A Quasi-local, Functional Analytic Detection Method for Stationary Limit Surfaces of Black Hole Spacetimes
Christian Röken
TL;DR
This paper tackles the challenge of locally identifying stationary limit surfaces in black hole spacetimes by introducing a quasi-local, functional-analytic detector based on ellipticity-hyperbolicity transitions of the Dirac, Klein–Gordon, Maxwell, and Fierz–Pauli Hamiltonians on spacelike slices. By computing the principal-symbol determinants, the authors show that these transitions occur precisely at stationary limit surfaces, enabling a slice-local criterion that is coordinate-invariant on time-independent sections. They derive explicit determinant expressions for the Dirac and Klein–Gordon Hamiltonians and apply the method to Kerr–Newman, Schwarzschild–de Sitter, and Taub–NUT spacetimes, locating outer/inner ergosurfaces and horizons, respectively, in a quasi-local framework. For spacetimes with static regions, the approach provides a quasi-local event-horizon detector and introduces a relational notion of black-hole entropy $S_{\text{BH}} = \mathscr{C}_0 \max(\det(\sigma_{\mathcal{H}})^{-1}(0))^2$, which with $\mathscr{C}_0=\pi$ recovers the familiar $S_{\text{BH}} = A/4$, linking horizon localization to entropy in a novel, computationally efficient manner.
Abstract
We present a quasi-local, functional analytic method to locate and invariantly characterize the stationary limit surfaces of black hole spacetimes with stationary regions. The method is based on ellipticity-hyperbolicity transitions of the Dirac, Klein-Gordon, Maxwell, and Fierz-Pauli Hamiltonians defined on spacelike hypersurfaces of such black hole spacetimes, which occur only at the locations of stationary limit surfaces and can be ascertained from the behaviors of the principal symbols of the Hamiltonians. Therefore, since it relates solely to the effects that stationary limit surfaces have on the time evolutions of the corresponding elementary fermions and bosons, this method is profoundly different from the usual detection procedures that employ either scalar polynomial curvature invariants or Cartan invariants, which, in contrast, make use of the local geometries of the underlying black hole spacetimes. As an application, we determine the locations of the stationary limit surfaces of the Kerr-Newman, Schwarzschild-de Sitter, and Taub-NUT black hole spacetimes. Finally, we show that for black hole spacetimes with static regions, our functional analytic method serves as a quasi-local event horizon detector and gives rise to relational concepts of event horizons and black hole entropy.