Prolate horizons and the Penrose inequality

The Penrose inequality has so far been proven in cases of spherical symmetry and in cases of zero extrinsic curvature. The next simplest case worth exploring would be nonspherical, nonrotating black holes with nonzero extrinsic curvature. Following Karkowski et al.’s construction of prolate black holes, we define initial data on an asymptotically flat spacelike 3-surface with nonzero extrinsic curvature that may be chosen freely. This gives us the freedom to define the location of the apparent horizon such that the Penrose inequality is violated. We show that the dominant energy condition is violated at the poles for all cases considered.

Figure 1

${\theta }_{-}$ is plotted as a function of the radius of the marginally trapped surface $\widehat{\sigma }$. The surface will be outer-trapped and our data will represent a black hole when ${\theta }_{-}<0$.

Figure 2

In this diagram we look at $\rho -|J|$ on the equator of an radius $\widehat{\sigma }=1.4$ black hole, as a function of the parameters $a$ and $b$ from $K_{ij}$. From this we set $a\ll 0$ and $b$ small.

Figure 3

$\rho$ for all values of angle $\tau$ on the surface of the horizon where $a=-100$, $b=2$ for a variety of radii $\widehat{\sigma }$. This plot shows that the energy density on the horizon is generally positive for the solutions we are considering.

Figure 4

$\rho -|J|$ for all values of angle $\tau$ on the surface of the horizon where $a=-100$, $b=2$ for a variety of possible radii $\widehat{\sigma }$. This plot shows that the DEC violations will remain near the poles.

Figure 5

$\rho -|J|$ for all values of angle $\tau$ on the surface of the horizon, for horizon at $\widehat{\sigma }=1.4$, $a=-100$, $b=2$. The negative divergence at the poles indicate that the DEC has been violated.