Hawking Temperature as the Total Gauss-Bonnet Invariant of the Region Outside a Black Hole (2208.12993v2)

Abstract: We provide two novel ways to compute the surface gravity ($\kappa$) and the Hawking temperature $(T_{H})$ of a stationary black hole: in the first method $T_{H}$ is given as the three-volume integral of the Gauss-Bonnet invariant (or the Kretschmann scalar for Ricci-flat metrics) in the total region outside the event horizon; in the second method it is given as the surface integral of the Riemann tensor contracted with the covariant derivative of a Killing vector on the event horizon. To arrive at these new formulas for the black hole temperature (and the related surface gravity), we first construct a new differential geometric identity using the Bianchi identity and an antisymmetric rank-$2$ tensor, valid for spacetimes with at least one Killing vector field. The Gauss-Bonnet tensor and the Gauss-Bonnet scalar play a particular role in this geometric identity. We calculate the surface gravity and the Hawking temperature of the Kerr and the extremal Reissner-Nordstr\"om holes as examples.

• The paper presents novel methods linking Gauss-Bonnet invariants with black hole thermodynamics to compute surface gravity and Hawking temperature.
• It employs both volume and surface integrals to derive equivalent expressions, validated through Kerr and extremal Reissner-Nordström examples.
• The findings offer versatile tools for extending black hole thermodynamics to alternative gravitational theories and quantum gravity scenarios.

Novel Formulations to Compute Hawking Temperature and Surface Gravity of Black Holes

Introduction

In the domain of theoretical physics, the paper of black holes serves as a compelling intersection of quantum mechanics, thermodynamics, and general relativity. Within this context, the surface gravity ($κ$) and the Hawking temperature ($T_H$) of black holes present intriguing physical quantities that bridge thermodynamic properties with geometric characteristics of spacetime. The research conducted by Emel Altas extends our understanding by introducing innovative methodologies to compute these quantities, offering insights into their geometric underpinnings. The foundation of these calculations is set on the differential geometric identities involving the Gauss-Bonnet invariant and the Riemann tensor, applicable in a general n-dimensional spacetime framework.

New Geometric Identities and Calculations

At the heart of this paper lies the development of new geometric identities utilizing the Gauss-Bonnet scalar and tensor. These identities leverage the Bianchi identity and an antisymmetric rank-2 tensor that is valid in spacetimes hosting at least one Killing vector field. A remarkable feature of these formulations is their ability to express the surface gravity ($κ$) in two distinct yet equivalent manners:

• Through a three-volume integral: Here, $κ$ manifests as a volume integral of the Kretschmann scalar or its equivalent expressions depending on the spacetime's curvature properties, specifically allocated in the domain exterior to a black hole's event horizon.
• Via a surface integral: This formulation pivots on a surface integral at the event horizon, incorporating the Riemann tensor and the covariant derivative of a Killing vector, outlining a direct geometrical avenue to $κ$ and subsequently, $T_H$.

These methods reinforce a geometric perspective, implying that any gravitational theory can, in principle, adopt these identities to derive $κ$ and $T_H$, with the specifics of a theory entering the equations subsequently.

Applications and Insights

The paper meticulously applies these novel formulations to the Kerr and the extremal Reissner-Nordström black holes as pertinent examples, showcasing the utility and accuracy of the proposed methods. For these cases, the derived $κ$ and $T_H$ not only align with existing paradigms but also encapsulate the inherent geometric nature of these quantities. Specifically, for the Kerr black hole, the computation through the volume integral of the Kretschmann scalar outside the event horizon yields a result that elegantly converges with the known value, illustrating the method's efficacy. Additionally, the extremal Reissner-Nordström metric serves to demonstrate the applicability of these formulas in a charged, non-vacuum context, further broadening the scope of this framework.

Theoretical and Practical Implications

The implications of Altas's work are far-reaching, both from theoretical and practical standpoints. Theoretically, this research enriches our understanding of black hole thermodynamics by anchoring the computations of $κ$ and $T_H$ in the spacetime geometry outside the event horizon. This novel viewpoint potentially paves the way for deeper insights into the thermodynamic nature of black holes, especially when extended beyond the field of classical gravity to incorporate quantum effects.

Practically, these formulations offer a robust and versatile toolset for exploring black hole properties in various gravitational theories. By grounding the calculations in geometric identities, this approach allows for a more unified understanding of black hole thermodynamics across different theoretical frameworks.

Future Directions

Looking ahead, the methodologies developed by Altas open numerous avenues for further research. One intriguing prospect is the extension of these geometric identities to alternative theories of gravity, potentially offering new perspectives on black hole thermodynamics in these contexts. Moreover, the quantum gravity implications of tying surface gravity and Hawking temperature to spacetime geometry merit exploration, particularly in understanding the microscopic origins of black hole entropy and temperature.

Conclusion

In conclusion, the work presented by Emel Altas introduces groundbreaking methodologies for calculating the surface gravity and Hawking temperature of black holes, deeply rooted in the geometry of spacetime. These developments not only enhance our comprehension of black hole thermodynamics but also provide a versatile framework for exploring these enigmatic objects across different gravitational theories. As we explore the mysteries of black holes, the geometric pathways charted by these novel formulations promise to unfold new horizons in our quest to understand the fundamental laws governing the universe.