Kerr-de Sitter spacetime, Penrose process, and the generalized area theorem

Abstract

We investigate various aspects of energy extraction via the Penrose process in the Kerr-de Sitter spacetime. We show that the increase in the value of a positive cosmological constant, Λ, always reduces the efficiency of this process. The Kerr-de Sitter spacetime has two ergospheres -associated with the black hole and the cosmological event horizons. We prove by analysing turning points of the trajectory that the Penrose process in the cosmological ergoregion is never possible. We next show that in this process both black hole and cosmological event horizons' areas increase, the later becomes possible when the particle coming from the black hole ergoregion escapes through the cosmological event horizon. We identify a new, local mass function instead of the mass parameter, to prove this generalized area theorem. This mass function takes care of the local spacetime energy due to the cosmological constant as well, including that arises due to the frame dragging effect due to spacetime rotation. While the current observed value of Λ is much tiny, its effect in this process could be considerable in the early universe scenario endowed with a rather high value of it, where the two horizons could have comparable sizes. In particular, the various results we obtain here are also evaluated in a triply degenerate limit of the Kerr-de Sitter spacetime we find, in which radial values of the inner, the black hole and the cosmological event horizons are nearly coincident.The existence of such negative energy particles gives rise to a classical mechanism for energy extraction from a black hole, namely the Penrose process, as follows [1,2,3], also references therein. Let a particle carrying positive energy enters the ergoregion of a Kerr black hole. We imagine that it breaks into two pieces there -one carrying negative energy and the other, positive. The negative energy particle enters the BEH and the positive energy particle, after reaching a turning point, comes out of the ergosphere and finally gets intercepted by an outside observer. With respect to such an observer the usual (positive) energy conservation must be valid. Thus it is clear that the ejecta will carry more energy than the initial incoming particle, effectively extracting energy from the black hole. It turns out that the energy thus extracted is largely rotational and the process can only continue until the black hole settles down into the Schwarzschild spacetime, where no ergosphere is present.Since the Penrose process reduces more the rotation of a black hole than its mass, a Kerr black hole can never become a naked curvature singularity under this process, see [4] for a formal proof of this. In [5], it was proved that the black hole horizon area must increase under this process, thereby providing evidence in favour of the second law of black hole mechanics. We further refer our reader to [6,7] for various inequalities (respectively, the Wald and the Bardeen-Press-Teukolsky inequality) regarding the local speed of the fragments within the ergoregion. In order tha...

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