Violation of the third law of black hole mechanics in vacuum gravity

TL;DR

The paper demonstrates that in five-dimensional vacuum gravity, one can form an extremal rotating black hole in finite time by gluing together Schwarzschild and extremal Myers-Perry spacetimes, thereby violating the third law of black hole mechanics in vacuum. The authors construct cohomogeneity-2 solutions with $SU(2)\times \mathbb{Z}_4$ symmetry via characteristic gluing on null hypersurfaces, and implement a gluing ansatz using fields $\Phi$, $\mathcal{B}$, and a quasi-local angular momentum $J$, solved numerically with a DG-based approach. They present two solution types: (i) a tiny Schwarzschild black hole absorbing gravitational waves to become an extremal Myers-Perry black hole in finite time, and (ii) the finite-time formation of an extremal black hole from vacuum gravitational waves with no initial black hole, both analyzed for regularity and convergence. This work provides explicit vacuum counterexamples to the third law in 5D, suggesting the law is not universal across matter models and motivating future extensions toward 4D Kerr and broader gluing constructions.

Abstract

We demonstrate numerically the existence of solutions of five-dimensional vacuum gravity describing the formation, in finite time, of an extremal rotating black hole from a pre-existing Schwarzschild black hole. This is the first example of a violation of the third law of black hole mechanics in vacuum gravity and demonstrates that the third law is false independently of any matter model. We also demonstrate the existence of solutions describing the formation, in finite time, of an extremal rotating black hole from vacuum initial data that does not contain a black hole.

Figures (7)

• Figure 1: Penrose diagrams for gluing constructions of type (i) (left) and (ii) (right). The dark regions are exactly isometric to subsets of the Schwarzschild, EMP and Minkowski spacetimes. Light-shaded regions are local solutions obtained by solving characteristic initial/final value problems. The dash-dotted curves are sample Cauchy surfaces. In the left panel, the Schwarzschild region occupies $[U_0,U_f]\times [V_i,0]$; we do not extend it to the singularity at $r=0$. The striped pattern denotes the trapped region, its boundary is the apparent horizon. In the right panel, Cauchy stability ensures the solution in the light shaded region can be extended to the centre of symmetry $r=0$ using an argument in Kehle:2022uvc.
• Figure 2: A $k=2$ gluing solution of type (i). Left: $\Phi$ and $\mathcal{B}$, with the red dot marking the EMP limit of $\mathcal{B}$. Right: $r$, $J$ and $m_{\rm H}$. The inset plot on the right panel shows that $J=0$ but $m_{\rm H}>0$ at $V=0$.
• Figure 3: A $k=4$ gluing solution of type (ii).
• Figure 4: Type (i) solution: plots of $U$-derivatives of $\Phi$, the fields $\mathcal{B}^j$, and $\partial_U r$. The $\Phi$ derivatives stay small, while $\mathcal{B}^j$ shows larger variations; red dots indicate EMP matching values.
• Figure 5: Type (ii) solution: plots of $U$-derivatives of $\Phi$ and the fields $\mathcal{B}^j$. The $|\mathcal{B}^j|$ magnitudes are shown on a logarithmic scale and reach very large values; red disks denote EMP matching values.