Abstract
The Hessian of either the entropy or the energy function can be regarded as a metric on a Gibbs surface. For two parameter families of asymptotically flat black holes in arbitrary dimension one or the other of these metrics are flat, and the state space is a flat wedge. The mathematical reason for this is traced back to the scale invariance of the Einstein–Maxwell equations. The picture of state space that we obtain makes some properties such as the occurence of divergent specific heats transparent.
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References
Hawking S.W.(1975). Particle creation by black holes. Commun. Math. Phys. 43, 199
Weinhold F.(1976). Thermodynamics and geometry. Phys. Today March 23
Ruppeiner, G.: Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys. 67, 605 (1995); 68, 313(E) (1996)
Å man J.E., Bengtsson I., Pidokrajt N.(2003). Geometry of black hole thermodynamics. Gen. Rel. Grav. 35: 1733
Å man J.E., Pidokrajt N.(2006). Geometry of higher-dimensional black hole thermodynamics. Phys. Rev. D 73: 024017
Johnston D.A., Janke W., Kenna R.(2003). Information geometry, one, two, three (and four). Acta Phys. Polon. B 34: 4923
Arcioni G., Lozano-Tellechea E.(2005). Stability and critical phenomena of black holes and black rings. Phys. Rev. D 72: 104021
Ferrara S., Gibbons G.W., Kallosh R.(1997). Black holes and critical points in moduli space. Nucl. Phys. B 500: 75
Shen, J., Cai, R.G., Wang, B., Su, R.K.: Thermodynamic geometry and critical behaviour of black holes. arXiv preprint gr-gc/0512035
Bengtsson I., Życzkowski K.(2006). Geometry of Quantum States. Cambridge University Press, Cambridge
Hankey A., Stanley H.E.(1972). Systematic application of generalized homogeneous functions to static scaling, dynamic scaling, and universality. Phys. Rev. B 6: 3515
Dubrovin B.: Geometry of 2D topological field theories. In: Francaviglia, M., Greco, S.(eds) Integrable Systems and Quantum Groups. Lecture Notes in Mathematics, vol. 1620. Berlin (1996)
Emparan R., Reall H.S.(2002). A rotating black ring in five dimensions. Phys. Rev. Lett. 88: 101101
Myers R.C., Perry M.J.(1986). Black holes in higherdimensional space-times. Ann. Phys. 172: 304
Bañados M., Teitelboim C., Zanelli J.(1992). The black hole in three-dimensional spacetime. Phys. Rev. Lett. 69: 1849
Nulton J.D., Salamon P.(1985). Geometry of the ideal gas. Phys. Rev. A 31: 2520
Davies P.C.W.(1977). The thermodynamic theory of black holes. Proc. Roy. Soc. A 353: 499
Tranah D., Landsberg P.T.(1980). Thermodynamics of non-extensive entropies II. Collect Phenom. 3: 73
Penrose R.(2004). The Road to Reality. Jonathan Cape, London
Sorkin R.(1982). A stability criterion for many-parameter families. Astrophys. J. 257: 847
Katz J., Okamoto I., Kaburaki O.(1993). Thermodynamic stability of pure black holes. Class. Quant. Grav. 10: 1323
Lynden-Bell D.(1998). Negative specific heat in astronomy, physics and chemistry. In: Proceeding of XXth IUPAP Conference on Statistical Physics. Kluwer, Dordrecht
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Åman, J.E., Bengtsson, I. & Pidokrajt, N. Flat Information Geometries in Black Hole Thermodynamics. Gen Relativ Gravit 38, 1305–1315 (2006). https://doi.org/10.1007/s10714-006-0306-1
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DOI: https://doi.org/10.1007/s10714-006-0306-1