[Paper Review] Hawking Radiation via Gauss-Bonnet Theorem

\"Ovg\"un, A., İzzet Sakallı|arXiv (Cornell University)|Feb 12, 2019

Black Holes and Theoretical PhysicsPhysics and Astronomy62 references41 citations

TL;DR

This paper proposes a topological method to compute Hawking temperature using the Gauss-Bonnet theorem and Euler characteristic in 2D Euclidean spacetime, demonstrating its consistency across diverse black hole solutions including dyonic Reissner-Nordström, Schwarzschild-electromagnetic, quantum-corrected, bumblebee gravity, and linear dilaton black holes. The method yields exact Hawking temperatures matching standard results, confirming its robustness and universality for spherically symmetric black holes.

ABSTRACT

In this paper, we apply the topological method to the various black holes to derive their Hawking temperature. We show that the this method can easily be employed to compute the Hawking temperature of black holes having spherically symmetric topology. Therefore, we conclude that the topological method provides a consistent formula to achieve the Hawking temperature.

Motivation & Objective

• To establish a topological framework based on the Gauss-Bonnet theorem for deriving Hawking temperature without relying on traditional quantum field theory.
• To extend the applicability of the topological method to various black hole solutions with spherically symmetric topology.
• To validate the method across multiple black hole types, including charged, quantum-corrected, and modified gravity models.
• To demonstrate that the Euler characteristic and Ricci scalar in 2D Euclidean spacetime encode the full thermodynamic information of 4D black holes.
• To provide a unified, geometric approach to Hawking temperature that is consistent with established results across different black hole families.

Proposed method

• Apply Wick rotation to 4D spherically symmetric black hole metrics to obtain 2D Euclidean metrics, transforming Lorentzian signature to Riemannian.
• Use the Gauss-Bonnet theorem in 2D Euclidean spacetime, where the topological invariant is the Euler characteristic χ = 1 for R² topology.
• Compute the Ricci scalar R from the 2D metric, which depends only on the radial coordinate r.
• Apply the topological formula: TH = (1/4π) ∫_rh √g R dr, with √g as the square root of the metric determinant.
• Evaluate the radial integral over the event horizon rh, using the appropriate f(r) for each black hole type.
• Simplify physical constants to unity for analytical clarity, then recover physical units in final results.

Experimental results

Research questions

• RQ1 Can the Gauss-Bonnet theorem and Euler characteristic provide a consistent and universal method to compute Hawking temperature across diverse black hole solutions?
• RQ2 Does the topological method reproduce the standard Hawking temperature for dyonic Reissner-Nordström black holes?
• RQ3 How does the method perform for black holes with quantum corrections, such as the Reissner-Nordström-like black hole with quantum potential?
• RQ4 Can the method be extended to black holes in modified gravity theories, such as bumblebee gravity or linear dilaton gravity?
• RQ5 What is the role of the 2D Euclidean geometry in preserving the thermodynamic properties of 4D black holes?

Key findings

• The topological method successfully computes the Hawking temperature of the dyonic Reissner-Nordström black hole as TH = √(M² - Q² - P²) / [2π (M + √(M² - Q² - P²))²], matching the standard result.
• For the Schwarzschild-electromagnetic black hole, the method yields TH = (2a) / [4π M (a + 1)²], which exactly reproduces the known Hawking temperature.
• The quantum-corrected Reissner-Nordström-like black hole has a temperature TH = √(M² - ℏη) / [2π (√(M² - ℏη) + M)²], consistent with prior work.
• In the bumblebee gravity model, the method gives TH = 1 / (8Mπ √(1 + l)), matching the standard Hawking temperature for this Schwarzschild-like solution.
• For the linear dilaton black hole, the temperature is TH = 1 / (4π r₀), independent of mass b, confirming isothermal radiation as predicted by earlier studies.
• The method consistently reproduces known results across all tested black hole types, validating its universality and reliability for spherically symmetric spacetimes.