# What kind of volume does the event horizon of a Kerr black hole enclose?

I'm sorry if this is a naive question, I'm not too good with General Relativity.

I'm aware that a rotating black hole is described by the Kerr Metric, and black holes of this kind have ring singularities. For a normal, Schwarzschild metric governed black hole, the singularity is a 'point' singularity, and the event horizon encloses a spherical region centered at this 'point'.

Would the event horizon of a Kerr black hole enclose a sphere too? Or would it be toroidal, like my intuition leads me to believe?

Imagine putting some soap on the ring singularity then that whole disk is what is enclosed inside the event horizon and if you passed through that ring you'd end up in a strange unstable place where you can travel in time just by moving around. So its kind of nice that the whole thing is protected by an event horizon, two actually just to be on the safe side.

So you can imagine covering the disk with an oblate spheroid. And covering that with an even larger oblate spheroid. Each of those is an event horizon. But they aren't oblate spheroids, they are just sphere-like in the sense that they aren't toroidal.

However none of those event horizons actually looks like that, in some sense they can be more like a vase with a rounded lid. It's just that some coordinates you might want to use for the region around the event horizon have sets of coordinates for the event horizons where they look like an oblate spheroid. Really it's just useful to have the two surfaces so we can divide things up into three regions. Outside both horizons, between the horizon, and inside both horizons.

You can use an r variable implicitly defined by the actual coordinates to keep track of where you are. Sorta keep track or where you are. But only sorta because for instance there are really two different in between regions and possibly more than one of the other regions. Well, we don't really know for sure what happens inside the horizons since we can't see them, but we can investigate the math. Here is a useful picture, the one we want is on the right

To read the diagram imagine that you can always go straight up and light can go at most 45 degrees right and at most 45 degrees left and matter has to go between them. But distances don't really mean anything for instance on the lower left diamond thats the region outside both horizons and the bottom left line if that diamond is infinitely far in the past and infinity far away from the black hole the upper left line of the diamond is infinitely far in the future and infinitely far away from the black hole and everything in the diamond is outside both horizons so you can have an infinite past and future as long as you stay outside the outer horizon and nothing weird happens. That where we live.

Locations of constant r are sorta like open parenthesis on the left and sorta like close parenthesis on the right. As the close parenthesis get closer and closer to the outer horizon they start to line up with the upper right line which is literally the outer event horizon (as is the lower right) but notice that the lower right is like a white whole event horizon things only come out and did so long ago and the upper right is like a black hole event horizon things only go in.

In this sense event horizons are simply not at all like spatial surfaces. They are weird. For instance they have tangents that are also orthogonal to them and they can only be crossed one way because of how they angle with the direction light goes.

Let's not worry about the infinite past for now we can have a smile near the bottom of the diamond and ignore everything below that for now.

If you are inside both event horizons then you can go through that ring (remember the one we put soap on) and go to a weird place, a place of negative r. Plus you can see the ring singularity. That's weird. The Schwarzschild singularity hits you before you see it so you turn to spaghetti before you even hit it. But this one you can see it and you can run away from it. And seeing it is the first encounter you have with it.

In the diagram this region through the ring singularity is on the left of the jagged line in the diamond above the diamond labelled our universe (the label calls it an antigravity universe but time travel is allowed there and before you get there you'd see the singularity so you shouldn't take it too seriously). In the diagram it looks like you have to hit the singularity but really everything we covered in soap at the first step is what is covered by that vertical line, remember that a surface of constant r is two dimensional and the singularity is 1d and you can avoid hitting the singularity even when you cross through the disk be threading the ring (even though you can't avoid seeing the ring singularity to cross through the ring).

If you want to run away from the singularity you can (we call the singularity time like to refer to this fact and if you look at the Schwartzschild picture of the left of the picture you see the singularity there is horizontal so unavoidable). And by run away I mean you can literally increase your value of r after you get so close you get past both horizons. Seriously if you are inside both horizons you can cross that inner horizon again (this time while increasing r) to the in between region.

Outside both is pretty normal. Inside both is kinda normal except you can see singularities and who knows what looking at a singularity does to you and that region on the other side is definitely weird. But when you are between them you get the kind of weirdness you normally see inside a spherical non rotating black hole.

Specifically, in the in between region r is not too small and not too large and you are in between the two horizons, that's the diagram region in the diamond to the upper right of the original diamond where you and I live. (Or the one above that, we'll get to that later.)

This is much more like being inside the Schwarzschild event horizon where your r coordinate decreased no matter what you do. But for the Schwarzschild your r had to decrease because it decreased when you crossed it. Here it is like you have two different in betweens. If you go in between on you way in then you are forced to keep going in until you hit the inner horizon.

This is because lines of constant r in this region are smiles on the bottom and frowns on the top and you are still forced to go between the 45 degrees left and 45 degrees right in an upwards way. So you don't have a choice about going in or going out just a choice about heading towards the singularity you were heading to or turning to go towards the singularity things entering from that lower right universe were heading to or steering between (which is different than steering through the ring into the weird place with time travel)

So once you cross that outer horizon you can see a whole other universe which can enter your in between region from a whole different outside. And in here r is like a time coordinate which is why you can't go out, and why trying to go out just heads towards the inside that the other universe naturally was heading towards.

The lower left line is the outer horizon where things enter from the left universe's outside. The lower right is the outer horizon where things enter from the right universe's outside. The upper left line is the inner horizon where things from the right universe's outside start out heading The upper right is the inner horizon where things from the left universe's outside start out heading.

But note there is a whole other in between region right above this diamond. This in between is entered from below not from an outside but either from the in between right below it (which corresponds to a path that just touched the inner horizon and starts seeing two whole singularities from two insides with two rings and two weird time travel places through the two rings) but didn't go through the horizon). And again you can't help but go to increasing r trying to turn around just sends you towards the outside the other inside is naturally pointing towards.

The other way to get to this other in between place is to cross through the inner horizon at which point you enter one of the insides and see just one singularity. And it sees you. Which could potentially be weird too, this also doesn't happen with the Schwarzschild solution. You might never get to the other in between if you do this, but if after you cross over you turn around then you can enter the other in between this time heading towards increasing r.

So outside both event horizons you are in a normal universe, no time travel, and the only weird things are the things coming out of the outer event horizon. You can cross the outer event horizon and then you are doomed to touch the inner one and then it is possible to climb up even up all the way back to the outer horizon.

So in the in between region (both of them) the lines of constant r are smiles and frowns and so r is really a surface of constant time morally speaking. So calling is a spatial surface is pretty misleading. In the upper in between the lower lines are the inner horizon and the upper lines are the outer horizon. In both the in between regions your r must keep decreasing or jeep increasing (just like time), it really is your time coordinate in absolutely every sense.

And by the point you cross the outer horizon you have seen the entire history of both singularities (including seeing them seeing you if you crossed through the inner horizon instead if touching it) and by then you can clearly aim at one of the outsides if you want bursting out of the event horizon like it was a white hole.

So now we are done ignoring the past, since whether we are in the same universe or another isn't so clear by now we've seen an entire history of a universe beyond the ring that has time travel so it seems less weird to be back in time to our original universe.

So imagining the ring and the soap and the oblate spheroids and the different regions was helpful to know what is out there. And if the black hole doesn't rotate much the event horizons can look pretty close to those oblate spheroids. But they don't really look at all like that. And you can tell because of how weird that in between region was. And really those surfaces are light like. And really the in between region has r as a time coordinate in every sense.

No picture of two nested oblate spheroids is going to indicate how weird it is that you can't turn from going from one to other. And you don't see the fact that whole new worlds you've never interacted with can start to affect you when you cross one.

But the picture I posted is a penrose diagram and it can show you these things. They are like generalizations of Kruskal-Szekeres diagrams where light moves at 45 degrees upwards and matter can go between the 45 degrees left and 45 degrees right in an upwards way.

So what do these event horizons look like? They look like the path of light, like light trying to escape forever but going nowhere. And you can also look at the surfaces as intrinsic surfaces in their own right. And then if the black hole rotates enough (e.g. a/m > 99% of critical value where you get naked singularities) you see that there are regions of negative curvature (like in a vase that has a part that has a narrowing part that widens) and oblate spheroids don't have that.

They are not toroidal, they block the whole disk and that is great since there is time travel on the other side of that disk. For their shape check out Kerr Spacetime by Visser http://arxiv.org/abs/0706.0622

The ring singularity would be in the plane of rotation, therefore it would not form a toroid. As Horus mentions, the shape would be an oblate spheroid. How "flat" it is, depends on the angular speed of rotation. To find its volume, find its elliptical area and "rotate" it (integrate) around an axis.