Rapidly rotating spacetimes and collisional super-Penrose process

Abstract

We consider generic axially symmetric rotating spacetimes and examine particle collisions in the ergoregion. The results are generic and agree with those obtained in the particular case of the rotating Teo wormhole in Tsukamoto and Bambi, Phys Rev D 91:104040, 2015. It is shown that for sufficiently rapid rotation, the energy of a particle escaping to infinity can become arbitrary large (so-called super-Penrose process). Moreover, this energy is typically much larger than the center-of mass energy of colliding particles. In this sense the situation differs radically from that for collisions near black holes.

Appendix

Here, we prove inequality (35). As $L_{1}\le 0$, $K\ge K_{+}$ where $K_{+}$ corresponds to the maximum possible value of $E_{3}=E_{+}+s$. Therefore, it is sufficient to show that $K_{+}>0$. Using (15), (33), (39) one finds

$$\begin{aligned} K_{+}=a_{0}+a_{1}L_{1}+a_{2}L_{2}^{2}\text {,} \end{aligned}$$

(81)

where

$$\begin{aligned} a_{0}= & {} \omega E_{1}^{2}>0\text {,} \end{aligned}$$

(82)

$$\begin{aligned} a_{1}= & {} \frac{N^{2}E_{1}}{g_{\phi }}\left( 1+\frac{\omega \sqrt{g_{\phi }B}}{N} \right) -2E_{1}\omega ^{2}\text {.} \end{aligned}$$

(83)

$$\begin{aligned} a_{2}= & {} \frac{g_{00}}{g_{\phi }}\left( \omega -\frac{N\sqrt{B}}{\sqrt{g_{\phi }}}\right) . \end{aligned}$$

(84)

By assumpion, the point of collision is located inside the ergoregion, where (29) is satisfied, hence

$$\begin{aligned} \omega \sqrt{g_{\phi }}>N\text {.} \end{aligned}$$

(85)

It is seen from (57) that $B<1$. Then,

$$\begin{aligned} a_{2}>\frac{g_{00}}{g_{\phi }}\left( \omega -\frac{N}{\sqrt{g_{\phi }}}\right) >0. \end{aligned}$$

(86)

The coefficent $a_{1}$ can be rewritten as

$$\begin{aligned} a_{1}=E_{1}c\text {,} \end{aligned}$$

(87)

where

$$\begin{aligned} c=\frac{N^{2}}{g_{\phi }}-2\omega ^{2}+\frac{N}{g_{\phi }}\omega \sqrt{ g_{\phi }B}. \end{aligned}$$

(88)

Using the inequality $B<1$ again,

$$\begin{aligned} c\le \frac{N^{2}}{g_{\phi }}-2\omega ^{2}+\frac{N}{\sqrt{g_{\phi }}}\omega . \end{aligned}$$

(89)

It follows from (85) that

$$\begin{aligned} c\le \frac{N^{2}}{g_{\phi }}-\omega ^{2}=-\frac{g_{00}}{g_{\phi }}<0. \end{aligned}$$

(90)

Thus, since $L_{1}\le 0$, all terms in (81) are positive or, at least, non-negative. This completes the proof.