general relativity - How do we expect distance measurements to compare inside and outside the event horizon of a black hole?

I've read that as one approaches the event horizon of a black hole, time is dilated relative to time measured farther away from the event horizon (clocks tick slower near the event horizon).

I've also read that within the event horizon, the curvature of space becomes very large and approaches infinity near the singularity.

So I'm trying to understand with the time dilation and high curvature of space (or perhaps rather space-time) what an observer within the event horizon would measure as distance.

For example - from the outside of the black hole one might estimate the black hole's diameter to be 500 km. Would an observer inside the event horizon also estimate this distance or would the diameter appear to be more extensive?

Answer

When you take a Schwarzschild (i.e. static, noncharged and non-rotating) black hole, it does not make much sense to talk about an observer trying to map the interior of the black hole because the singularity is fundamentally in the near future of the observer and the event horizon is fundamentally in the past of the observer.

When you are even above the event horizon the black hole seems much bigger than you would predict from naive Euclidean considerations. In Euclidean geometry you would predict the surface of a ball to fill a half of your view only when it is very large compared to you and only when you are on the surface itself. In the case of a black hole, the blackness covers a half of your view much earlier and when you are on "the surface" - on the event horizon, the blackness has covered your whole view and the whole sky is squeezed into one very bright point.

Once you plunge into the black hole, the featureless blackness has covered everything. You do not see anything, no instrument can reach the horizon and the surrounding seems isotropic, just the slap forces seem to be dynamically increasing. You send out light signals but they never return. In this sense, the answer most close to truth would be that the event horizon seems infinite from inside, because you cannot reach it or find any trace of it no matter how hard you try.

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Just btw., the apparent shadow of a black hole from far away is actually about ten times larger than that of a planet with a radius equal to the event horizon radius. This is because the radius $2GM/c^2$ is fatal to photons going in the purely radial direction but photons which are just passing by are devoured much earlier.

I believe it is not very instructive to talk about the size of the black hole because it often lures us into thinking about it in terms of a black planet with a definite surface. When you study relativity further, you actually find out it even isn't a surface in any good sense, rather a very peculiar light-cone.