Energy formula, surface geometry and energy extraction for Kerr-Sen black hole

Abstract

We evaluate the surface energy ($\mathcal {E}_{s}^{\pm }$), rotational energy ($\mathcal { E}_{r}^{\pm }$) and electromagnetic energy ($\mathcal { E}_{em}^{\pm }$) for a Kerr-Sen black hole (BH) having the event horizon ($\mathcal { H}^{+}$) and the Cauchy horizon ($\mathcal { H}^{-}$). Interestingly, we find that the sum of these three energies is equal to the mass parameter i.e. $\mathcal { E}_{s}^{\pm }+\mathcal { E}_{r}^{\pm }+\mathcal { E}_{em}^{\pm }=\mathcal { M}$ . Moreover in terms of the scale parameter  $(\zeta _{\pm })$, the distortion parameter ($\xi _{\pm }$) and a new parameter $(\sigma _{\pm })$ which corresponds to the area ($\mathcal { A}_{\pm }$), the angular momentum  (J) and the charge parameter (Q), we find that the mass parameter in a compact form $\mathcal { E}_{s}^{\pm }+\mathcal { E}_{r}^{\pm }+\mathcal { E}_{em}^{\pm }=\mathcal { M} =\frac{\zeta _{\pm } }{2} \sqrt{\frac{1+2\,\sigma _{\pm }^2}{1-\xi _{\pm }^2}}$ which is valid through all the horizons ($\mathcal { H}^{\pm }$). We also compute the 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 equatorial circumference increases while the polar circumference decreases. We show that there exist two classes of geometry separated by $\xi _{\pm }=\frac{1}{2}$ in Kerr-Sen BH. In the regime $\frac{1}{2}<\xi _{\pm }\le \frac{1}{\sqrt{2}}$, the Gaussian curvature is negative and there exist two polar caps on the surface. While for $\xi _{\pm }<\frac{1}{2}$, the Gaussian curvature is positive and the surface will be an oblate deformed sphere. Furthermore, we compute the exact expression of rotational energy that should be extracted from the BH via Penrose process. The maximum value of rotational energy which is extractable should occur for extremal Kerr-Sen BH i.e. $\mathcal { E}_{r}^{+} =\left( \sqrt{2}-1\right) \sqrt{\frac{J}{2}}$.

Appendices

Appendix-A

To compute the Gaussian curvature we have to write the metric in the following form by using Eq. (35)

$$\begin{aligned} d\mathcal { S}^2= & {} \zeta _{\pm }^2 \left[ \frac{dy^2}{f(y)}+f(y)\,d\phi ^2 \right] ~. \end{aligned}$$

(63)

where

$$\begin{aligned} f(y)= & {} \frac{1-y^2}{1-\xi _{\pm }^2(1-y^2)}~. \end{aligned}$$

(64)

and the value of $y=\cos \theta$. We know that the Gaussian curvature [9] is defined for the above metric as

$$\begin{aligned} K \equiv K(y) = - \frac{f''(y)}{2\zeta _{\pm }^2}~. \end{aligned}$$

(65)

For our metric, the Gaussian curvature of $\mathcal { H}^{\pm }$ is calculated to be

$$\begin{aligned} K_{\pm }(\mathcal { M},a,q,\theta )= & {} \frac{\left( r_{\pm }^2+2qr_{\pm }+a^2 \right) \left( r_{\pm }^2+2qr_{\pm }-3a^2\cos ^2\theta \right) }{\left( r_{\pm }^2+2qr_{\pm }+a^2\cos ^2\theta \right) ^3}~. \end{aligned}$$

(66)

In terms of scale parameter and distortion parameter, the Gaussian curvature of $\mathcal { H}^{\pm }$ becomes

$$\begin{aligned} K_{\pm }(\zeta _{\pm },\xi _{\pm },\theta )= & {} \frac{\left[ 1-\xi _{\pm }^2\left( 1+3\cos ^2\theta \right) \right] }{\zeta _{\pm }^2 \left( 1-\xi _{\pm }^2 \sin ^2\theta \right) ^3}~. \end{aligned}$$

(67)

When the rotation is off i.e. $a=\xi _{\pm }=0$, we will get the spherically symmetric situation and in this situation the Gaussian curvature becomes

$$\begin{aligned} K_{\pm }(\zeta _{\pm },0,\theta )= & {} \frac{1}{\zeta _{\pm }^2}=\frac{1}{r_{\pm }^2+2qr_{\pm }} =K_{\pm }(\mathcal { M},0,q,\theta )~. \end{aligned}$$

(68)

If one could take the value of charge parameter becomes zero then we would get spherically symmetric Schwarzschild BH:

$$\begin{aligned} K_{\pm }(\zeta _{\pm },0,\theta )= & {} \frac{1}{\zeta _{+}^2}=\frac{1}{r_{+}^2} =K_{\pm }(\mathcal { M},0,0,\theta )~. \end{aligned}$$

(69)

It implies that the Gaussian curvature looks like a sphere. This is the limiting situation. For generalized case the Gaussian curvature is a function of polar angle $\theta$ when the the BH is rotating i.e. ($\xi _{\pm } \ne 0$).

Case-I: When $\xi _{\pm }=\frac{1}{2}$, the Gaussian curvature becomes zero at the poles ($\theta =0$). Alternatively this occurs when $a=\frac{\sqrt{3\mathcal { M}^2-2Q^2}}{2}$

Case-II: When $\frac{1}{2}<\xi _{\pm }\le \frac{1}{\sqrt{2}}$, the Gaussian curvature becomes negative and there exist two polar caps on the surface. Like Kerr-Newman BH, there exist two geometrically distinct classes of charged rotating BH in heterotic string theory when the value of distortion parameter satisfied the following criterion:

$$\begin{aligned} \xi _{\pm } \gtrless \frac{1}{2} \end{aligned}$$

Say for example when we choose the criterion $\xi _{\pm }>\frac{1}{2}$, the Gaussian curvature is computed to be

$$\begin{aligned} K_{\pm }(\zeta _{\pm },\frac{3}{5},\theta )= & {} \frac{625}{\zeta _{\pm }^2}\frac{(16-27\cos ^2\theta )}{(25-9\sin ^2\theta )^3}. \end{aligned}$$

(70)

Consequently at the pole it becomes

$$\begin{aligned} K_{\pm }(\zeta _{\pm },\frac{3}{5},0)= & {} - \frac{11}{25 \zeta _{\pm }^2}<0. \end{aligned}$$

(71)

This class of BH geometry having the Gaussian curvature is negative. That’s why the surfaces are likely to be a pseudosphere.

On the other hand if we choose the criterion $\xi _{\pm }<\frac{1}{2}$, the Gaussian curvature is computed to be

$$\begin{aligned} K_{\pm }(\zeta _{\pm },\frac{2}{5},\theta )= & {} \frac{625}{\zeta _{\pm }^2}\frac{(21-12\cos ^2\theta )}{(25-4\sin ^2\theta )^3}. \end{aligned}$$

(72)

Consequently at the pole it becomes

$$\begin{aligned} K_{\pm }(\zeta _{\pm },\frac{2}{5},0)= & {} \frac{9}{25 \zeta _{\pm }^2}>0. \end{aligned}$$

(73)

This class of BH geometry having the Gaussian curvature is positive everywhere on the surface. The surfaces are likely to be a oblate deformed sphere.

Note that as a special case at the equator, the curvature of $\mathcal { H}^{\pm }$ reduces to

$$\begin{aligned} K_{\pm }^{e}= & {} \frac{\left( r_{\pm }^2+2qr_{\pm }+a^2 \right) }{\left( r_{\pm }^2+2qr_{\pm }\right) ^2} =\frac{2\mathcal { M}}{r_{\pm }\left( r_{\pm }+2q\right) ^2}>0 ~. \end{aligned}$$

(74)

In summary, at the pole, there exist two polar caps while at the equator there does not exist such a polar cap.

Appendix-B: Topology, Euler number and Gauss-Bonnet theorem

Using the general formula of Gaussian curvature derived in Eq. (67), we can compute the topology of the surface. To evaluate the topology of the surface we can apply the Gauss-Bonnet theorem which states that

$$\begin{aligned} \iint _{M} K_{\pm }\,\, \omega _{\theta }\wedge \omega _{\phi }= & {} 2\pi \chi (M) ~. \end{aligned}$$

(75)

where $\chi$ is the Euler characteristic of the surface. For $\mathcal { H}^{\pm }$, we get

$$\begin{aligned} \iint _{M} K_{\pm }\,\, \omega _{\theta }\wedge \omega _{\phi }= & {} 4\pi ~. \end{aligned}$$

(76)

This implies that $\chi =2$. Like Kerr BH, the surface of the charged BH in heterotic string theory is topologically a 2-sphere.