[Paper Review] Quantum gravity of Kerr-Schild spacetimes and the logarithmic correction to Schwarzschild black hole entropy

Basem Kamal El-Menoufi|arXiv (Cornell University)|Nov 27, 2015

Black Holes and Theoretical Physics3 references4 citations

TL;DR

This paper derives the one-loop quantum gravity effective action for Kerr-Schild spacetimes using non-local heat kernel techniques and dimensional transmutation, revealing a logarithmic correction to the Bekenstein-Hawking entropy of Schwarzschild black holes. The correction arises from long-distance quantum fluctuations of massless particles and exhibits a non-trivial interplay between UV and IR physics in the effective field theory framework.

ABSTRACT

In the context of effective field theory, we consider quantum gravity with minimally coupled massless particles. Fixing the background geometry to be of the Kerr-Schild type, we fully determine the one-loop effective action of the theory whose finite non-local part is induced by the long-distance portion of quantum loops. This is accomplished using the non-local expansion of the heat kernel in addition to a non-linear completion technique through which the effective action is expanded in gravitational curvatures. Via Euclidean methods, we identify a logarithmic correction to the Bekenstein-Hawking entropy of Schwarzschild black hole. Using dimensional transmutation the result is shown to exhibit an interesting interplay between the UV and IR properties of quantum gravity.

Motivation & Objective

• To systematically compute the one-loop quantum gravity effective action in Kerr-Schild spacetimes using non-local heat kernel expansions.
• To identify finite, non-local quantum corrections induced by long-distance propagation of massless particles in curved spacetime.
• To derive the logarithmic correction to the Bekenstein-Hawking entropy of Schwarzschild black holes from quantum gravity effects.
• To explore the interplay between UV and IR physics in quantum gravity through dimensional transmutation in the effective action.

Proposed method

• Employing the non-local expansion of the heat kernel in curved spacetime to compute the one-loop effective action for minimally coupled massless particles.
• Using a non-linear completion technique to expand the effective action in powers of gravitational curvature invariants.
• Applying Euclidean path integral methods to compute the black hole partition function and extract entropy corrections.
• Utilizing dimensional transmutation to relate the UV cutoff scale to the IR behavior of the effective action.
• Deriving form factors via integrals over proper-time and momentum-space representations to express non-local terms.
• Expanding curvature invariants in momentum space using tensor integral identities and metric perturbations in the Kerr-Schild formalism.

Experimental results

Research questions

• RQ1 What is the structure of the one-loop quantum gravity effective action in Kerr-Schild spacetimes?
• RQ2 How do long-distance quantum fluctuations of massless particles modify the classical Einstein-Hilbert action in curved spacetime?
• RQ3 What is the origin and magnitude of the logarithmic correction to the Bekenstein-Hawking entropy in Schwarzschild black holes?
• RQ4 How does dimensional transmutation connect UV and IR physics in the quantum gravity effective action?
• RQ5 Can non-local corrections in the effective action be systematically computed and related to thermodynamic quantities like black hole entropy?

Key findings

• The one-loop effective action for quantum gravity in Kerr-Schild spacetimes contains a finite, non-local contribution arising from long-distance quantum loops of massless particles.
• The non-local structure is derived using the heat kernel expansion with proper-time ordering and form factor identities.
• A logarithmic correction to the Bekenstein-Hawking entropy of the Schwarzschild black hole is identified via Euclidean path integral methods.
• The correction is parameter-free and arises purely from quantum effects of massless minimally coupled fields in the effective field theory framework.
• Dimensional transmutation reveals a non-trivial interplay between UV and IR scales in the quantum gravity effective action.
• The result is consistent with the expectation that quantum gravity induces universal, low-energy corrections independent of UV completion.