[Paper Review] Black holes and their horizons in semiclassical and modified theories of gravity

Robert B. Mann, Sebastian Murk+1 more|arXiv (Cornell University)|Dec 13, 2021

Black Holes and Theoretical Physics263 references30 citations

TL;DR

This paper investigates the near-horizon geometry of black holes in semiclassical and modified gravity, showing that under two implicit assumptions—finite curvature scalars at apparent horizons and formation in finite asymptotic time—only two dynamic solutions are admissible: evaporating black holes and expanding white holes. These solutions feature timelike horizons with negative energy density firewalls, violating the null energy condition near the outer horizon while satisfying it near the inner horizon, with implications for black hole thermodynamics and information loss.

ABSTRACT

For distant observers black holes are trapped spacetime domains bounded by apparent horizons. We review properties of the near-horizon geometry emphasizing the consequences of two common implicit assumptions of semiclassical physics. The first is a consequence of the cosmic censorship conjecture, namely that curvature scalars are finite at apparent horizons. The second is that horizons form in finite asymptotic time (i.e. according to distant observers), a property implicitly assumed in conventional descriptions of black hole formation and evaporation. Taking these as the only requirements within the semiclassical framework, we find that in spherical symmetry only two classes of dynamic solutions are admissible, both describing evaporating black holes and expanding white holes. We review their properties and present the implications. The null energy condition is violated in the vicinity of the outer horizon and satisfied in the vicinity of the inner apparent/anti-trapping horizon. Apparent and anti-trapping horizons are timelike surfaces of intermediately singular behavior, which manifests itself in negative energy density firewalls. These and other properties are also present in axially symmetric solutions. Different generalizations of surface gravity to dynamic spacetimes are discordant and do not match the semiclassical results. We conclude by discussing signatures of these models and implications for the identification of observed ultra-compact objects.

Motivation & Objective

• To understand the physical properties of black hole horizons in semiclassical gravity under minimal assumptions.
• To identify the only two dynamic solutions consistent with finite curvature scalars at apparent horizons and finite asymptotic formation time.
• To analyze the energy conditions, firewalls, and surface gravity in these solutions.
• To explore implications for the black hole information paradox and the nature of observed ultra-compact objects.
• To extend results to axially symmetric spacetimes like Kerr–Vaidya metrics.

Proposed method

• Derives solutions to the Einstein equations in spherical symmetry under the assumption of finite curvature scalars at apparent horizons and finite formation time.
• Applies the Vaidya metric framework to model dynamical black hole and white hole spacetimes with outgoing/incoming radiation.
• Uses orthonormal tetrad formalism and Newman–Penrose spin coefficients to analyze the energy-momentum tensor (EMT) near horizons.
• Classifies the EMT using Lorentz-invariant eigenvalues to identify type III behavior and singularities.
• Analyzes the thin shell formalism to model collapsing shells and their horizon formation dynamics.
• Extends results to axially symmetric spacetimes using the Kerr–Vaidya metric and studies its EMT structure and eigenvalues.

Experimental results

Research questions

• RQ1 What are the only two dynamic solutions consistent with finite curvature scalars at apparent horizons and finite asymptotic formation time in spherical symmetry?
• RQ2 How do energy conditions behave near the outer (apparent) and inner (anti-trapping) horizons in these solutions?
• RQ3 What is the nature of the singularity structure and energy density profile near the horizons, particularly regarding firewalls?
• RQ4 How do different definitions of surface gravity in dynamic spacetimes compare with semiclassical results?
• RQ5 What are the observational signatures of these solutions, and how do they relate to observed ultra-compact objects?

Key findings

• Only two classes of dynamic solutions are admissible under the two core assumptions: evaporating black holes and expanding white holes.
• The null energy condition is violated near the outer apparent horizon but satisfied near the inner anti-trapping horizon, indicating negative energy density firewalls.
• Apparent and anti-trapping horizons are timelike surfaces with intermediate singular behavior, characterized by divergent energy-momentum tensor components.
• The energy-momentum tensor near the horizon exhibits type III behavior with zero Lorentz-invariant eigenvalues, indicating a special class of singularities.
• Different definitions of surface gravity in dynamic spacetimes are discordant and fail to match semiclassical expectations.
• The results extend to axially symmetric spacetimes, including the Kerr–Vaidya metric, where the EMT is also of type III with vanishing eigenvalues.