Abstract
This chapter starts with the origin and a brief history of the application of the thermodynamic laws to (static) black holes and its subsequent generalization to Cosmology. The concepts of the generalized second law of thermodynamics, surface gravity and its connection with Hawking temperature, and Bekenstein entropy are carefully explained. Finally, the implications of the modified and the generalized Hawking temperatures are discussed with reference to the cosmological event horizon.
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Notes
- 1.
The event horizon of a BH is the boundary which marks the limits of the BH. Anything (even light) that falls within the event horizon can never escape.
- 2.
The surface gravity can be defined as the local gravitational field strength experienced by a test particle at the surface of an astronomical body. This definition can be extended to include cosmological horizons as well. The surface gravity for a static or a stationary BH is a constant thanks to the zeroth law of BH thermodynamics, while in the case of a dynamic horizon relevant in the context of Cosmology, the surface gravity depends on the radius of the horizon as well as on its time derivative.
- 3.
Rindler horizons and Unruh temperatures are generally associated with accelerated observers in Minkowski spacetime of Special Theory of Relativity.
- 4.
A proof of this important fact can be found in Chap. 3 of the book by V. Faraoni (2015).
- 5.
The reader may go through Chap. 3 of the book by V. Faraoni for a detailed account of the MSH mass and its connection with the cosmological apparent horizon. This formalism was extended from BHs to the cosmological context by Bak and Rey (2000).
- 6.
We shall mostly consider the cosmological apparent and event horizons, however, arbitrary horizons may also be considered whenever necessary or relevant.
- 7.
The tensorial form of Eq. (3.9) is given by \(-dE_h=4\pi R_{h}^{2}T_{ab}\kappa ^{a}\kappa ^{b}dt\), where \(\kappa ^{\mu }\) is a null vector.
- 8.
A Bekenstein system is one which satisfies (in natural units) the Bekenstein entropy-mass bound \(S \le 2\pi RE\) as well as the Bekenstein entropy-area bound \(S \le \frac{A}{4}\), where S, R, E, and A are respectively the entropy, radius, energy, and area of the system.
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Suggested Further Reading
Black Hole Thermodynamics: Wald, R.M. 2001. The thermodynamics of black holes. Living Reviews in Relativity 4: 6; Carlip, S. 2014. Black hole thermodynamics. International Journal of Modern Physics D 23: 1430023.
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Saha, S. (2018). Cosmological Thermodynamics. In: Elements of Cosmological Thermodynamics. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-74706-4_3
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