Black holes die hard: Can one spin up a black hole past extremality?

A possible process to destroy a black hole consists on throwing point particles with sufficiently large angular momentum into the black hole. In the case of Kerr black holes, it was shown by Wald that particles with dangerously large angular momentum are simply not captured by the hole, and thus the event horizon is not destroyed. Here, we reconsider this gedanken experiment for a variety of black hole geometries, from black holes in higher dimensions to black rings. We show that this particular way of destroying a black hole does not succeed and that cosmic censorship is preserved.

Figure 1

This figure shows the maximum increase in spin, $\frac{M}{m_0}(\delta j{)}_{\max }$ caused by a particle with $E/m_0=100$ falling into a Myers-Perry black hole with a single rotation parameter. The dimensionless rotation parameter $a_{*}$ is defined as $a_{*}\equiv \frac{a}{M^{1/(D-3)}}$. Notice that it is not possible to spin up an extremal black hole (for $D=4$, 5, the extremal value is marked with a dotted line).

Figure 2

This figure shows the maximum increase in spin, $\frac{M}{m_0}(\delta j{)}_{\max }/E$ caused by a particle with energy $E=100$, 1.5, 1.2 falling into a Myers-Perry black hole with $D=6$ and for a single rotation parameter. The dimensionless rotation parameter $a_{*}$ is defined as $a_{*}\equiv \frac{a}{M^{1/(D-3)}}$. For $E=1.2$, $\frac{M}{m_0}\delta j$ becomes negative at $a_{*}=3.3$.

Figure 3

This figure shows the maximum increase in spin, $\frac{M}{m_0}(\delta j{)}_{\max }$ caused by $d$ particles with $E/m_0=100$ falling into a Myers-Perry black hole with all rotation parameters equal. The left panel considers even $D$; the right panel refers to odd spacetime dimensions $D$. It is not possible to spin up an extremal black hole, marked with a dotted line. The dimensionless rotation parameter $a_{*}\equiv \frac{a}{M^{1/(D-3)}}$.

Figure 4

This figure shows the maximum increase in spin, $\frac{R^2}{m_0}(\delta j{)}_{\max }$ caused by a massive particle with energies per unit mass $E=1.1$, 1.15, 1.2 falling into a singly spinning neutral black ring. For $1/\sqrt{\pi }\ge j_0<\sqrt{32/27\pi }$, there are two distinct black rings with the same charges. In this range the lower (upper) curve corresponds to the fat (thin) black ring. Notice that it is not possible to over spin a fat black ring. The vertical line (at $j_0=1/\sqrt{\pi }$) marks the minimum spin that (regular) black rings can possess.

Figure 5

This figure shows the maximum increase in spin, $\frac{R^2}{m_0}(\delta j{)}_{\max }$ caused by a massive particle with energies per unit mass $E=1.3$, 1.35, 1.38, 1.45 falling into a singly spinning neutral black ring. The fat black ring branch is not visible in this plot. Thin black rings with very large spins $j_0$ always lose dimensionless angular momentum when absorbing a massive particle with $E<\sqrt{2}$.

Figure 6

This figure shows the maximum increase in spin, $\frac{R^2}{m_0}(\delta j{)}_{\max }$ caused by a massive particle with energies per unit mass $E=1.2$ and $E=1.5$ falling into a dipole black ring. The dipole parameter is set to $\mu =0.01$ while in the horizontal axis the parameter $\nu$, which spans the interval [0,1), is shown.

Figure 7

This figure shows the final dimensionless angular momentum, $j_{\mathrm{fin}}$, caused by a massless particle with energy $E=1.5$ falling into an extremal dipole black ring. The horizontal axis corresponds to the dipole parameter $\mu$ of the initial ring. The value $R=10$ was employed to ensure that $E\ll {\mathcal{M}}_0$ is always satisfied, therefore avoiding backreaction effects. Also shown is the upper (extremal) bound on $j$ considering the final ring has dipole charge $q_{\mathrm{fin}}$. It is apparent that the dipole ring is never over spun.