Zero gravity at center of astronomical bodies

Question

All theories agree that the net gravity at center of all astronomical bodies is zero.

1. Are black holes exception to this?

2. Is gravity zero at the center:) of singularity?

3. Is anything like center of singularity even there?

Answer 1

You can show within Newtonian gravitation that the net gravitational force at the center of a continuous mass distribution with a nontrivial point group symmetry is zero. (If the body doesn't have a point group symmetry then there's no clear notion of a "center".) Black holes fail to satisfy two of these conditions - they're not Newtonian and the mass distribution isn't continuous (it's supported entirely at the singularity) so this result doesn't apply.

Answer 2

You start with the statement:

All theories agree that the net gravity at center of all astronomical bodies is zero

and I would guess that by this you mean the gravitational acceleration at the centre of a spherically symmetric mass is zero. When we do the calculation of the acceleration using Newtonian gravity we find the acceleration at a distance $r$ from the centre is:

$$ a(r) = \frac{GM(r)}{r^2} $$

where $M(r)$ is the total mass within a sphere or radius $r$. At first glance this is singular because as $r \rightarrow 0$ we get a factor of $1/0^2$, which is singular. Fortunately the expression for $M(r)$ is:

$$ M(r) = \rho\tfrac{4}{3}\pi r^3 $$

so our equation becomes:

$$ a(r) = G\rho\tfrac{4}{3}\pi r $$

And this is not singular at $r=0$ so the acceleration at the centre has the well defined value of zero.

But suppose our mass is a point mass (without worrying too much about the physical meaning of this) so that $M(r)$ is a constant $M$ that is independent of $r$. In that case we once again have an equation for $a(r)$ that is singular at $r=0$. The physical interpretation of a singular quantity is that it is undefined i.e. with a point mass the acceleration at the centre is undefined.

This is all scene setting really as with a black hole the situation is rather more subtle. Assuming that by black hole you mean the Schwarzschild metric, then a black hole does not have a point mass at the centre because a Schwarzschild black hole does not contain any mass at all. It is a vacuum solution i.e. it is a solution in which the stress-energy tensor is everywhere zero. The mass $M$ that appears in the Schwarzschild metric is the ADM mass, but this is a property of the spacetime as a whole rather than a mass in the usual Newtonian sense of the word.

This shouldn't worry us too much because the Schwarzschild metric is an idealised model of a black hole that cannot exist in our universe. In fact strictly speaking no black holes exist in our universe because a true event horizon takes an infinite time to develop. For more on this see Why does Stephen Hawking say black holes don't exist?, but don't get too hung up on this. When astrophysicists talk about black holes they mean objects that are in practice indistiguishable from the idealised Schwarzschild and Kerr geometries, and such objects most certainly exist.

Anyhow, the point I'm gradually getting around to is that your question simply doesn't have an answer. This is because if you are asking about the idealised Schwarzschild black holes then the geometry is undefined at $r=0$ so the gravitational acceleration is undefined there. If you are asking about real black holes then the gravitational acceleration at the centre is finite because they haven't finished collapsing yet, and indeed won't finish collapsing at least until an infinite time has passed, or indeed not at all if they evaporate due to Hawking radiation first.

One last comment, which I think is what you're getting at in your point 3. With the idealised black holes we usually consider the singularity to be not part of the manifold in which our spacetime geometry is defined. So the centre of a Schwarzschild black hole is not part of the spacetime and therefore it is meanigless to ask at the various physical quantities there.