Speed of light inside a black hole

[Gerinski]

We know that light's speed gets slowed down when traveling through a medium, and the more dense the medium the slower light can travel (of course c remains constant, but it takes longer to travel due to the continuous scatterings, absorbtions and re-emissions).

Inside a black hole, just below the Schwarzschild's radius, matter density should be high enough as to also slow down light? If so, besides the usual description that it's gravity's spacetime curvature which traps any light emitted from within the Schwarzschild radius preventing it from escaping, could the slowing down due to matter density be also a contributing factor in light being unable to escape? so in fact the hole could become black even with a mass a bit lower than that required for the spacetime curvature effect alone?

Of course I don't know if the interior of the black hole can be considered 'transparent' or 'opaque', but this is linked to some comment I read that the density at the Schwartzschild radius of some black holes can be similar to that of water:

https://www.physicsforums.com/threads/density-of-supermassive-black-holes.483191/

 

[phinds]

The "density" of a BH is a mathematical fiction without much physical meaning. It's just the mass divided by the volume surrounded by the Event Horizon, but the mass is all in the center so the density just inside the EH is essentially zero and so consequently your concept that this "density" would contribute anything to slowing anything down does not make sense.

 

[DEvens]

Lots of ideas flying around here, not always consistently.

Inside a black hole there is this interesting condition that all time-like and all null-like geodesics point inward. This means that inside the horizon all matter and all light is getting closer to the central singularity. And, as a homework assignment in my fourth year undergrad shows, the proper time matter experiences before it gets to the singularity is finite. This is the property that makes the black hole black.

So, absent matter continuously falling into a black hole, the idea you describe of dense matter just inside the horizon does not apply. Light that goes inside the horizon keeps going to the centre.

The nature of a very dense object that is not inside its horizon is very different. As long as the object is outside its horizon, then light can escape. Mass can escape. It may do it very slowly, very inefficiently, require a lot of kinetic energy to do it, etc. But in principle, it can escape.

The idea of a black hole with the density of water, or even less, goes like so. The mass required to be inside a given radius to produce a black hole is proportional to that radius. Twice the mass will disappear inside a radius twice as big. But the volume inside that radius is proportional to the cube of the radius. So the volume inside twice the radius is eight times as much. This means that, with uniform density, a larger mass disappears at a larger radius proportional to the radius. But with a volume proportional to the cube of the radius. And so a density proportional to one over the cube of the radius. So twice the mass will have one eighth the density when it disappears behind its own horizon. So with a mass in the range of a galaxy, it would disappear behind its horizon with a density about that of air.

The reason this last is important is because it tells us that a gravitational horizon could form in a range of densities that do not involve any unusual state of matter. So the concept of a black hole, at least the horizon part, is not affected by any potential exotic behaviour of matter at high densities. Something odd may happen at the singularity. Indeed, one would expect that something unusual might well happen before infinite density was achieved. But the horizon can happen at very mundane density.

 

[Gerinski]

The "density" of a BH is a mathematical fiction without much physical meaning. It's just the mass divided by the volume surrounded by the Event Horizon, but the mass is all in the center so the density just inside the EH is essentially zero and so consequently your concept that this "density" would contribute anything to slowing anything down does not make sense.

Inside a black hole there is this interesting condition that all time-like and all null-like geodesics point inward. This means that inside the horizon all matter and all light is getting closer to the central singularity. And, as a homework assignment in my fourth year undergrad shows, the proper time matter experiences before it gets to the singularity is finite. This is the property that makes the black hole black.

So, absent matter continuously falling into a black hole, the idea you describe of dense matter just inside the horizon does not apply. Light that goes inside the horizon keeps going to the centre.

Thanks. So first, you are both saying that unless new matter is falling into the hole, all the volume from the EH to the center is empty vacuum of heavily distorted spacetime, all the mass is at the center (let's not bother with the singularity itself). If so it's fine, but then the question itself becomes trivial, as no light can be emitted from within the hole if there is no matter there. Frequent popular pictures depict the fact that light can not escape by picturing some photon(s) being emitted close below the horizon and showing that their path bends inwards. The truth though according to you is that they are black not just because light can not escape, but because there is no light to escape, there's no matter inside which could radiate a photon, right? (besides the matter at the center, let's forget about that for simplicity).

Secondly, black holes do frequently have large accretion disks of infalling matter (well, if only because these are the ones we can detect most easily). So in these, around the disc rotation plane there is matter continuously falling in, so there must be certainly a 'density' in the region below the EH and beyond towards the center. All that matter must fall with a certain velocity, it does not suddenly disappear at the EH and reappear at the center. And that infalling matter may still radiate photons on its way from the EH towards the center. So in this case, those photons might be slowed down by the matter density, just as I was asking in the OP. Well at least this seems intuitively reasonable.

I understand that the geodesics point towards the center because of gravitational warping, I only asked whether it could be the case that even at a point where the geodesic still escapes the hole by a small fraction, in practice light from that point can not escape because it gets slowed down by the matter density in that region and ends up going also to the center. No more no less. BTW, if what I said makes sense, the fact that in black holes with infalling matter from an accretion disk, in the 'equator' plane there is matter all the way from EH down to the center, while in the 'polar' plane all the distance between EH and center is empty vacuum of distorted spacetime, shouldn't it mean that rotating black holes with infalling matter are not precisely spherical but slightly flattened as the Earth is?

 

[Drakkith]

Frequent popular pictures depict the fact that light can not escape by picturing some photon(s) being emitted close below the horizon and showing that their path bends inwards. The truth though according to you is that they are black not just because light can not escape, but because there is no light to escape, there's no matter inside which could radiate a photon, right? (besides the matter at the center, let's forget about that for simplicity).

I wouldn't say that. As far as I understand, matter takes some amount of time to fall from the EH to the singularity, which means that there is actually matter inside the EH that can emit light.

I understand that the geodesics point towards the center because of gravitational warping, I only asked whether it could be the case that even at a point where the geodesic still escapes the hole by a small fraction, in practice light from that point can not escape because it gets slowed down by the matter density in that region and ends up going also to the center. No more no less.

Geodesics cannot escape the black hole at all. A geodesic is a one-dimensional line with no width, so it cannot 'escape by a small fraction'. Once the geodesic is inside the EH, it stays there.