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

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. ${\cal E}_{r}^{+}%=\left(\frac{\sqrt{2}-1}{2}\right)\sqrt{2{\cal M}^2-Q^2} =\left(\sqrt{2}-1\right)\sqrt{\frac{J}{2}}$.