Schwarzschild’s equations & Newtonian Potential

[TheSicilianSa]

Would someone please help me in thinking the following out:

I was taught that the Newtonian gravitational potential was given as -Gm/r; and when examining a sphere of uniform density ρ and radius r, then at any point x ≤ r (within the sphere), m would refer to the effective mass of an equivalent sphere of radius x. -The logic given was that the effective mass of all points > x would “cancel each other out”.

I was content with that explanation, accepting that at the center of the sphere the gravitational potential would be zero. However, I was always uneasy with the “indeterminacy” at r=0 | m=0.

However, if we were to rephrase the “effective mass” (at any point x) in terms of the sphere’s density, would we not get :

(1) m(x) = 4π ρ { r δ(x – r) + x δ(r – x) } /3; where δ = delta function

And, were we to replace “m” with (1) in Schwarzschild’s equations (i.e. in lieu of m in the standard expression for the Newtonian potential), would this not eliminate the singularity at x=0?

And, further, does this not imply that for a black hole, the gravitational effect of its mass must go from a maximum at the event horizon to zero at the body’s center?

 

[marcus]

Ordinarily different people would answer. So you get a chance to try discussing with a variety. Just patiently keep asking good questions until a range of people (George Jones, Cristo, Russ Watters, Hellfire, Matt.O...to mention only a few) respond.

I'm probably not the best to reply to BH questions. But this has been on the board for a few hours and nobody has responded, so I'll take a shot.

What goes on inside the eventhorizon of a BH is necessarily model dependent.
What you think it looks like and what the falling-in experience is like depends on what model.

I respect and am interested by the fact that you set up your own mathematical model.

I'm not sure I understand but I think you have like a continuous density inside the eventhorizon (EH).

The prevailing model doesn't. It has a great emptiness inside the EH, all except for a tiny dot at the center which has very very high density.

The classical 1918 Schwarzschild model actually breaks down right at the center and can't say anything about conditions---it suffers from infinities

Vintage 2008 modifications of that model don't break down, they just achieve a small region of very very high density, but not infinite density.

Various things happen. It's my bedtime and I'm sleepy. Maybe I can discuss this tomorrow.
In a number of papers I've looked at, there is a runaway collapse which gets down to where the density is a substantial fraction of Planck density.

Some models of quantum gravity predict that as Planck density is approached certain correction terms become important and cancel out the attractiveness of gravity, the fundamental degrees of freedom (constituents) of geometry and matter fight back and halt the collapse. It's just a math model that happens to exhibit this strange feature where quantum corrections in the equations make gravity repell when the density gets high enough. An actual singularity (a math breakdown) is avoided.

In those models you don't get infinite density (which seems unreal so not to consider seriously) but you get high density in a very small region. Maybe they have something about this at the Einstein Online website. (link in sig at end of post). More about this tomorrow, hopefully from more besides myself.