Would someone please help me in thinking the following out:(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Schwarzschild’s equations & Newtonian Potential

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