Growth of a black hole requires the participation of a near-by accretion disk if it is to occur at a significant rate. The Kerr solution of Einstein’s equation is a vacuum solution, but the center of a realistic Kerr black hole is not a vacuum, so the predicted disk singularity does not exist. Instead, the center of a black hole is occupied by an ultra-dense, spheroidal core whose diameter is greater than that of the theoretical disk singularity. The surface of a black hole’s core is continually bombarded by energetic particles from the external universe. Hence the cold remnant of a gravitationally-collapsed star that has often been assumed to be present at the center of a black hole must be replaced conceptually by a quark-gluon plasma whose temperature is of the order of 10 12 K or more. The gravitational potential well of a black hole is extremely deep (TeV), but the number of discrete energy levels below the infinite-red-shift surface is finite. Information can be conveyed to observers in the external universe by thermally-excited fermions that escape from levels near the top of a black hole potential well.
A well-known result for a particle approaching a black hole in Schwartzschild or Kerr geometry [
The geometrical structure of a Kerr black hole of mass M and angular momentum a, as described, for example, by Padmanabhan [
1) A vertical symmetry axis where θ = 0 or π.
2) An outer ergosurface, given by
the mass of the black hole. This is a stationary limit surface, inside which it is impossible to be a stationary observer and within which the phenomenon of dragging of inertial frames occurs.
3) An outer event horizon given by
face touch one another at the poles of the vertical symmetry axis. The region between these two surfaces is called either the ergosphere or the ergoregion.
4) An inner event horizon given by
5) An inner ergosurface given by
6) A disk singularity in the plane perpendicular to the symmetry axis, given by
The event horizons are one-way membranes, analogous to a Schwartzschild event horizon. An incoming particle that passes through the outer event horizon must continue moving inwards until it reaches the quasi-solid core of the black hole.
The structures (2) to (6) derive from calculations rather than observations and therefore are not absolutely trustworthy, but the only serious doubt attaches to the disk singularity of item (6). This item is incompatible with the Heisenberg uncertainty principle, for the standard reason that the value of a positional variable x, y or z cannot have zero uncertainty in the physical universe without introducing an unacceptable infinite uncertainty into the conjugate momentum. The Generalized Uncertainty Principle of Vaganas and coworkers [
An interesting point here is that the solution of the Einstein equation for the interior of a Kerr black hole in terms of items (2) to (6) is incomplete [
In previous communications [
A Kerr black hole grows by capturing particles of matter inside the outer event horizon and forcibly merging them with the core. A polar on-axis approach would appear to be most effective because it reduces the likeli- hood of an incoming particle being scattered off the rotating core, but even scattered particles must eventually merge. For an off-axis incoming particle the angular momentum of the particle can either be co-rotating or coun- ter-rotating relative to rotation of the core. Padmanabhan [
The ultimate nature of the growth process is dependent on the depth of the gravitational potential well that has to be traversed by an incoming particle before it collides with the core. In previous communications [
where G is Newton’s gravitational constant, M is the mass of the black hole core, and m is the neutrino mass. This gave a set of approximate energy levels as
where the sum of the energy levels from n = 1 to infinity is the well depth −Ewell. One of the several approximations involved here consists of ignoring the mc2 contribution of the rest-mass of the neutrino. In practice the value of −Ewell is numerically so large (1.4 × 109 TeV for a neutrino in the gravitational well of a black hole of 106 solar masses; 1.5 × 108 TeV for the same neutrino in a black hole of 102 solar masses) that this particular approximation is valid for all likely particles. Here we are not considering the merger of two black-hole-sized objects, nor particles that arrive at the event horizon at relativistic speeds. For a very old and very large black hole the quasi-continuum of energy levels between the event horizon and the top of the potential well is available for occupation by thermally excited neutrinos and other particles that can be exchanged with the surrounding universe as part of an information flow to and from the interior of the black hole.
A particle that falls into a black hole core from the innermost stable orbit of the associated accretion disk can convey a major fraction of the energy of the well depth to the core, in addition to the kinetic energy associated with orbital motion in the accretion disk [
Neutrinos near the top of the thermal distribution within the well are able to escape to the external universe (
the core or distributed over discrete energy levels inside the potential well. Therefore the extremely cold mathematical singularity that has been presumed to exist at the center of a black hole must be replaced conceptually by a quark-gluon plasma whose temperature is of the order of 1012 K or more [
A growing Kerr black hole has an ultra-dense, quasi-solid, spheroidal core, whose radius is greater than the radius of the theoretical ring singularity, and an associated accretion disk, whose presence is required if growth is to occur at a measurable rate. Because a black hole core is not a vacuum, the predicted ring singularity is absent. The gravitational potential wells associated with black holes are extremely deep and the kinetic energy of in- falling particles that strike the core is correspondingly large. It is proposed that bombardment of a black hole core by incoming particles of matter from the external universe can generate a quark-gluon plasma. The thermal distribution of neutrinos and other light particles over discrete energy levels in the gravitational potential well can serve as both a storehouse for entropy and a channel for transfer of information between the black hole and the external universe.
This work was supported in part by the University of Canterbury.
Leon F.Phillips, (2015) Growing a Kerr Black Hole. Journal of Modern Physics,06,1789-1792. doi: 10.4236/jmp.2015.613181