Rotating Black Holes and Thermodynamics

Abstract

This chapter deals with rotating black-holes and the Kerr-Newman metric. The usually skipped form of the spin connection and of the Riemann tensor of this metric is calculated and presented in full detail, together with the electric and magnetic field strengths associated with it in the case of a charged hole. This is followed by a careful discussion of the static limit, of locally non-rotating observers, of the horizon and of the ergosphere. In a subsequent section the geodesics of the Kerr metric are studied by using the Hamilton Jacobi method and the system is shown to be Liouville integrable with the derivation of the fourth Hamiltonian (the Carter constant) completing the needed shell of four, together with the energy, the angular momentum and the mass. The last section contains a discussion of the analogy between the Laws of Thermodynamics and those of Black Hole dynamics including the Bekenstein-Hawking entropy interpretation of the horizon area.

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)