Classical static final state of collapse with supertranslation memory

TL;DR

The paper argues that the classical final state of gravitational collapse is not necessarily Schwarzschild but can carry a finite, angle-dependent supertranslation memory, encoded by a nontrivial asymptotic supertranslation field C. It derives an angle-dependent energy conservation law linking the final C to collapse fluxes and constructs a static Schwarzschild-like final state obtained by a finite BMS diffeomorphism, complete with the bulk realization of the supertranslation field and associated charges. The analysis introduces the concept of a supertranslation horizon and a cosmic censorship bound that constrains the possible memory; it also shows that global, beyond-patch measurements (via integrated superrotation charges) are required to detect the memory. Overall, the work provides a classical mechanism by which black hole final states can retain memory of their collapse history, with clear avenues for numerical and astrophysical testing.

Abstract

The Kerr metric models the final classical black hole state after gravitational collapse of matter and radiation. Any stationary metric which is close to the Kerr metric has been proven to be diffeomorphic to it. Now, finite supertranslation diffeomorphisms are symmetries which map solutions to inequivalent solutions as such diffeomorphisms generate conserved superrotation charges. The final state of gravitational collapse is therefore parameterized by its mass, angular momentum and supertranslation field, signaled by its conserved superrotation charges. In this paper, we first derive the angle-dependent energy conservation law relating the asymptotic value of the supertranslation field of the final state to the details of the collapse and subsequent evolution of the system. We then generate the static solution with an asymptotic supertranslation field and we study some of its properties. Up to a caveat, the deviation from the Schwarzschild metric could therefore be predicted on a case-by-case basis from accurate modeling of the angular dependence of the ingoing and outgoing energy fluxes leading to the final state.

Figures (5)

• Figure 1: Isometric embedding of the supertranslation horizon $\rho = \rho_{SH}(\theta,\phi)$ in 3-dimensional Euclidean space $ds_{(3)}^2 = dx_s^2+dy_s^2+dz_s^2$ as defined from \ref{['eq:89']} and \ref{['defrhoD']}. The supertranslation field is chosen to be the lowest non-trivial axisymmetric $l=2$, $m=0$ spherical harmonic $C(\theta,\phi)=Y_{2,0}(\theta,\phi)$.
• Figure 2: Isometric embedding of the supertranslation horizon in the case $C(\theta,\phi)=Y_{2,1}(\theta,\phi)-Y_{2,-1}(\theta,\phi)$.
• Figure 3: Isometric embedding of the supertranslation horizon in the case $C(\theta,\phi)=-i(Y_{2,1}(\theta,\phi)+Y_{2,-1}(\theta,\phi))$.
• Figure 4: Relative positions of the supertranslation horizon $\rho_{SH}$ and the Killing horizon $\rho_H$ as a function of the spherical angles $(\theta,\phi)$. On the left figure, $C(\theta,\phi)=\frac{M}{6}(3\cos^2\theta-1)$. This corresponds to the toy model \ref{['toy1']} with $\alpha=1$. The Killing horizon is reached close to the north and south poles but the supertranslation horizon is reached first close to the equator. On the right figure, $C(\theta,\phi)= -\frac{M}{12} (3\cos^2\theta-1)$. This corresponds to the toy model \ref{['toy1']} with $\alpha=-\frac{1}{2}$. The supertranslation horizon is entirely shielded by the Killing horizon.
• Figure 5: Relative positions of the supertranslation horizon and the Killing horizon. On the left figure: $C(\theta,\phi)=\frac{M}{12}(3\cos^2\theta-1)$ ($\alpha = \frac{1}{2}$ in \ref{['toy1']} ). The supertranslation horizon is entirely shielded by the Killing horizon. On the right figure: $C(\theta,\phi)=\frac{M}{6} \sin 2\theta \cos\phi$. This corresponds to the toy model \ref{['toy2']} with $\alpha=1$. Depending on the angle, one first encounters the Killing horizon or the supertranslation horizon.