Chapter 3

Rotating Black Holes and Thermodynamics

Tu vedresti ’l Zodiaco rubecchio

Ancora all’Orse piú stretto rotare

Se non uscisse fuor dal cammin vecchio.

Sí ch’ambo e due hann’un solo orizzon,

E diversi emisperi: . . .

Dante Alighieri (Purgatorio Canto IV, 64)

3.1 Introduction

In this chapter we study in considerable detail the quite intriguing and challenging properties of rotating black-holes encoded in the Kerr-Newman metric which contains only three parameters (m, J, q) corresponding, respectively, to the mass, to the angular momentum and to the charge of the hole. As anticipated in the previous chapter, irrespectively from all the details of its initial structure, a gravitational collapsing body sets down to a final equilibrium state described by the Kerr-Newman metric, which is the unique one, in D = 4, to be static, stationary, axial symmetric and asymptotically flat. The geodesic problem for this metric is still a completely integrable one, since there are enough first integrals, yet the explicit integration is very much laborious because it involves higher transcendental functions and the classification of trajectories turns out very complicated. We will derive the final integration formulae but we will present only a simple example of their application in view of such a complexity. We will instead dwell on the general new properties displayed by rotating black-holes that allow for a mechanism of energy extraction whose features have a surprising analogy with the laws of thermodynamics. Such an analogy is actually only the tip of an iceberg. The horizon area of the black holes behaves as an entropy and this makes it clear that, in a fundamental quantum theory of gravity, black holes must necessarily be endowed with a statistical interpretation in terms of some kind of microstates.

3.2 The Kerr-Newman Metric

Let us consider the standard set up of polar coordinates r, θ , φ for R3 plus the parameter t for time. For the angular variables θ , φ labeling the points of each 2-

symmetric Schwarzschild metric. Indeed, under these assumptions we have:

Reissner-Nordström If we put α = 0 keeping both m and q non-vanishing we obtain the so-called Reissner-Nordström metric which is spherical symmetric but not Ricci-flat. As we shall discuss later on, this metric corresponds to an electrovac solution namely to a solution of the coupled system of Maxwell-Einstein equations. This solution describes the gravitational field generated by an electrically or magnetically charged monopole of mass m and charge q. If α = 0, we have

Kerr If we put the charge parameter to zero, namely q = 0, the Kerr-Newman metric degenerates into the Kerr metric which is Ricci-flat but not spherically symmetric. It is only axial-symmetric and it describes a rotating black-hole of mass

3.2.1 Riemann and Ricci Curvatures of the Kerr-Newman Metric

The next step in the analysis of the proposed metric (3.2.3) is the construction of the corresponding curvature forms. As usual we adopt the vielbein formalism and we

46 3 Rotating Black Holes and Thermodynamics

aim at the construction first of the spin connection ωab , secondly of the curvature 2-form Rab , from which we will extract the Riemann and Ricci tensors.

As written in (3.2.3) the Kerr-Newman metric is already presented as a sum of four squares so that singling out the vielbein 1-forms is an immediate task. Indeed

Next we consider the construction of the torsionless spin-connection defined by:

Relying on the above result we can proceed to the calculation of the curvature 2- form, defined by:

and we obtain the following result:

We wonder which kind of matter can produce a stress-energy tensor of the form (3.2.20) so that the constructed metric might be an exact solution of Einstein field equations. The answer is very simple: an electromagnetic field!

Let us consider the general form of the stress energy tensor for a Maxwell field. From the Maxwell action:

The stress-energy tensor Tμν(Maxw) is traceless (gμν Tμν(Maxw) = 0) and in flat indices takes the same form as in curved indices:

For the particular case of an electromagnetic field of the form:

Due to tracelessness of the stress-energy tensor, Einstein field equations reduce to: