Understanding Black Hole Shapes & Forms

[davidge]

Sorry, I'm not sure what is the more appropriate word to use: shape or form. Let's to the question:

How do we know what the shape of a given black hole is? I mean, how do we know whether it is spherical or whatever other form it has? Specifically, where do we look on the equations to get this information?

 

[.Scott]

The angular momentum of the black hole will affect its shape. With no angular momentum, it is spherical.

 

[davidge]

The angular momentum of the black hole will affect its shape. With no angular momentum, it is spherical.

Thanks, @.Scott.
And an expression for the angular momentum can be derived from the metric + Einstein Equations, correct?

 

[Eliasben]

Depending on the black hole in question, it can have a spherical shape or if a black hole is rotating, then it will be shaped as an oblate spheroid, slightly larger around the equator than in the direction of the poles. The terms in the equations of general relativity tell us that there are multiple radii, including the inner event horizon and the oblate spheroidal exterior surface on the outside where the region in between is the ergosphere. This is for the case of a spinning (Kerr) black hole.

Most studies of static Schwarzschild black holes suggest a spherical 4D shape i think.

 

[davidge]

@Eliasben, your response was very helpful. Thanks

 

[vanhees71]

The question is which "shape" you mean. A Schwarzschild black hole, e.g., is just a point at $r=0$ in the usual Schwarzschild coordinates. The event horizon, which you might also consider to define a black hole's shape is a sphere.

 

[davidge]

Thanks vanhees71

The event horizon, which you might also consider to define a black hole's shape is a sphere

Is it a sphere regardless of angular momentum?

 

[vanhees71]

I talked about a Schwarzschild black hole, i.e., a spherical symmetric static solution. A rotating black hole is described by another exact solution of the Einstein equations, the Kerr solution:

https://en.wikipedia.org/wiki/Rotating_black_hole

 

[Ibix]

Most studies of static Schwarzschild black holes suggest a spherical 4D shape i think.

Is that right? I'd think more like a 4-cylinder - a sphere in 3d but extended in time.

 

[PeterDonis]

Most studies of static Schwarzschild black holes suggest a spherical 4D shape i think.

No, they don't. The 2-sphere (or 2-spheroid) shape you mention is the shape of the horizon at a single instant. The full spacetime shape of the horizon, taking into account all instants, is a 3-cylinder--an infinite connected series of 2-spheres (or 2-spheroids).

If you want to interpret the "shape of the hole" to mean the "shape" of the interior as well as the horizon, things get even more complicated, because the 4-volume of the interior of the hole (whether it's rotating or not) is infinite. What's more, even the "shape" of the volume at a single "instant" is not invariant--it depends on your choice of coordinates. You can choose coordinates in which, at a single "instant", the volume is finite (basically the interior of the 2-sphere or 2-spheroid of the horizon), but you can also choose coordinates in which, at a single "instant", the volume is infinite. So the "shape" of the hole's interior isn't even well defined.

I'd think more like a 4-cylinder - a sphere in 3d but extended in time.

It's not even that simple. See above.

 

[Eliasben]

Is that right? I'd think more like a 4-cylinder - a sphere in 3d but extended in time.

Thanks for the clarification, it sounds similar to the Einstein static universe (my avatar), which is a cylinder when projected onto a screen (removing angular dimensions) and infinite in time vertically. I meant simply speaking the black hole is 4 dimensional meaning it lives in a four dimensional spacetime - three spatial dimensions and one time dimension, though I guess that's technically wrong to say!

 

[Eliasben]

No, they don't. The 2-sphere (or 2-spheroid) shape you mention is the shape of the horizon at a single instant. The full spacetime shape of the horizon, taking into account all instants, is a 3-cylinder--an infinite connected series of 2-spheres (or 2-spheroids).

If you want to interpret the "shape of the hole" to mean the "shape" of the interior as well as the horizon, things get even more complicated, because the 4-volume of the interior of the hole (whether it's rotating or not) is infinite. What's more, even the "shape" of the volume at a single "instant" is not invariant--it depends on your choice of coordinates. You can choose coordinates in which, at a single "instant", the volume is finite (basically the interior of the 2-sphere or 2-spheroid of the horizon), but you can also choose coordinates in which, at a single "instant", the volume is infinite. So the "shape" of the hole's interior isn't even well defined.
It's not even that simple. See above.

Sorry I meant to quote you on my previous reply too^