A Maximum Entropy Conjecture for Black Hole Mergers

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.

💡 Research Summary

The paper investigates whether a simple thermodynamic principle can determine the final black‑hole remnant produced by a binary black‑hole merger. While numerical relativity (NR) can predict the final mass and spin of the remnant with exquisite precision, the underlying physical rationale remains obscure. The authors propose that the merger outcome may be governed by an entropy‑maximization principle that is already encoded in the dynamics of general relativity.

The analysis proceeds in two stages. First, the authors use post‑Newtonian (PN) theory for an equal‑mass, non‑spinning binary, expanding the binding energy and orbital angular momentum up to fourth‑order (4PN). They introduce a dimensionless angular‑momentum parameter (j = J/(M_{\rm tot}^2 \nu)) (with (\nu) the symmetric mass ratio) and obtain an explicit expression for the total mass (M(j)) as a series in (j). This expression is valid in the weak‑field, slow‑motion regime but is extrapolated to higher frequencies where the merger occurs.

Second, they map each instantaneous pair ((M(j),J(j))) onto a hypothetical Kerr black hole and compute its Bekenstein–Hawking entropy (S_{\rm Kerr}=A(M,J)/4\hbar), where the horizon area (A) is the usual Kerr formula. As the binary inspirals, (j) decreases, and the corresponding Kerr entropy first rises, reaches a maximum at a particular value (j^\ast), and then declines. Remarkably, the mass (M(j^\ast)) and spin (\chi^\ast = J(j^\ast)/M(j^\ast)^2) are extremely close to the NR‑determined final mass (M_f) and spin (\chi_f). Table I and Figure 1 show that at 4PN the predicted spin (\chi^\ast\approx0.699) differs from the NR value (\chi_f=0.68646) by less than two percent, and similar agreement holds at lower PN orders.

To move beyond a purely kinematic mapping, the authors incorporate the balance laws for energy and angular momentum carried away by gravitational waves. Using Bondi fluxes (F_E(u)=-\dot M(u)) and (F_J(u)=-\dot J(u)), they rewrite the evolution in terms of the angular momentum as a time parameter, yielding an integral relation for the final mass and spin: \