Thermodynamic curvature of the BTZ black hole

Abstract

In this paper we apply the concept of thermodynamic geometry to the Bañados-Teitelboim-Zanelli ͑BTZ͒ black hole. We find the thermodynamic curvature diverges at the extremal limit of the black hole, which means the extremal black hole is the critical point with the temperature zero. We also study the effective dimensionality of the underlying statistical model. Near the critical point, the picture is clear; the spatial dimension of the underlying statistical model is just one, which agrees with other results. However, far from the critical point, the dimension becomes less than one and even negative. In order to interpret this result, we resort to a qualitative analogy with the Takahashi gas model. ͓S0556-2821͑99͒04516-6͔ PACS number͑s͒: 04.70.Dy, 04.60. Kz, 05.70.Jk Over the decades the statistical interpretation of blackhole entropy has been one of the most fascinating subjects. There have been many approaches to the problem, although nothing was completely successful. One curiosity about black-hole thermodynamics is that it looks different from an ordinary thermodynamical system, due to the negative heat capacity. This makes it hard to compose its thermodynamic ensemble and to make the underlying statistical model. One way to study the statistical aspects is to assume the microcanonical ensemble for an isolated black hole. Actually, in this way one can understand the critical behavior for the near extremal black hole. The critical exponents satisfying the scaling law even tells us the dimensionality of the underlying statistical model.In this paper, we suggest another tool useful for studying the statistical properties including the fluctuation, the critical behavior, and so on. This is the thermodynamic geometry. It defines a metric on the space of thermodynamic variables. ͑Of course, it has nothing to do with the geometry of space and time.͒ Here, the thermodynamic potential becomes the geometrical potential generating the metric components. The thermodynamic variables constitute the coordinates for the geometry. Details will be given below through the example of the Bañados-Teitelboim-Zanelli ͑BTZ͒ black hole ͓2͔. As is known, the BTZ black hole could play an important role in understanding entropy and some dynamical properties of certain five-and four-dimensional black holes in supergravity theories, because of the U duality between the BTZ black hole and those high-dimensional black holes ͓3͔. For a review see ͓4͔.In general, it is technically difficult to define the zerotemperature critical point, as is the case with the black hole. Conventional definition for the zero-temperature critical point is the point where at least one of the second derivatives of some thermodynamic potential diverges ͓1͔. However, in the language of thermodynamic geometry one can define it unambiguously as the point where the thermodynamic curvature ͑the curvature with respect to the thermodynamic metric͒ diverges. This is based on the fact that thermodynamic curvature is proportional to the correlation volume ͓5͔:where 2 is ...

Thermodynamic geometry of charged rotating BTZ black holes

“…The third singularity located at ln r + = 0 can be interpreted as a critical point that is not determined by the heat capacity (19). In fact, at r + = 1 the second derivative of the mass ∂ 2 M/∂Q 2 = 0, indicating either the transition into a region of instability or a second order phase transition.…”

“…Nevertheless, a change of the thermodynamic potential affects the Ruppeiner geometry in such a way that the resulting curvature singularity does not correspond to a phase transition. Another example is provided by the Bañados-Teitelboim-Zanelli (BTZ) black hole thermodynamics for which the curvature of the equilibrium space turns out to be flat [19][20][21]. This flatness is usually interpreted as a consequence of the lack of thermodynamic interaction.…”

“…The Weinhold and Ruppeiner geometries have been analyzed in a number of black hole families to study phase space, critical behavior, and stability properties [12][13][14][15][16][17][18][19][20][21][22]. In some particular cases, it was found that Weinhold and Ruppeiner geometries carry information about the phase transitions structure.…”

Phase transition and thermodynamical geometry for Schwarzschild AdS black hole in AdS5 × S5 spacetime

“…To the best of our knowledge, applying the thermodynamical geometry to black hole thermodynamics was initiated in [58], there it was found that the Weinhold metric is proportional to the metric on the moduli space for supersymmetric extremal black holes, whose Hawking temperature is zero, and the Ruppeiner metric governing fluctuations naively diverges, which is consistent with the argument that near the extremal limit, the thermodynamical description breaks down. Applying the thermodynamical geometry approach to phase transition of black holes was followed in [59][60][61], and for more relevant references see the recent review [62] and references therein. In particular, refs.…”

Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature

“…The Riemannian-geometric approach to thermodynamics has been applied to a variety of systems, among which black holes stand out (this approach to black hole thermodynamics was pioneered by Cai et al [16]). The purpose of this viewpoint is twofold: on the one hand, it intends to relate the divergences of heat capacities (often regarded as thermodynamic critical points) to geometric singularities of Weinhold's metric structure; on the other, it is aimed to determine the nature of the interactions involved in microscopic models of gravity via Ruppeiner's conjecture.…”