Black holes are generically formed by gravitational collapse of a dust cloud under its own gravity or accretion of matter onto a gravitating centre. For rotating black hole, matter with angular momentum should accrete, and for that to happen, particles should have angular momentum less than the threshold value defined by the innermost stable circular orbit (ISCO). Thus existence of ISCO becomes the necessary condition for rotating black hole formation. It is known that, what to talk of ISCO, even bound orbits cannot exist in dimensions greater than the usual four. That’s how arises the question in the mind of authors of recent study, how do rotating black holes form in higher dimensions?
The primary requirement for formation of astrophysical black hole is accretion of matter under its own gravity onto a gravitating centre. This would work fine for a non-rotating black hole. However the situation would be quite different for rotating black hole where accreting matter should have angular momentum. The latter would produce centrifugal barrier which has to be overcome or bypassed. In general relativity (GR) where the barrier pinches off as radius decreases, and more importantly there exists the threshold bound on angular momentum defined by the existence of innermost stable circular orbit (ISCO). Thus a particle with angular momentum less than this threshold (which for the non-rotating Schwarzschild black hole is L² = 12M²), particle would fall in and deposit its angular momentum to the accreting centre. This is how the barrier could be bypassed in GR.
Thus existence of ISCO becomes the necessary condition – a key requirement for formation of rotating black hole by gravitational collapse.
It further turns out that in dimensions greater than four, there cannot occur bound orbits for particles. The reason is however very simple, gravitational potential in D-dimension goes as 1/r^D–3 while the centrifugal potential as always fall off as 1/r². In D > 4, the latter would not be able to counter balance the former to produce a potential well so as to harbour bound orbits. That is, effective potential has no minimum and consequently no stable circular orbit. That is, ISCO cannot exist in higher dimensions greater than the usual four. It would therefore raise the question, how does then rotating black holes form in higher dimensions?
Dadhich and Shaymatov, in their paper showed that there can exist neither bound orbits nor ISCO in higher dimensions. This latter fact was however known for non rotating black holes in higher dimensions. This feature is also carried over to rotating black holes.
Since there occur no bound orbits in higher dimensions, there can neither occur ISCO defining the minimum threshold for the angular momentum nor accretion disk even for non-rotating gravitating centre. Therefore no particle with non-zero angular momentum can fall in and posit angular momentum on the gravitating centre. Thus not only rotating black hole, no compact object like a star with rotation could be formed in higher dimensions by gravitational collapse. This is a very important conclusion they made in their paper that follows from the very simple and straightforward analysis.
But, this also raises the question about accretion process for non rotating black hole. Accretion is mediated through accretion disk which cannot occur in higher dimensions because there exist no bound orbits. Of course a gas cloud free of angular momentum could collapse under its own gravity — purely radial accretion, to form a non-rotating black hole or any compact object like a star. As shown in the upper panels of Figs 1 and 2, accretion with zero angular momentum could proceed unhindered onto a non-rotating black hole without formation of accretion disk. But such an object cannot acquire any rotation.
The situation for the rotating black hole is even worse because there are two repulsive components in the potential, one due to gravitational interaction of black hole rotation and the other due to centrifugal force due to particle’s angular momentum and both falling as 1/r² in the leading order. Fig. 2 shows that for zero angular momentum motion, overall force however remains attractive all through (upper panel) while for non-zero angular momentum it is repulsive except very close to horizon. It is this critical property as well as non-occurrence of ISCO that distinguishes the four dimension from the higher dimensions for gravitational collapse with angular momentum.
Accretion disk provides avenue for other interactions involving viscosity and collisions between particles from which particle can lose angular momentum and keep on falling inward until it reaches ISCO. Any further decrease in angular momentum would then take it down to the hole. This is how rotation could be imparted to the central object. This is however not possible in higher dimensions because there could exist no accretion disk. That is, in higher dimensions an accretion process cannot ensue for formation of black hole with a spin.
This however does not rule out a possibility of particle with angular momentum and energy exceeding the threshold determined by the maximum of potential barrier falling into black hole (as studied in previous studies) for overspinning of higher dimensional black hole. It was shown that overspinning was not possible because particles with overspinning angular momentum were not able to reach the horizon. Since Veff > 1 always, hence they would not be able to reach rotating black hole horizon unless they were somehow energised to a value overriding maximum of the potential barrier. According to authors, the only possibility of such an energising process could perhaps be collision with other compact objects, like neutron stars or black holes. This process may as well have to obey the constraints of potential barrier and hence may not be very effective.
How about taking the question to generalized theories of gravity? The most natural generalization of General relativity in higher dimensions is the Lovelock theory which is quintessentially higher dimensional. It is only the pure Lovelock theory, which has only one Nth order term without sum over lower orders in the Lovelock Lagrangian, that admits bound orbits in the dimension range 2N + 1 < D < 4N + 1, where N is the degree of Lovelock polynomial. For N = 1, Einstein gravity bound orbits exist only in four dimension while for N = 2 pure Gauss-Bonnet (GB) gravity, they exist for D = 6, 7, 8. On the other hand non rotating black hole is stable only in dimensions D ≥ 3N + 1; i.e. for pure Gauss-Bonnet in seven and eight dimensions. That is, in these dimensions there would occur ISCO and hence accretion disk could exist and thereby accretion process could ensue leading to formation of rotating black holes. Like Kerr black hole in four dimension, rotating black holes could be formed by gravitational collapse through the usual accretion process with accretion disk in pure GB gravity in D = 6, 7, 8 (black hole would stable only in dimension greater than six; i.e. in seven and eight dimensions). In general for the dimension window, 2N + 1 < D < 4N + 1, pure Lovelock rotating black holes could in principle be formed. It is however another matter that there does not yet exist an exact solution for pure Gauss-Bonnet vacuum equation describing a rotating black hole.
This is indeed a very elementary and straightforward analysis which identifies the critical requirement for collapse – existence of ISCO. This necessary condition is not satisfied in higher dimensions. This is the key consideration for setting in of accretion process. Added to that is also the fact that overall gravitational force for particles with angular momentum turns out to be repulsive as shown in the lower panel of Fig. 2. This is reflected in Veff ≥ 1 ( see Fig 1). This clearly indicates that collapse cannot commence from infinity unless it is pushed or impulsed by some external agency. No such agency could be invoked at infinity.
All other questions of detailed analysis of accretion process become pertinent only when this necessary condition is satisfied. This is exactly the case like non-existence of bound orbits around a static object in higher dimensions, which simply follows from the fact that gravitational potential falls off faster than the centrifugal potential. The same is the case here that gravitational potential due to mass falls off faster than that due to rotation of black hole, and thereby overall gravitational force turns repulsive in dimensions greater than five. It is interesting that such a simple consideration leads to a truly non-trivial result.
Finally, they concluded that higher dimensional rotating black holes cannot be formed in Einstein gravity by collapse and accretion process, they could however be formed only in pure Lovelock gravity, in particular pure Gauss-Bonnet gravity in dimensions D = 6, 7, 8. Besides kinematicity of gravity in all critical odd dimensions D = 2N + 1 and existence of bound orbits, the formation of rotating black holes in higher dimensions is yet another property that singles out pure Lovelock gravity.
Reference: Naresh Dadhich, Sanjar Shaymatov, “Could higher dimensional rotating black holes be formed by gravitational collapse?”, Astronomical Journal, pp. 1-6, 2021. https://arxiv.org/abs/2009.10528
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