A Quasi-local, Functional Analytic Detection Method for Stationary Limit Surfaces of Black Hole Spacetimes

TL;DR

This paper introduces a novel quasi-local, functional analytic method to detect and characterize stationary limit surfaces of black hole spacetimes by analyzing Hamiltonian transitions, offering a new approach distinct from traditional geometric invariants.

Contribution

The paper develops a new method based on ellipticity-hyperbolicity transitions of Hamiltonians to locate stationary limit surfaces, applicable to various black hole spacetimes and related to event horizon detection.

Findings

Successfully located stationary limit surfaces in Kerr-Newman, Schwarzschild-de Sitter, and Taub-NUT spacetimes.

Proposed a quasi-local method that differs from scalar polynomial and Cartan invariants.

Extended the method to serve as a quasi-local event horizon detector.

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.