Hoop conjecture in five dimensions: Violation of cosmic censorship

We study the condition of black hole formation in five-dimensional space-time. We analytically solve the constraint equations of five-dimensional Einstein equations for momentarily static and conformally flat initial data of a spheroidal mass. We numerically search for an apparent horizon in various initial hypersurfaces and find both necessary and sufficient conditions for the horizon formation in terms of inequalities relating a geometric quantity and a mass defined in an appropriate manner. In the case of infinitely thin spheroid, our results suggest a possibility of naked singularity formation by the spindle gravitational collapse in five-dimensional space-time.

Figure 1

The logarithm to the base 10 of the Kretschmann invariant $I$ is plotted as function of the coordinates $R$ and $z$ in the case $a=0$ and $b=2r_{\mathrm{s}}$.

Figure 2

The logarithm to the base 10 of the Kretschmann invariant $I$ on the polar axis $(R=0)$ is plotted as function of $z$ in the case $a=0$ and $b=2r_{\mathrm{s}}$. The value of $I$ is diverge at only $(0,b)$.

Figure 3

Apparent horizons for each shape of the four-dimensional spheroid are depicted in $(R,z)$-plane. The solid line shows the surface of the spheroid. The dashed line shows the apparent horizon if it is present.

Figure 4

In the case of $a=0$, apparent horizons for each $b$ are depicted.

Figure 5

The connecting points (left figure) and the hyperhoops which we calculate for a value of $z_0$ (right figure) in first search.

Figure 6

The hyperhoop having the smallest value of $\Gamma$ in first search (left figure) and the hyperhoops which we calculate for a value of $q_0$ (right figure) in second search .

Figure 7

The value of ${\Gamma }_{\min }$ is plotted as a function of $b$. The dashed line bound this value above. This line is the corresponding quantity for the infinitely long spindle source which have line density $M/2b$.