The Black Hole Problem

Question

Consider a spherical body of uniform density $\rho$ and initial radius R. You can imagine this body containing another sphere of radius R/2 which touches the center and the periphery of the larger sphere. The smaller sphere has 8 times less mass, since the mass goes as the radius cubed.

If you let the big sphere collapse gravitationally along with the smaller one, keeping the density uniform, the whole thing just shrinks proportionally. The Schwarzschild radius is proportional to the mass, so $R_S= 2GM/c^2$ for the larger mass, and for the smaller sphere $r_S= 2GM/8c^2$, one eigth as big. So when the large mass collapses, the smaller one hasn’t.

It follows that the smaller sphere can still communicate with things outside, but the larger sphere should prevent it from doing so. The outgoing world lines at P due to the smaller sphere must be trapped by the spacetime effects of the residual portion ie, the portion represented by the larger sphere minus the smaller one.It seems that the remaining larger irregular shaped mass has a greater attractive power than the regular shape of greater mass at specific points like P.Is it really so?

For a sphere of a larger fraction of the size, say 3/5R, which touches the boundary of the larger sphere at some point and includes the center [of the larger sphere, the smaller Schwarzschild radius is about .2 of the larger Schwarzschild radius.

When the larger sphere collapses the smaller one has not collapsed---it has world lines in the outward direction.The smaller sphere is not supposed to have a singularity on its own.But it contains the singularity of the bigger mass , though not at its center.Is the singularity, [of the larger sphere], produced by the effect of the residual portion?

Related question: Can we conclude solely on the basis of these “Experiments’ that different parts of the body can exchange signals after it has collapsed? [QM effects are not being considered in this problem]

Answer 1

These points are well known and correct--- the scaling of the Schwarzschild radius is linear in mass, while the scaling of mass is cubic in the radius, so there is a counterintuitive mismatch in scaling which tells you the collapse radius, and big black holes have much less mass per radius cubed than small black holes, so a big container will collapse before any of its parts will.

This mismatch means, as you deduced, that objects in the interior of a collapsing star stay in communication with each other for a while after the big object has collapsed, they all collapse together to the center, and they can't talk to the outside, but they can talk to each other for a while as they collapse, and if the surrounding collapsing sphere is enormous, the time it takes to collapse after the horizon is formed is of the order of the time it takes light to go from one end to the other, which can be so long that they might not notice any collapse has occured. This is another way of saying that you can fall into an enormous black hole without anything strange happening as you cross the horizon.

The first question is true--- the remaining mass (regardless of shape) can block communication to the outside world from the extreme point of the inner smaller sphere. This doesn't mean its gravitational effect is somehow super-enhanced, it just is nonlocally attractive, and blocks the outgoing light.

For your second example, it doesn't matter that the subsphere contains the center. As the collapse happens, everything is squeezed down nonuniformly with the bound provided by the collapsing light-cone. The collapse is global looking, and depends on the whole mass inside the horizon, not on some parts and not others.

There is no paradox here, but there is an unusual scaling, which makes it clear that gravity is acting in a strange nonlocal way during collapse. The ultimate reason for the unusual scaling is the holographic principle, although of course, this was discovered the other way around, the unusual scaling suggested the holography.