(I could be phrasing my question incorrectly, so bear with me as I try to explain it.) Recently I've been very interested in the geometry and physics of black holes, so I started messing around with various gravity-related equations and geometric equations pertaining to n dimensions, only to find myself wanting more. I wondered if there was an equation for the amount of pressure (energy) on the center point of the gravitational force of a collapsing star, eventually creating a black hole. Does this equation exist? If not, who wants to figure it out?
There is no such equation I'm afraid.
Even for a static star the calculation is hard. The pressure is calculated from the equation of state but while we have a pretty good handle on the equation of state for ordinary stars the equation of state for neutron stars is not fully understood. The problem is it's impossible to reproduce such extreme conditions in the laboratory so we have no way of determining the equation of state experimentally. In practice we use a best guess and the calculation is done numerically on a (very) large computer.
For a collapsing star the calculation is even more complicated. The problem is that for a collapsing star we also need to consider the momentum of the infalling gas because the rate of change of momentum of this gas creates an additional force and hence contributes to the pressure. As with the static star we have to resort to numerical models and a great deal of computer time.
Googling will find you lots of articles on modelling stars, both static and collapsing, but be warned that this has evolved into a very complex discipline that will be impenetrable for the beginner.
At the final fraction of a nanosecond before a collection of particles of, say, mass = M, collapses to form a black-hole, one may perhaps assume that the initial mass, M, has been transformed into E = mc^2. If the math (below) is correct, then the force of gravity at the Schwarzschild boundary would be constant for all black-holes and all black-holes would have the same energy-density...2E/r:
Since r = 2GM/c^2, and c^2 = 2GM/r(Schwarzschild) = constant, (and M/r = constant), one might assume that the escape velocity for all Black-holes should be constant; and the gravitational force at the Schwarzschild boundary should also be constant:
r(s) = 2GM/c^2 *radius of BH; or
c^2 = 2GM/r = constant
Thus if c = constant, then M/r = constant, and multiplying be C^2…
c^4 = 2GMc^2/r = 2GE/r = constant; and dividing by G….
c^4/G = 2E/r = constant; note that c^4/G = the Planck force (F) = 1.2 Newtons X 10^44: then
F = 2E/r = constant. = The gravitational force of a black-hole.
Thus it would appear that all black-holes have a maximum force (very close to the EH) equal to the Planck force. This would suggest that all objects located at (near) the surface of a BH would "weigh" the same, regardless of the radius (size) of the BH.
Additionally...Multiplying by 2pi gives:
F = 4pi E/Schwarzschild circumference (C) = constant. Thus it would also appear that the energy density per unit (Schwarzschild) circumference is the same for all black-holes, regardless of their size. This would suggest that all black-holes have the same "temperature"...the Planck temperature.