B Schwarzschild Radius: Compression Inside Black Holes

[stoomart]

Learning about Schwarzschild radius from Wikipedia:

the radius of a sphere such that, if all the mass of an object were to be compressed within that sphere, the escape velocity from the surface of the sphere would equal the speed of light

Is it accurate to say any object of mass crossing the event horizon of a black hole is compressed sufficiently to have its own Schwarzschild radius, becoming a black hole itside of a black hole?

 

[Dale]

Is it accurate to say any object of mass crossing the event horizon of a black hole is compressed sufficiently to have its own Schwarzschild radius, becoming a black hole itside of a black hole?

No. For a super massive black hole nothing particularly bad happens at the event horizon in standard GR.

 

[stoomart]

Is there a known equation to calculate the gravitational or external force required to compress an object to the volume of its Schwatzchild radius?

 

[PeterDonis]

Is it accurate to say any object of mass crossing the event horizon of a black hole is compressed

No. (This is true even if we leave out the rest of the sentence.)

Is there a known equation to calculate the gravitational or external force required to compress an object to the volume of its Schwatzchild radius?

You can't; before you reach that point, the object's internal structure will be destroyed and it will collapse. The limit before that happens is actually 9/8 of the Schwarzschild radius; at that point, even infinite internal pressure is insufficient to keep the object static.

Also, asking what force is required to achieve this is a bit misleading, because there is no way to exert force without a source of energy, and that energy will itself gravitate and contribute to the total mass of the system. So if, for example, we rig up a big piston and use it to start pushing an object together, we will find that what eventually becomes a black hole if things get compact enough is not just the object inside the piston, but the whole assembly of object, piston, and the piston's energy source.

 

[stoomart]

Also, asking what force is required to achieve this is a bit misleading, because there is no way to exert force without a source of energy, and that energy will itself gravitate and contribute to the total mass of the system. So if, for example, we rig up a big piston and use it to start pushing an object together, we will find that what eventually becomes a black hole if things get compact enough is not just the object inside the piston, but the whole assembly of object, piston, and the piston's energy source.

I had two scenarios in mind: near/at the singularity in a common black hole, and inside a particle accelerator.

 

[PeterDonis]

near/at the singularity in a common black hole

Here the matter is already inside a black hole, so it already is compressed enough to form a black hole. Compressing it more makes no difference.

inside a particle accelerator.

We are still many orders of magnitude away from being able to probe the Planck regime in particle accelerators, which is what it would take to have a significant chance of making a black hole inside of one.

 

[stoomart]

Here the matter is already inside a black hole, so it already is compressed enough to form a black hole. Compressing it more makes no difference.

So then a foreign object entering a black hole does get compressed to/past its Schwartzchild radius? I interpreted Dale's response as saying this doesn't happen.

 

[Dale]

So then a foreign object entering a black hole does get compressed to/past its Schwartzchild radius? I interpreted Dale's response as saying this doesn't happen.

You do realize that the event horizon and the singularity are entirely different, right? I was answering a question about the event horizon, and @PeterDonis was answering a question about the singularity.

Again, for a sufficiently large black hole nothing happens to an object crossing the event horizon. At the singularity our theories break down, but long before that tidal forces would shred any matter.

 

[stoomart]

You do realize that the event horizon and the singularity are entirely different, right? I was answering a question about the event horizon

Yes, maybe I was too vague (or wordy) in my OP. When I said "crossing the event horizon", I was referring generally to entering a black hole. Is it currently known where in a black hole matter is compressed?

 

[Dale]

Is it currently known where in a black hole matter is compressed?

Usually the word used is "spaghettified". It is compressed in the horizontal directions and stretched in the vertical direction. For a large black hole that would be close to the singularity. For a small black hole it could be well outside the event horizon.

That is for a test object free falling. For a static structure surrounding the black hole, see @PeterDonis answer above.

 

[PeterDonis]

So then a foreign object entering a black hole does get compressed to/past its Schwartzchild radius?

No. See below.

Is it currently known where in a black hole matter is compressed?

As Dale said, what actually happens is compression horizontally and stretching vertically. (Actually, this is only true in a highly idealized black hole that formed from a perfectly spherically symmetrical collapsing object. In a real hole, the stretching/compressing would be highly chaotic--google "BKL singularity" if you want the gory details.)

However, as I said before, thinking of this process as eventually "compressing matter inside its Schwarzschild radius" (even along just one dimension) is not correct. The matter is already inside a black hole, so it is already inside its Schwarzschild radius. In other words, once a small object falls into a black hole, you can't really think of it as having a separate "Schwarzschild radius" of its own. It's part of the black hole in that respect, and its Schwarzschild radius is the same as that of the hole.