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?
Best Answer
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.