Gravitational collapse of rotating shells and cosmic censorship

Summary

The study of gravitational collapse of stars leading to black holes has a long history dating back to the work of Oppenheimer and Snyder, but investigations of collapsing matter with rotation only became possible with the advent of numerical relativity in the ’80s.

Besides uncovering the fate of realistic stars, such studies have an important impact at a more fundamental level, concerning Penrose's cosmic censorship hypothesis. Despite many attempts to (dis)prove it over more than 40 years, this remains a conjecture and an issue of intense debate. In essence, it asserts that any curvature singularities forming from generic collapse of reasonable matter must remain hidden inside black holes. In other words, the conjecture forbids the development of so-called naked singularities, preventing quantum gravity effects to be seen by exterior observers. The conjecture is motivated by the hope that Einstein's General Relativity is self-contained as a classical theory. If singularities were to be exposed, any description of the future evolution would require a theory of quantum gravity, for which we still don't have a fully satisfactory framework.

Numerical simulations of stellar collapse have shown that whether or not a back hole forms depends strongly on the amount of rotation. If the initial configuration has sufficiently large angular momentum, there is the danger of collapsing into an (over-spinning) naked singularity. Instead, what has been observed is that either matter disperses and a singularity does not form or any excessive angular momentum is radiated away until a final rotating black hole is produced.

The topic of gravitational collapse in the presence of rotation has hardly been explored with analytic methods. Nevertheless, a path to tackle this problem (without restricting to lower dimensions) was identified a couple of years ago. The idea relies on the consideration of equal angular momenta (EAM) spacetimes. This possibility arises in higher odd dimensions (D = 5, 7, 9, ...), allowing rotating geometries to depend on a single coordinate — for this reason they are referred to as cohomogeneity-1 geometries. This results from an enhanced symmetry, in practice making them resemble spherically symmetric spacetimes. The original framework for 5D, developed in T. Delsate et al. (2014), has now been extended to arbitrary (odd) higher dimensions, see this paper.

Figure: Collapsing shell in 5D. The left panel shows two of the angular variables in which the motion is trivial (plus the radial motion). The right panel shows the third angular direction which incorporates all rotational effects (plus radial motion).

This made possible the investigation of the collapse of rotating shells onto black holes, with a large set of tuneable parameters: the mass of the shell, its spin and radial velocity at infinity, the dimensionality of spacetime D, and equation-of-state parameters characterising the imperfect fluid that makes up the shell. I have now conducted, in collaboration with R. Santarelli, an extensive scan of the parameter space for D = 5, 7, 9, 11 and with fluids described by a linear equation-of-state. Details can be found in this paper.

Figure: Artistic rendition of a thin shell collapsing onto a rotating black hole.

For pressureless fluids, i.e., dust shells, that start off from rest at infinity our results do not show any violation of the cosmic censorship conjecture during the collapse of such equal angular momenta thin shells: If the shell plunges past the (preexistent) horizon one always gets another black hole. The only way a (over-spinning) naked singularity is formed is if standard energy conditions are violated by the matter shell. Not even endowing the shells with tension or with large radial boosts at infinity allows the formation of naked singularities. We have confirmed this holds when considering both the null and the weak energy condition, which impose indistinct constraints for this class of solutions.

The more restrictive dominant energy condition is typically violated for rotating dust shells when they are sufficiently compressed (already inside the horizon) but pressurized shells can produce bouncing trajectories that satisfy the dominant energy condition throughout.

Figure: Example plots of the parameter space (shell mass m₀, rotation parameter Ma²) for dust shells (w=0) starting from rest at infinity (E = 1). These are results for four different spacetime dimensions, namely D = 5, 7, 9, 11, corresponding to N = 1, 2, 3, 4. The blue regions (FP1) indicate full plunges satisfying the weak energy condition (WEC), though all of them violate the dominant energy conditions (DEC). The green regions (FP2) correspond to full plunges violating the WEC. Light-purple regions (TWO1) indicate two-world orbits satisfying the WEC but not the DEC, and light-orange regions (TWO2) correspond to two-world orbits satisfying the WEC and the DEC. True bounces respecting both the WEC and the DEC are indicated by dark-orange domains (TB1). The light-gray regions (TWO3) correspond to two-world orbits that violate the WEC and those colored dark-gray (TB2) identify true bounces violating the WEC. The dashed black curve indicates the maximum value of Ma² for which the external geometry possesses a horizon (and therefore corresponds to an extremal black hole). The horizontal dashed red line indicates a similar situation but for the interior geometry.