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Comment by cyberax | Hacker News Reader

So just to add another note, the infinite volume of black holes is likely an artifact of the classical (not quantum) representation. Just like the ultraviolet catastrophe.

Think about volume like this: it's measured by the amount of "liquid" that fills the given shape. This liquid consists of infinitely small idealized "atoms". Volume in curved space is similar, except that the "liquid" has different "densities" at different points in space.

This breaks down in case of black holes, as the central singularity can "compress" this "liquid" infinitely.

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For a notion of density to have a meaning, the atoms need to be finitely small.

Why? Density can be defined for a classic fluid, without any need to make it atomic.

> The inside looks very different, however. The spherical volume formula that you learned in grade school doesn’t apply. The problem is that spatial volume is defined at one moment in time. To calculate it, you have to slice up the space-time continuum into “space” and “time,” and inside a black hole there is no unique way to do that.

Susskind argued that the most natural choice is a slicing process that maximizes the spatial volume at every moment; by the logic of relativity, it amounts to the shortest distance across the hole. “It’s a natural volume analogue of the shortest-line rule,” said Adam Brown, a physicist at Stanford. And because the interior space-time is so warped, the volume by this measure grows with time forever. “The slice on which I measure this volume gets deformed more and more,” said Luca Iliesiu, a physicist also at Stanford."

Yes, that's what I'm talking about. You need to take curvature into account, and it blows up into infinity near the black hole singularity.

But I believe that this is purely an artifact of using classic mechanics for that.

The similar thing happened in classic electrodynamics with the "ultraviolet catastrophe". Basically, you measure how much radiation would be inside a box at a temperature T, and you get an infinite amount. So any closed box should become a source of unlimited power.

And for very similar reasons, there's no low limit on the wavelength, so there is an infinite number of wavelengths that can fit inside a box. And more importantly, each wavelength should still carry some power, so the total sum diverges if you try to integrate all of their contributions.

The fix was to assume that light can only exist in discrete wavelengths, thus setting the lower limit for the wavelength. The resulting total sum then converges, removing the paradox.

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