I know the question has been asked about how an event horizon is distinguishable from a singularity given that time must come to a stop at the event horizon, but I haven't been fully satisfied by the answers I've seen and I have another related question that to me deepens the issue.

If the entropy of a black hole is related to the surface area instead of the volume, doesn't this suggest that the interior doesn't exist to an outside observer? If we think of entropy as disorder, or 'inability to do work' (if we take a pragmatic approach), then we should expect the entropy to be proportional to the surface area instead of the volume as everything inside of the horizon is unavailable to that which is outside.

This line of thought is in line with black hole complementarity. I find this to be an interesting example of David Hilbert's belief that,

"The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought… The role that remains for the infinite to play is solely that of an idea."

If the equivalence principle requires that a falling observer passing through the event horizon would not notice that they did so, but an outside observer must witness them incinerated by a firewall (or at least for them to take infinitely long to pass the event horizon), then can't we say that these are simply two different perspectives of the same event? Is the event horizon not a projection of the singularity? A projection that can't be experienced because it disappears as one approaches it?

Time comes to a stop at the singularity and at the event horizon because they are the same thing, that just look different depending on the motion and location of the observer. This 'infinity that is nowhere to be found in reality' (the experience of infinite time dilation) is the singularity/event horizon which always remains as an inexperiencable idea as it remains a distant horizon to all observers.

I'll add one more point to make it clear: If the equivalence principle requires that someone passes through an event horizon without being able to detect that they did so (all free fall is indistinguishable), then in what way is the singularity anything more than a breakdown in our ability to describe what is happening? Again, anything beyond the event horizon must be outside the domain of relativity as it is a place where spacetime warps beyond c. If we know someone can pass through the horizon and not notice a thing, but according to an outside observer this would take infinitely long, then it is either our theory or the connectedness of spacetime that comes to an end, but not reality itself. If event horizons are analogous to cosmological horizons, we have further reason to believe that singularities are inexperiencable as we ourselves are within the expanding cosmological horizon and are not crushed into a singularity. How can something which falls through an event horizon distinguish between an infinitely expanding horizon (what would appear to them as a cosmological horizon) and a singularity that they are supposedly approaching? I don't see how they can, and it seems the resolution of the problem of infinities in singularities is this matter of perspective. The singularity never comes, as it appears as an event horizon from the outside perspective, and an expanding cosmological horizon from the inside. Both infinities are never reached. Do singularities not sound like 'end of the earth' flat earth thinking where everything is just believed to come to an end?

  • $\begingroup$ "then can't we say that these are simply two different perspectives of the same event?" We can and do. Note the event horizon is relative to the observer; for simplicity we assume the observer to be at infinity. If you were falling into a black hole your event horizon would differ from that determined by a distant observer. Event horizons are relative, not absolute. $\endgroup$ – Peter Webb Mar 3 '15 at 6:32
  • $\begingroup$ Thanks for the reply. From what I understand (and what Moonraker wrote below), the Schwarzschild radius is not considered to be relative. Also, assuming the observer to be at infinity is not helpful in defining event horizons and singularities since, as said in my post, this is not a possible perspective. $\endgroup$ – JTT Mar 3 '15 at 16:45
  • $\begingroup$ @JTT In an asymptotically flat spacetime, there can be a natural surface to associate with the asymptotically flat region, and that is often called the event horizon. But there are also apparent horizons, causality horizons, particle horizons, trapped surfaces, and so forth. Individual observers can have horizons even incompletely flat spacetimes, the simplest example is uniform hyperbolic motion in special relativity. Something far enough away will never reach you even if it moves at the speed of light, and there is a cut off distance and everything beyond the cut off will never reach you. $\endgroup$ – Timaeus Mar 3 '15 at 19:20

First, I worry that you might confuse a singularity, which is a region (possibly pointlike) where the curvature is undefined (or infinite), versus a black hole, which is a special surface with a particular global property in an asymptotically flat spacetime.

There is a famous conjecture (the cosmic censorship conjecture) that every singularity is within an event horizon. However, there are known solutions to Einstein's Equation where this is not the case, so really it comes down to trying to rule out the solutions with a naked singularity and then justify the reasonableness of ruling them out.

So, it is possible to have a singularity with no event horizon. It's just that it is frowned upon, and no consensus has yet been reached as to exactly why we want to frown upon it, and for instance, some people seriously consider big bang cosmologies that have a naked singularity as a region of earliest time in their model.

Since you can have a singularity without an event horizon (mathematically you can also have event horizons without singularities, though the lack of singularity is unstable), the two ideas are most definitely and most certainly two different things.

That said, many of your points have merit, black hole complementarity takes seriously the idea of a duality of observers: 1) those that cross the horizon who see nothing special and head towards the region that classical general relativity predicts to have a singularity 2) those that never cross, and to them it is similar to a $t=+\infty$ surface in that they never see anything cross it or interact with anything that reports as having crossed it. This last option happens regardless of the fact that there are paths with a finite affine parameterization that takes them all the way to the surface.

  • $\begingroup$ Thanks for the reply. Don't you think the most interesting part though is that the entropy of a black hole is strictly on the surface simply because the volume does not exist to an outside observer? In other words, if an event horizon depends on the motion of an observer (which I am saying does because an outside observer will see time stop at the event horizon instead of the singularity...which really I am saying are the same thing from different perspectives), then the entropy of the black hole must also be relative. Someone who falls past the horizon still sees the horizon ahead, and sees $\endgroup$ – JTT Mar 4 '15 at 0:24
  • $\begingroup$ time come to a stop for things sufficiently far ahead 'toward the singularity', just as someone further from the black hole would see time come to a stop for the infalling observer who still experiences time them self...and as someone outside the cosmological horizon would see our entire universe. $\endgroup$ – JTT Mar 4 '15 at 0:25
  • $\begingroup$ "First, I worry that you might confuse a singularity, which is a region (possibly pointlike) where the curvature is undefined (or infinite), versus a black hole, which is a special surface with a particular global property in an asymptotically flat spacetime." I don't think he was confusing the two, specifically asking if the two are actually the same thing. $\endgroup$ – Shufflepants Aug 4 '16 at 18:57

To take your question literally, we may speak of it because our speech is not bound by nature or reality - one walk down the 'Fiction' aisle of a bookstore will demonstrate that.

We write of and speak of and theorize about many things we cannot, or will not experience (such as other multiverses, interiors of stars, etc.).

Why would you think we would be prevented from "speaking about the interior of a black hole", and what mechanism do you postulate to limit speech in this regard?

  • $\begingroup$ Ok, don't take my question literally. I think it is clear that it was an intro to the questions in the body. $\endgroup$ – JTT Mar 3 '15 at 6:09
  • $\begingroup$ I changed the title to better reflect my questions. $\endgroup$ – JTT Mar 3 '15 at 17:48

The event horizon of static black holes without charge is the same for every observer, and it is located at the Schwarzschild radius.

Even for an observer falling into the black hole the event horizon is the same, but at the difference that he will not notice the presence of any event horizon when falling through it. In the meanwhile outside observers will notice the event horizon because for them the time it takes for the infalling person to reach the event horizon is dilated to eternity. As a result, where the infalling person takes one minute to reach the Schwarzschild radius, this single minute is dilated to eternity for outside observers, for them the infalling person does never reach the event horizon.

Accordingly, we may still observe around the event horizon all infalling objects which had fallen into the black hole since the beginning of the existence of the black hole, thus from our point of view there are plenty of objects marking the event horizon, which becomes observable.

There is no correspondence between the singularity and the event horizon. There is no way to observe any objects which have passed across the event horizon, from this moment they are definitely gone for observers.

  • $\begingroup$ I appreciate your response, though I don't think you have cleared up the issue or addressed the points I brought up. If the event horizon is where time comes to a stop (to an outside observer) then this must be the end of where relativity can be applied as its assumption of c limits it to this domain. If an infalling observer doesn't see the event horizon as a special place, then the equivalence principle says that this should always be the case and the event horizon is a place that is forever in the distance. Patches of space can become isolated from each other (such as cosmic horizons) but a $\endgroup$ – JTT Mar 3 '15 at 15:14
  • $\begingroup$ singularity can never be experienced as infinity is always unattainable (just as acceleration to c is). Imagine a black hole inside an expanding universe. The location of that expanding cosmic horizon is of the same nature as the event horizon of the black hole. Each are 'disconnected' to observers who are at rest to each, but must be said to grow to those moving away from each respectively. $\endgroup$ – JTT Mar 3 '15 at 15:19

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