Force in the Centre of a Black Hole: Explained

[Matthew Goldman]

The force in the centre of the earth, assuming it was a perfect sphere and the density was the same everywhere, would be zero as the pull from all directions would cancel. Why isn't it like this for a black hole? Surely the forces from each direction should cancel leaving zero resultant force at the centre rather than a singularity.
Thanks,
Matt

 

[PeterDonis]

Why isn't it like this for a black hole?

Because a black hole is not an ordinary object like the Earth. The interior of a black hole is not static; it's not possible for anything to stay at the same radius. Everything has to fall into the singularity. (Also, the interior is vacuum; it's not filled with matter.)

 

[ChrisVer]

(Also, the interior is vacuum; it's not filled with matter.)

I guess that intuitively the best explanation. If you enter a black hole, you won't need a drill to drive you in the center... in fact you will be free-falling into the vacuum (until you'd be able to feel the tidal forces)...

 

[bcrowell]

From #1 it sounds like you're convinced that the force of gravity at the center of a black hole is large (or maybe infinite). Actually there is no reason to think this. The center of a black hole is the singularity, which is not a point or region of spacetime -- it's like something that's been cut out of the spacetime. Therefore we can't say what it would mean to put an object there and measure the force on it.

A separate issue is that in GR we don't describe gravity as a force, and we describe free-falling objects as not accelerating, so in that sense all gravitational fields can be said to be zero everywhere.

What you can say is that if you have some measure of tidal forces T, then for a black hole \lim_{r\rightarrow0}T=\infty. (A typical measure of tidal forces would be a curvature tensor or a curvature polynomial such as the Kretschmann invariant.) A tidal force isn't the same thing as a Newtonian gravitational force.

Note that even in Newtonian mechanics, a gravitational force can't crush an object. When a fragile object such as a house of cards collapses, it's because there are two different forces acting on it, one from the table and one from the Earth's gravity. These forces are in opposite directions and are not applied to the cards at the same points.

 

[Matthew Goldman]

Thanks everyone for replying. I feel like I have things a bit straighter in my head now. Particularly the idea of free falling in.
Cheers,
Matt