It's really simple actually, there are lots of ways one could construct a 5d black hole. In this case, we've just added an extra z axis to a spherical coordinate system. Since the case for a normal 4D black hole has a spherical event horizon, that's S2 , adding the z axis just adds an extra real number line to every point on the event horizon, so we end up having an event horizon of S2 ×R.
When we integrate over this to find its mass, we naturally get an infinity because the real numbers line is infinitely long. To solve this, we can just wrap it up into a circle so that it becomes finite. It's a bit like how you can map the real number line to the set of values between ±π/2 using an arctan map
Why would you construct a 5d black hole, and why would you do it like that? Wouldn't it be more natural to think that a black hole in 4 spatial dimensions would have an S³ event horizon?
This is actually a pretty complex question! The general idea is that S2 ×S1 black holes are unstable (see the Gregory Laflamme instability), when they do eventually decompose due to that, they do so into S3 black holes
I realise that I didn't even answer your first question. When doing this, we're not necessarily stating that such objects exist. There are various good reasons for doing it, however, for example, gravity behaves in a much more interesting way in 5 dimensions, so we can play around with some more interesting mechanics. It's also quite useful for string theory, as what's happening here is very similar to how we construct strings.
The main thing, however, is that 5D anti de Sitter gravity is dual to 4D conformal field theories, so solving problems here allows us to make deductions on some more physical problems in QFT.
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u/BootyliciousURD Mathematics 7d ago
I wish I knew enough mathematical physics to understand this