The transformation of the rotational energy of a Kerr black hole

Abstract

This paper analyzes the feedback of the rotational energy extraction from a Kerr black hole (BH) by the ``ballistic method'', i.e., the test particle decay in the BH ergosphere pioneered by Roger Penrose. The focus is on the negative energy counterrotating particles (which can be massive or massless) going in towards the horizon, and the feedback on the BH irreducible mass is assessed. Generally, the change in irreducible mass is a function of the conserved quantities of the particle. For extreme Kerr BH and in the test particle limit, all the reduced transformable energy goes into the irreducible mass (i.e. $\Delta M_{\rm irr}/|E_1| \to \infty$), resulting in high irreversibility. The amount of extracted energy from a Kerr BH using test particles is much lower than the change of transformable energy. For non-extreme Kerr BHs, the effective potential of particle motion on the equatorial plane in Kerr spacetime is analyzed, and it is demonstrated that the Penrose process can only be undergone by BHs with a dimensionless spin $\hat{a} > 1/\sqrt{2}$ if the decay point coincides with the turning point. Based on that, the lower limit of the change in irreducible mass is provided as a function of the dimensionless spin of the BH. The significance of the increase in the irreducible mass of the BH during the energy extraction process is generally and concisely illustrated by introducing the concept of transformable energy of the BH. The feedback from the Penrose process on the irreducible mass demonstrates the irreversibility of energy extraction and highlights that the total amount of energy that can be extracted from a BH is less than previously anticipated.