Understanding Black Hole Shapes & Forms

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Discussion Overview

The discussion revolves around the shapes and forms of black holes, specifically addressing how their characteristics are determined by factors such as angular momentum and the equations of general relativity. Participants explore the implications of different black hole solutions, including Schwarzschild and Kerr black holes, and the complexities involved in defining their shapes in a four-dimensional spacetime context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that the shape of a black hole is influenced by its angular momentum, with non-rotating black holes being spherical.
  • Others argue that rotating black holes take on an oblate spheroid shape, with varying dimensions depending on their rotation.
  • A participant notes that the event horizon of a Schwarzschild black hole can be considered spherical, while questioning if this holds true regardless of angular momentum.
  • There is a discussion about the interpretation of the shape of black holes, with some proposing that the full spacetime shape is more complex than a simple sphere, suggesting it resembles a 3-cylinder when considering all instants.
  • Concerns are raised about the definition of the "shape" of a black hole's interior, which may be infinite and dependent on the choice of coordinates.
  • One participant expresses uncertainty about the dimensionality of black holes, suggesting they might be better described as a four-dimensional object rather than simply a sphere.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definitions and implications of black hole shapes, with multiple competing views and interpretations remaining unresolved throughout the discussion.

Contextual Notes

The discussion highlights limitations in defining the shape of black holes, including the dependence on angular momentum, the choice of coordinates, and the distinction between the event horizon and the interior volume.

davidge
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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?
 
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The angular momentum of the black hole will affect its shape. With no angular momentum, it is spherical.
 
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.Scott said:
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?
 
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.
 
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@Eliasben, your response was very helpful. Thanks
 
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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.
 
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Thanks vanhees71

vanhees71 said:
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?
 
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
 
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Eliasben said:
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.
 
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  • #10
Eliasben said:
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.

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

It's not even that simple. See above.
 
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  • #11
Ibix said:
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!
 
  • #12
PeterDonis said:
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^
 

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