The Fourth Law of Black Hole Thermodynamics

We show that black holes fulfill the scaling laws arising in critical transitions. In particular, we find that in the transition from negative to positive values the heat capacities CJQ, CΩQ and CJΦ give rise to critical exponents satisfying the scaling laws. The three transitions have the same critical exponents as predicted by the universality Hypothesis. We also briefly discuss the implications of this result with regards to the connections among gravitation, quantum mechanics and statistical physics. ∗ Permanent Address: IAFE, Cas. Corr. 67, Suc. 28, 1428 Buenos Aires, ARGENTINA. e-mail: [email protected]. 1. The four laws of Black Hole Thermodynamics In the work of Bardeen, Carter and Hawking [1] it was established a remarkable mathematical analogy between the laws of thermodynamics and the laws of black hole mechanics derived from General Relativity. If one makes the formal replacements E → M , T → Cκ, and S → A/8πC (where C is a constant) in the laws of the thermodynamics, one obtains the laws that govern the mechanics of black holes . The physical analogy seemed to have problems due to the fact that in classical General Relativity the thermodynamic temperature of a black hole appears to be absolute zero. However, Hawking [3] found that when quantum effects are taken into account, a black hole absorbs and emits particles as a body at temperature T = κ/2π, and this resolved that puzzle. Here and throughout this paper we take units in which G = h̄ = c = κB = 1. The four laws of black hole thermodynamics can be briefly formulated as follows: The Zeroth Law: The surface gravity, κ, of a stationary black hole (at equilibrium) is constant on the entire surface of the event horizon. This property is proved as a theorem in ref. [1]. Landsberg [4] gives however a more precise formulation: “In the absence of adiabatic partitions and long range fields, an equilibrium system exhibits a unique temperature”. If the above conditions are not satisfied, it can be shown [5] that a system with a built-in adiabatic partition can have two different temperatures even in thermal equilibrium. For a Kerr-Newman black hole endowed with mass M , charge Q and angular momentum ~ J , the surface gravity is given by [6] κ = 1 2 r+ − r− (r2 + + a 2) = √ M2 − a2 −Q2 2M2 −Q2 + 2M √ M2 − a2 −Q2 , (1) where a = | ~ J |2/M2 and r± = M ± √ M2 − a2 −Q2 , (2) are the event and internal horizons respectively. The First Law: This is just an expression that states that in an isolated system, including black holes, the total energy of the system is conserved. In ref.[1] it is derived, in a general context, the following differential mass formula for stationary black holes, δM = κ 8π δA+ ~ Ω · δ ~ J +ΦδQ , (3) where ~ Ω is the angular velocity and Φ the electric potential of the event horizon, and A = 4π(r + + a ) = 8π { M − Q 2 2 + √ M4 − J2 −M2Q2 }