Abstract
We study the thermodynamics of a $d$-dimensional Schwarzschild black hole in the canonical ensemble. This generalizes York's formalism to any number $d$ of dimensions. The canonical ensemble, characterized by a cavity of fixed radius $r$ and fixed temperature $T$ at the boundary, allows for two possible solutions in thermal equilibrium, a small and a large black hole. From the Euclidean action and the path integral approach, we obtain the free energy, the thermodynamic energy, the pressure, and the entropy, of the black hole plus cavity system. The entropy is given by the Bekenstein-Hawking area law. The heat capacity shows that the smaller black hole is in unstable equilibrium and the larger is stable. The photon sphere radius divides the stability criterion. To study perturbations, a generalized free energy function is obtained that allows to understand the possible phase transitions between classical hot flat space and the black holes. The Buchdahl radius, that appears naturally in the general relativistic study of star structure, also shows up in our context, the free energy is zero when the cavity's radius has the Buchdahl radius value. Then, if the cavity's radius is smaller than the Buchdahl radius classical hot flat space can nucleate a black hole. It is also pointed out the link between the canonical analysis performed and the direct perturbation of the path integral. Since gravitational hot flat space is a quantum system made purely of gravitons it is of interest to compare the free energies of quantum hot flat space and the stable black hole to find for which ranges of $r$ and $T$ one phase predominates over the other. Phase diagrams are displayed. The density of states at a given energy is found. Further calculations and comments are carried out, notably, a connection to thin shells in $d$ spacetime dimensions which are systems that are also apt to rigorous thermodynamics.
Highlights
Black holes are physical systems that possess thermodynamic properties
This entropy stems from the contribution of the classical Euclidean Einstein action of the black hole metric to the partition function, and its cause is the nontrivial topology with a nonzero Euler characteristic of the Euclidean four-dimensional Schwarzschild black hole, in contrast to ordinary quantum field theories, where the classical contribution to the action is absorbed into the normalization of the functional integral [2]
In the path integral approach to quantum gravity, i.e., the Hartle-Hawking approach, integration of the Euclidean Einstein action over the space Rof metrics g yields the canonical partition function Z 1⁄4 D1⁄2g expð−I1⁄2gÞ Taking a black hole solution as the background metric, the leading term in the expansion will be that of the classical action, Z 1⁄4 e−I: ð10Þ
Summary
Black holes are physical systems that possess thermodynamic properties. The path-integral approach to quantum gravity is a powerful technique that when applied to black holes displays them clearly as thermodynamic systems. 3 2 rþ where rþ is the gravitational radius, and since rþ 1⁄4 2m, one can write r 1⁄4 3m, where m is the spacetime mass This special radius appears in the thermodynamic study of the black hole in the canonical ensemble, as York noticed. 9 4 m, where again m spacetime mass This special radius appears in the thermodynamic study of the black hole in the canonical ensemble, as we noticed here. [24] and for a shell in a d-dimensional spacetime we give here When studying the black hole in the canonical ensemble, one needs to use the thermodynamic properties of a radiation gas in d spacetime dimensions as given in Ref. In Appendix F, we present a synopsis and further additions
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