On the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon

TL;DR

This work removes analyticity from key rigidity results for spacetimes with horizons by proving the existence of a smooth Killing field $k^a$ in a one-sided neighborhood of the horizon for smooth electrovac spacetimes. The authors connect Hawking's Killing-horizon theorem and Isenberg–Moncrief's compact Cauchy horizon result by constructing a discrete horizon isometry, unwrapping to Gaussian null coordinates, and extending to a bifurcate null surface to obtain the Killing field without analytic assumptions. The findings reinforce horizon symmetry as a robust feature in the smooth setting, strengthening implications for black hole rigidity, uniqueness, and cosmic censorship in non-analytic spacetimes. Collectively, the results bridge analytic and smooth approaches, showing that horizon-induced Killing fields persist beyond analyticity and informing rigidity arguments in general relativity.

Abstract

We consider smooth electrovac spacetimes which represent either (A) an asymptotically flat, stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics. The black hole event horizon or, respectively, the compact Cauchy horizon of these spacetimes is assumed to be a smooth null hypersurface which is non-degenerate in the sense that its null geodesic generators are geodesically incomplete in one direction. In both cases, it is shown that there exists a Killing vector field in a one-sided neighborhood of the horizon which is normal to the horizon. We thereby generalize theorems of Hawking (for case (A)) and Isenberg and Moncrief (for case (B)) to the non-analytic case.

Theorems & Definitions (21)

• Definition 2.1
• Remark 2.1
• Lemma 2.1
• Remark 2.2
• Proposition 2.1
• Definition 3.1
• Proposition 3.1
• Remark 3.1
• Remark 3.2
• Proposition 4.1