Energy Formula, Surface geometry and Energy Extraction for Kerr-Sen Black Hole

TL;DR

This paper analyzes the energy components of Kerr-Sen black holes, deriving a unified energy formula, examining surface deformations, and calculating maximum extractable rotational energy via the Penrose process.

Contribution

It introduces a compact energy formula valid at all horizons and explores surface deformation effects and energy extraction limits for Kerr-Sen black holes.

Findings

Sum of surface, rotational, and electromagnetic energies equals the mass parameter.

Surface deformation varies with black hole spin, increasing equatorial and decreasing polar circumference.

Maximum rotational energy extractable occurs at extremal Kerr-Sen black holes.

Abstract

We evaluate the \emph{surface energy~(${\cal E}_{s}^{\pm}$), rotational energy~(${\cal E}_{r}^{\pm}$) and electromagnetic energy~(${\cal E}_{em}^{\pm}$)} for a \emph{Kerr-Sen black hole~(BH)} having the event horizon~(${\cal H}^{+}$) and the Cauchy horizon~(${\cal H}^{-}$). Interestingly, we find that the \emph{sum of these three energies is equal to the mass parameter i.e. ${\cal E}_{s}^{\pm}+{\cal E}_{r}^{\pm}+{\cal E}_{em}^{\pm}={\cal M}$}. Moreover in terms of the \emph{ scale parameter ~$(\zeta_{\pm})$, the distortion parameter~($\xi_{\pm}$) and a new parameter~$(\sigma_{\pm})$} which corresponds to the area~(${\cal A}_{\pm}$), the angular momentum ~$(J)$ and the charge parameter~($Q$), we find that the \emph{mass parameter in a compact form} ${\cal E}_{s}^{\pm}+{\cal E}_{r}^{\pm}+{\cal E}_{em}^{\pm}={\cal M} =\frac{\zeta_{\pm} }{2} \sqrt{\frac{1+2\,\sigma_{\pm}^2}{1-\xi_{\pm}^2}}$ %\begin{eqnarray} %{\cal E}_{s}^{\pm}+{\cal E}_{r}^{\pm}+{\cal E}_{em}^{\pm}={\cal M} =\frac{\zeta_{\pm} }{2} %\sqrt{\frac{1+2\,\sigma_{\pm}^2}{1-\xi_{\pm}^2}} \nonumber %\end{eqnarray} which is valid {through all the horizons} (${\cal H}^{\pm}$). We also compute the \emph{equatorial circumference and polar circumference} which is a gross measure of the BH surface deformation. It is shown that when the spinning rate of the BH increases, the \emph{equatorial circumference increases} while the \emph{polar circumference decreases}. Furthermore, we compute the exact expression of \emph{rotational energy that should be extracted from the BH via the Penrose process}. The maximum value of rotational energy which is extractable should occur for \emph{extremal Kerr-Sen BH} i.e. ${\cal E}_{r}^{+}%=\left(\frac{\sqrt{2}-1}{2}\right)\sqrt{2{\cal M}^2-Q^2} =\left(\sqrt{2}-1\right)\sqrt{\frac{J}{2}}$.