[Paper Review] Some remarks on non-singular spherically symmetric space-times

Lorenzo Sebastiani, Sergio Zerbini|arXiv (Cornell University)|Jun 8, 2022

Cosmology and Gravitation Theories4 citations

TL;DR

This paper investigates non-singular spherically symmetric space-times, focusing on regular black holes with de Sitter cores and black bounce solutions that avoid Cauchy horizons. It applies the Sakarov criterion to identify curvature-invariant bounded solutions and explores generalized Painlevé gauges to analyze Hawking radiation, demonstrating that the Hawking temperature remains invariant under metric reparameterizations, thus providing a robust framework for singularity-free black hole models beyond general relativity.

ABSTRACT

A short review on spherically symmetric static regular black holes and spherically symmetric non singular cosmological space-time is presented. Several models of regular black holes, including new ones, are considered. First, a large class of regular black holes having an inner de Sitter core with the related issue of Cauchy horizon is investigated. Then, black bounce space-times, where the Cauchy horizon and therefore the related instabilities are absent, are discussed as valid alternatives of regular black holes with inner de Sitter core. Friedman-Lemaitre-Robertson-Walker space-times admitting regular bounce solutions are also discussed. In the general analysis concerning the presence or absence of singularities in the equations of motion, the role of a theorem due to Osgood is stressed.

Motivation & Objective

• To analyze spherically symmetric static regular black holes and non-singular cosmological models as alternatives to singular solutions in general relativity.
• To investigate the role of inner de Sitter cores in regular black holes and the associated Cauchy horizon instabilities.
• To propose black bounce space-times as viable alternatives that eliminate Cauchy horizons while preserving asymptotic flatness and regularity.
• To apply the Sakarov criterion to identify solutions with bounded curvature invariants and their covariant derivatives.
• To generalize the Painlevé gauge for static black holes and wormholes to study Hawking radiation in a coordinate-singularity-free manner.

Proposed method

• Uses the Kodama-Hayward formalism to define trapping horizons via the scalar χ(xᵃ) = γᵃᵇ∂ₐr∂ᵦr, with χ=0 at the horizon.
• Applies the Sakarov criterion to select regular black hole solutions where all curvature invariants and their covariant derivatives remain bounded.
• Introduces a generalized Painlevé gauge with arbitrary positive function g(r) to avoid coordinate singularities, ensuring metric regularity across horizons.
• Derives the Hawking radiation spectrum via the covariant Hamilton-Jacobi tunneling method, computing the imaginary part of the action I to obtain the tunneling probability Γ = e⁻²ᴵᵐᴵ.
• Utilizes the metric transformation dT = dt + √[(1−B(r)g(r))/(A(r)B(r))] dr to extend the Painlevé gauge to cases where standard gauge fails.
• Demonstrates that the Hawking temperature TH = √[A′(rₕ)B′(rₕ)]/(4π) is independent of the choice of g(r), ensuring physical consistency.

Experimental results

Research questions

• RQ1 Can regular black hole solutions with an inner de Sitter core be constructed such that curvature invariants remain bounded everywhere?
• RQ2 What are the implications of the Cauchy horizon for the stability and physical viability of regular black holes with de Sitter cores?
• RQ3 How do black bounce space-times, which lack a Cauchy horizon, compare to standard regular black holes in terms of metric regularity and physical consistency?
• RQ4 To what extent does the generalized Painlevé gauge preserve the physical predictions of Hawking radiation in singular-free black hole models?
• RQ5 Does the Hawking temperature remain invariant under metric reparameterizations in the generalized Painlevé gauge, ensuring robustness of the radiation spectrum?

Key findings

• The Sakarov criterion successfully identifies a class of regular black hole solutions where all curvature invariants and their covariant derivatives are bounded, ensuring physical regularity.
• Black bounce space-times, which avoid the Cauchy horizon, provide a stable alternative to regular black holes with de Sitter cores, eliminating associated instabilities.
• The generalized Painlevé gauge allows for a well-defined metric formulation across horizons even when the standard gauge fails, particularly in Reissner-Nordström and Schwarzschild-AdS-like solutions.
• The Hawking temperature derived via the Hamilton-Jacobi tunneling method is independent of the arbitrary function g(r), confirming the robustness of the radiation spectrum.
• The imaginary part of the action Im[I] = 2πE / √[A′(rₕ)B′(rₕ)] leads to a consistent Hawking radiation formula Γ = e⁻⁴ᵖᴱ⁄√[ᴬ′ᴮ′], valid across different regular metric formulations.
• All physical invariants, including the Hawking temperature, are independent of the gauge choice g(r), confirming the physical consistency of the tunneling approach in non-singular space-times.