[Paper Review] Deformations of the Schwarzschild Black Hole

Daniel Grumiller|ArXiv.org|Nov 4, 2003

Noncommutative and Quantum Gravity Theories17 references3 citations

TL;DR

This paper investigates deformations of the Schwarzschild black hole within a 2D dilaton gravity framework, analyzing both Lorentz-symmetry deformations and dynamical evolutions during long-time evaporation. It derives a finite S-matrix for s-wave scattering via coherent sum over virtual black holes and identifies a Planck-scale remnant mass from an attractor solution, suggesting a final 'thunderbolt' emission at M_Planck/2.

ABSTRACT

Due to its large number of symmetries the Schwarzschild Black Hole can be described by a specific two-dimensional dilaton gravity model. After reviewing classical, semi-classical and quantum properties and a brief discussion of virtual black holes deformations are studied: the first part is devoted to deformations of the Lorentz-symmetry, the second part to dynamical deformations and its role for the long time evaporation of the Schwarzschild Black Hole.

Motivation & Objective

• To explore deformations of the Schwarzschild black hole using 2D dilaton gravity as a tractable model for quantum gravity.
• To analyze the role of virtual black holes in scattering amplitudes via path integral quantization of geometry.
• To investigate dynamical deformations during black hole evaporation and their implications for the final state.
• To assess the viability of Lorentz-symmetry deformations in this framework and their physical consistency.
• To derive an evolution equation for deformation parameters and determine the long-term fate of the black hole.

Proposed method

• Formulates the Schwarzschild black hole as a 2D first-order dilaton gravity model with auxiliary fields X and X^a as Lagrange multipliers.
• Applies the first-order formalism to construct Carter-Penrose diagrams and analyze geometric structures, including virtual black hole configurations.
• Performs path integral quantization of geometry, integrating out spacetime degrees of freedom to obtain effective nonlocal vertices and scattering amplitudes.
• Derives the S-matrix for s-wave scattering of massless scalars using a coherent sum over virtual black holes with localized mass and acceleration on light-like cuts.
• Uses the most general consistent deformation framework of 2D BF-theory to derive an evolution equation for deformation parameters under asymptotic and bounded flux conditions.
• Identifies an attractor solution corresponding to flat spacetime, with evaporation ending at a remnant mass of M_Planck/2.

Experimental results

Research questions

• RQ1 Can Lorentz-symmetry deformations in 2D dilaton gravity lead to physically viable black hole geometries?
• RQ2 How does the coherent sum over virtual black holes with localized mass and acceleration yield a finite scattering amplitude?
• RQ3 What is the long-term evolution of the Schwarzschild black hole under consistent dynamical deformations?
• RQ4 Does the evaporation process terminate in a stable remnant, and if so, what is its mass?
• RQ5 How do boundary conditions and asymptotic structure influence the emergence of classical spacetime in the virtual black hole picture?

Key findings

• The scattering amplitude for s-waves is finite and encoded in a scale-independent factor T̃, with forward poles at Π=0, indicating self-energy effects.
• The effective geometry of virtual black holes is nonlocal and localized on light-like cuts, with mass and Rindler acceleration functions m(u,r), a(u,r), d(u,r) of compact support.
• The S-matrix is constructed via a sum over all virtual black hole configurations, enabling a Fock space description at future and past null infinity despite non-classical core geometry.
• Lorentz-symmetry deformations via Mignemi’s model lead to either trivial metrics or pathological non-invariance, rendering them unphysical in this context.
• A dynamical deformation model yields an attractor solution corresponding to flat spacetime, with evaporation ending at a remnant mass of M_Planck/2.
• The final stage of evaporation is characterized by a 'thunderbolt' emission, consistent with the Frolov-Vilkovisky model's conformal diagram structure.