I am looking for a mathematical solution of the following "paradox" regarding the Oppenheimer-Snyder collapse of a finite sphere of dust.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose the sphere of mass M has just collapsed to its Schwarzschild radius

[tex]R_S(M) = \frac{2GM}{c^2}[/tex]

1) Suppose there is another thin shell of dust just outside the Schwarzschild radius at ##[R, R + dr], \, R > R_S## which is falling into the black hole. Using the Schwarzschild time coordinate t it is clear that this shell never reaches the horizon at finite Schwarzschild time and therefore the mass flow through the event horizon is zero.

2) Suppose there is a stationary observer just outside the Schwarzschild radius at constant ##R > R_S## who sees the free falling dust. This shell has some mass dm. The shell will not fall through the Schwarzschild radius, but it will certainly fall through the radius R of the stationary observer and will reach ##[R^\prime, R^\prime + dr^\prime]## with ##R_S < R^\prime < R^\prime + dr^\prime < R## in finite Schwarzschild time.

(b/c the observer is stationary we can simply rescale Schwarzschild time to his proper time using a finite constant)

From the mass m one can derive the new Schwarzschild radius

[tex]R_S(M+dm) = \frac{2G(M+dm)}{c^2} = R_S(M) + R_S(dm) = R_S + dR_S[/tex]

Of course for large enough mass dm we find for this new Schwarzschild radius ##R_S < R^\prime < R^\prime + dr^\prime < R_S + dR_S##. Therefore the mass flow through the event horizon at ##R_S^\prime## is not zero - which contradicts the conclusion drawn from (1) that the mass flow is zero.

My question is whether there is a realistic solution for infalling matter of finite mass from which the growth of the event horizon can be derived analytically as a function of Schwarzschild time as some function ##R_S(t)##

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# Oppenheimer-Snyder collapse

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