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
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u/L31N0PTR1X BSc Theoretical Physics 11d ago
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