[Paper Review] Entropy and topology for manifolds with boundaries

Stefano Liberati, Giuseppe Pollifrone|ArXiv.org|Sep 15, 1995

Black Holes and Theoretical Physics1 references4 citations

TL;DR

This paper establishes a direct link between topology and thermodynamics in manifolds with boundaries by generalizing the Bekenstein-Hawking entropy formula to $ S = \chi A / 8 $, where $ \chi $ is the Euler characteristic and $ A $ is the boundary area. The formula correctly reproduces zero entropy for extreme black holes—where the standard formula fails—unifying black hole entropy across all cases via topological invariants derived from the Gauss-Bonnet theorem.

ABSTRACT

In this work a deep relation between topology and thermodynamical features of manifolds with boundaries is shown. The expression for the Euler characteristic, through the Gauss- Bonnet integral, and the one for the entropy of gravitational instantons are proposed in a form which makes the relation between them self-evident. A generalization of Bekenstein-Hawking formula, in which entropy and Euler characteristic are related in the form $S=χA/8$, is obtained. This formula reproduces the correct result for extreme black hole, where the Bekenstein-Hawking one fails ($S=0$ but $A eq 0$). In such a way it recovers a unified picture for the black hole entropy law. Moreover, it is proved that such a relation can be generalized to a wide class of manifolds with boundaries which are described by spherically symmetric metrics (e.g. Schwarzschild, Reissner-Nordström, static de Sitter).

Motivation & Objective

• To establish a fundamental relationship between topological invariants and thermodynamic quantities in manifolds with boundaries.
• To resolve inconsistencies in the Bekenstein-Hawking entropy formula for extreme black holes, where entropy is zero despite non-zero area.
• To generalize the entropy-area law using the Euler characteristic derived from the Gauss-Bonnet integral.
• To demonstrate the validity of the generalized formula across spherically symmetric spacetimes, including Schwarzschild, Reissner-Nordström, and static de Sitter metrics.
• To unify the description of black hole entropy through topological invariants, providing a geometrically consistent framework.

Proposed method

• Derives the Euler characteristic $ \chi $ using the Gauss-Bonnet theorem applied to manifolds with boundaries.
• Expresses gravitational instanton entropy in a form that explicitly reveals its dependence on $ \chi $ and boundary area $ A $.
• Proposes a generalized entropy formula $ S = \chi A / 8 $, replacing the standard $ S = A/4 $ in the Bekenstein-Hawking formula.
• Applies the formula to spherically symmetric metrics, verifying consistency across different black hole and cosmological solutions.
• Uses differential geometry and topological field theory techniques to relate curvature invariants to thermodynamic quantities.
• Demonstrates that the new formula reduces to the standard Bekenstein-Hawking result for non-extremal cases and correctly yields $ S = 0 $ for extreme black holes.

Experimental results

Research questions

• RQ1 How can the Euler characteristic of a manifold with boundary be related to its thermodynamic entropy?
• RQ2 Why does the standard Bekenstein-Hawking formula fail for extreme black holes, and how can it be corrected?
• RQ3 Can a unified entropy formula be derived that applies consistently across all types of black holes, including extremal ones?
• RQ4 What role does the Gauss-Bonnet integral play in connecting topology and gravitational thermodynamics?
• RQ5 Is the generalized entropy formula $ S = \chi A / 8 $ valid for a broad class of spherically symmetric spacetimes?

Key findings

• The generalized entropy formula $ S = \chi A / 8 $ correctly predicts zero entropy for extreme black holes, where $ A \neq 0 $, resolving a key inconsistency in the standard Bekenstein-Hawking formula.
• The Euler characteristic $ \chi $, derived from the Gauss-Bonnet integral, provides a topological invariant that directly determines entropy in the presence of boundaries.
• The formula reproduces the standard Bekenstein-Hawking result $ S = A/4 $ for non-extremal black holes, confirming consistency in the non-extremal limit.
• The relation $ S = \chi A / 8 $ is validated for a wide class of spherically symmetric spacetimes, including Schwarzschild, Reissner-Nordström, and static de Sitter metrics.
• The topological origin of entropy is demonstrated through the explicit geometric derivation of $ \chi $, linking curvature invariants to thermodynamic quantities.
• The work provides a unified framework where entropy is fundamentally tied to the global topology of spacetime, not just local geometry.