Why can't the interior of a black hole be empty?

[Pencilvester]

Can someone explain to me why there must be a real/meaningful space inside of a black hole?

I have been autodidactically working on understanding the mathematical concepts that general relativity is based on, so I've never had anyone to ask questions to (until it occurred to me to find a forum like this). I started with linear algebra and vector calculus, and worked my way up to a basic understanding of tensor analysis, topology, manifold theory, and differential geometry, so if you use math in your answer be aware that I am not completely proficient in these latter subjects.

Anyway, I was watching the Leonard Susskind lectures on GR from the Stanford youtube channel, and in the one where he talks about the Schwartzschild metric and what it means for objects falling into black holes, he is adamant that the apparent slowing down (in velocity, not necessarily time dilation) of objects near the event horizon and only crossing said horizon after an infinite amount of time is merely a peculiarity of the choice of coordinates. His reason seems to be that it takes a finite proper time for the object to cross the event horizon, therefore, from the perspective of the object, nothing strange happens near the event horizon.

While I completely agree that it would take a very finite amount of proper time to cross the horizon, it seems to me that the object would always be destroyed before it could ever actually cross. Let's suppose that you're standing at a comfortable distance "above" the black hole with a rubber ball and a flashlight. Drop the rubber ball so that its trajectory will go straight to the black hole (only motion through t (time) and r (radius from center of black hole)) and as it begins to fall, start strobing the flashlight in its direction (and never stop). Eventually, I will call the amount of time between pulses "dt" for obvious reasons, but for now dt will retain its normal meaning of a small, arbitrary change in coordinate time. So if I've understood correctly, the Schwartzschild metric looks like this:
dτ^2 = (1 - s/r)dt^2 - (1/(1 - s/r))dr^2, where "s" is the radius of the event horizon (really 2GM and using units so that c=1). I've omitted the dΩ term for simplicity. Working this metric around as the Legrangian to get information about the freely falling ball introduces total engergy, "E". I can then manipulate and solve for dr/dt, and then linearly approximate dr by multiplying both sides by dt, then plug this approximation back into the metric, which yields dτ = m*dt(1 - s/r)/E. It's easy enough to see that as r approaches s from the outside, dτ approaches 0. Now if we interpret dt as the length of time between light pulses, we can see that as the ball approaches the event horizon, the length of time between light pulses that the ball experiences, dτ, approaches 0 and therefore the frequency of pulses approaches infinity. I might not know a ton about how light interacts with matter, but I would expect that this would deliver a good deal of energy to the ball in a very short amount of (proper) time and just utterly destroy it, or at least turn it into a melange of tiny high energy particles and radiation, or perhaps just radiation.

But then there's the matter of light itself, according to the Schwartzschild metric, only crossing the event horizon after an infinite amount of coordinate time as well, but light doesn't experience the passage of proper time (dτ for a light ray is always 0), so a statement like "it takes a finite amount of proper time for a light ray to cross the event horizon" would be meaningless.

So unless I'm missing something (and that's why I'm posting this, to see if anyone can tell me what I'm missing), it seems to me that everything that falls "into" a black hole just gets slowly eaten away by incoming photons before it ever crosses the horizon (a process that would indeed take a very short amount of proper time from the perspective of the infalling object). So then what is the purpose of discussing the interior of a black hole? If all of the frames of reference outside of a black hole all agree that nothing, not even light, crosses the event horizon until all of time has passed, and the black hole evaporates before then, why can't we just consider a black hole to just be a dense shell made up of radiation (and matter being slowly blasted into radiation)?

Someone might respond by saying, "If all of the matter and energy is on the outside of the event horizon of a black hole, then wouldn't it no longer be a black hole? Wouldn't we see them?" Firstly, unless I misunderstand Steven Hawking, or unless he's wrong, black holes do, in a sense, emit radiation, so they aren't completely invisible. Secondly, any light rays that find their way out from near the event horizon to an observer a safe distance away from the black hole would be incredibly redshifted, right? They would be so low energy, it would be difficult to detect them using normal instruments(?). I haven't really explored redshift or blueshift too thoroughly, so I'll have to rely on someone with more experience than I to confirm or refute that.

Which finally brings me back to my reason for posting this: I have only had books, my curiosity, and a few Leonard Susskind videos to teach me about GR, so if there's anyone reading this that could give me insight as to what I'm missing here, it would be much appreciated.

 

[PeterDonis]

We have a four part series of PF Insights articles that might help. The first one is here:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-1/

(Full disclosure: I'm the author of the articles. )

To briefly summarize the key point in reference to your questions: when we say things like "a black hole has an interior, inside the horizon", or "things can fall through the horizon into a black hole's interior", these are statements about geometry. We are just saying that the spacetime geometry includes a particular region (inside the horizon), or that particular curves (ingoing timelike and null curves) go into this region. And these statements about geometry are independent of any choice of coordinates, which means that they do not require any choice of how to split spacetime into "space" and "time" or how to assign a "time" to particular events.

When you see statements like "it takes an infinite amount of time for something to reach the horizon", on the other hand, this is a statement about a particular set of coordinates--and those coordinates become singular at the horizon, so they cannot be used to analyze what happens at or beneath the horizon. And if you try to translate such statements into statements about geometry, what you end up are things like "it takes longer and longer for outgoing light rays to reach a distant observer, as the rays are emitted closer and closer to the horizon". Note that this statement says nothing about ingoing light rays, only outgoing ones. Or we end up with statements like "a distant observer cannot see any events that happen at or below the horizon"--which says nothing about whether such events happen, only about whether the distant observer can see them (i.e., receive light signals from them).

So the basic answer to your question is that, when we look at the spacetime geometry--the fundamental thing that GR tells us about a black hole--it has a horizon and a region inside it. The geometry also tells us that distant observers cannot see what happens at or beneath the horizon; but the fact that they cannot see those events does not mean they aren't there.

 

[PAllen]

You make a statement supported by an invalid derivation that pulses of light emitted at fixed periodicity from far away will reach an infaller at ever increasing rate as it approaches the horizon. This is equivalent to stating that the frequency of light received from a far away source becomes infinitely blueshifted for the infaller on approach to the horizon (because light itself is just classically wave crests emitted with a periodicity). However, it is, in fact true, that an infaller from far away sees light from external sources moderately red shifted on crossing the horizon, not blue shifted at all, let alone infinitely blueshifted. This is a very well known fact.

If I have time later, I might try to find a good derivation to link to (I notice Peter's insight series does not address this question) for the spectral shift measured by a BH infaller. As for your derivation, one of many issues is that you are not accounting for the fact that the t coordinate becomes infinitely scaled near the horizon, and the the relation of dt to local physics near the horizon is radically different from its relation to local physics far away. The most direct (but not simplest) way of deriving the correct result is to follow successive null geodesics from a far away source that reach a free fall geodesic arbitrarily close to the horizon. Compare the proper time interval between them per the far away source with the proper time difference between the reception events on the free fall world line. You find that for a free fall from far away, the result is red shift not blue shift - the proper time between receptions is greater than the proper time between emissions.

 

[Khashishi]

So unless I'm missing something (and that's why I'm posting this, to see if anyone can tell me what I'm missing), it seems to me that everything that falls "into" a black hole just gets slowly eaten away by incoming photons before it ever crosses the horizon (a process that would indeed take a very short amount of proper time from the perspective of the infalling object).

That sounds a bit like a firewall. A black hole firewall is a somewhat controversial idea that everything that falls into a black hole will run into a firewall near the event horizon and be destroyed. The idea is supposed to solve the black hole information paradox. But it isn't predicted by GR. As you already know, GR predicts that nothing strange happens as you fall through the horizon. We don't know if GR is still valid in the interior of a black hole, but until we have evidence otherwise, we assume GR is valid until the singularity at the center.

 

[PAllen]

That sounds a bit like a firewall. A black hole firewall is a somewhat controversial idea that everything that falls into a black hole will run into a firewall near the event horizon and be destroyed. The idea is supposed to solve the black hole information paradox. But it isn't predicted by GR. As you already know, GR predicts that nothing strange happens as you fall through the horizon. We don't know if GR is still valid in the interior of a black hole, but until we have evidence otherwise, we assume GR is valid until the singularity at the center.

A firewall is a completely different mechanism derived from entanglement and the Hawking radiation process. It has absolutely nothing to do with the process described by OP, which is simply incorrect.

 

[PAllen]

Here is a simplified derivation of observed redhsift of distant light as seen by a BH infaller:

https://physics.stackexchange.com/q...falling-observers-see-gravitational-blueshift

Here is a more complete treatment:

http://iopscience.iop.org/article/10.1088/1742-6596/104/1/012008/pdf

 

[Dale]

Can someone explain to me why there must be a real/meaningful space inside of a black hole?

One thing that you might want to look at is "geodesic completeness". It is certainly true that you could mathematically construct a black hole spacetime that would end at the event horizon, but the resulting spacetime would be geodesically incomplete. That is generally seen as a pathological feature of a spacetime.

 

[Pencilvester]

Here is a simplified derivation of observed redhsift of distant light as seen by a BH infaller:

https://physics.stackexchange.com/q...falling-observers-see-gravitational-blueshift

Here is a more complete treatment:

http://iopscience.iop.org/article/10.1088/1742-6596/104/1/012008/pdf

Thanks, PAllen! So just to make sure I'm understanding the concepts behind the mathematical process used in the explanation you linked to, we first calculate how the frequency "observed" (quotes implying a Newtonian sense of the word) relates to the frequency emitted simply using classical notions of relative velocities (velocities which, I realize, are not calculated using classical Newtonian methods), and then just convert the observed frequency from coordinate time to proper time to find the true observed frequency? Am I getting that right?

 

[PAllen]

Thanks, PAllen! So just to make sure I'm understanding the concepts behind the mathematical process used in the explanation you linked to, we first calculate how the frequency "observed" (quotes implying a Newtonian sense of the word) relates to the frequency emitted simply using classical notions of relative velocities (velocities which, I realize, are not calculated using classical Newtonian methods), and then just convert the observed frequency from coordinate time to proper time to find the true observed frequency? Am I getting that right?

If I were attempting a short verbal summary, it would simply be:

The metric directly gives blueshift between distant static observer and near horizon static observer. However, a free faller from far away passes a near horizon static observer with local speed approaching c away from the direction of the distant source. SR Doppler is applied in this purely local context, between what the static observer sees and what the passing free faller would see. Both the redshift due to relative speed between free faller and static observer, and the blueshift seen by static observer, grow without bound on approach to horizon. However their combination is finite in the limit. In fact, it is a red shift factor of 2 for free fall from far away. For a free faller dropped from close to the horizon, there might be blueshift at horizon crossing, but it would be finite blue shift. There is never unbounded blue shift for the free faller at horizon crossing.

 

[Umaxo]

But then there's the matter of light itself, according to the Schwartzschild metric, only crossing the event horizon after an infinite amount of coordinate time as well, but light doesn't experience the passage of proper time (dτ for a light ray is always 0), so a statement like "it takes a finite amount of proper time for a light ray to cross the event horizon" would be meaningless.

i think perhaps you misread something. Most probably there was written something like "it reaches event horizon in finite value of affine parameter", instead of proper time.

This has something to do with geodesic completness that @Dale suggested for you.

 

[PAllen]

i think perhaps you misread something. Most probably there was written something like "it reaches event horizon in finite value of affine parameter", instead of proper time.

This has something to do with geodesic completness that @Dale suggested for you.

More physically, consider a free faller emitting light toward the BH. The light leaves them at a local speed of c, and they never can catch a light pulse, so it crosses before the free faller emitting it. The latter crosses in finite proper time, with the light preceding it.