR}=\chi_{\mathrm{c}}\,M_{\mathrm{NR}}^2$, for a black hole of mass $M_\mathrm{NR}=0.9516\, M_\mathrm{tot}$, where $\chi_c = \sqrt{2\sqrt3-3}\approx 0.681$Davies:1977bgr.
• Figure 2: The spin $\chi_f$ (left) and mass $M_f$ (right) of the remnant black hole obtained from the integral form of the maximum entropy conjecture applied using the PN approximation at different PN orders. Also shown as a dashed line are the predictions of NR simulations. The $4$PN remnant spin is pretty close to the one predicted by full general relativity.
• Figure 3: Left: The entropy versus dimensionless spin of a Kerr black hole with the same energy and angular momentum as that at a particular point of a numerical relativity binary black hole simulation, along with the $4$PN prediction, all shown for nonspinning systems with mass ratios $q=M_1/M_2\ge 1$ of up to $8$ (SXS:BBH:3617, 1167, 2499, and 2707). The merger point corresponds to the peak of the amplitude of the dominant quadrupolar mode of the waveform, as in Nagar:2015xqa. Right: The scaled binding energy versus angular momentum curve from the NR simulation for the equal-mass nonspinning case, with the merger and final state marked, compared with the PN predictions at the available orders.
• Figure 4: The final mass (top panel) and dimensionless spin (bottom panel) of the NR simulations versus the values one would obtain at the point where a Kerr black hole would have the maximum entropy, along with the identity map (dashed line). The bottom panel also shows the fractional difference in entropy between the maximum Kerr entropy and that of the final state. Colors give the binary's mass ratio.