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I have always been curious about what black holes (or their event horizons) would look like if we could directly observe them. I notice that they tend to be depicted as flat holes in space but it seems to me that they would be spheres. Is there a scientific consensus on this. Or is it a vague question?

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The event horizon would be spherical. What is usually depicted in drawings is the accretion disk, where all the debris gather before entering the event horizon.

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It's complicated, and it depends on what you mean by "black hole".

In general, for a distant observer, the "black hole" is the event horizon - the surface within which there is no escape to the outside. The more massive the black hole, the bigger the event horizon.

If the black hole is not rotating, the event horizon is a sphere, plain and simple. In a perfect vacuum, and if the black hole is big enough (is not microscopic), looking at it you'll see a perfectly black, round "thing", with the stellar background behind it.

enter image description here

However, the background image near the black hole would be distorted as if seen through a bad lens. This would become apparent as you move around it. This is because the powerful gravity of the BH bends the rays of light. See the first half of this video:

If the black hole is not in a perfect vacuum, but is surrounded by various pieces of matter, it may capture that matter, and an accretion disk will grow around it.

enter image description here

If the black hole is rotating, things are a bit more complicated. This is an object described by the Kerr metric, and it has two important surfaces: the usual event horizon, and the outer skin of the ergosphere:

enter image description here

Nothing can escape from within the event horizon. It is possible to escape from within the ergosphere if you rotate with the black hole - this is known as the Penrose process.

I was not able to find visualisations of a Kerr (rotating) black hole, but it should look somewhat similar to a non-rotating one, except it would appear "flattened" at the poles, and bulging at the equator.

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Thanks! That pretty much answers my question. I always saw images like this (…), where the object is shown as sinking down into a field, but it always made more sense if the "black" of a black hole looked more like your second image. – rougon May 1 '14 at 17:50
That image is a map of the gravity field around a black hole, and it's quite misleading to a lot of people. There's no "funnel" there in reality. It's just a way of saying: the closer you get to the BH, the stronger the gravity; the depth of the "funnel" is meant as a measure of the field. But the "funnel" is not a thing. – Florin Andrei May 1 '14 at 17:56
Well, actually we don't know it, right? Otherwise it wouldn't be a singularity, by definition. – Py-ser May 2 '14 at 1:29

It depends on whether the black hole is rotating or not. If it is, The ergosphere and singularity are oblate spheroids. If not, both would be spherical. The event horizon is always spherical.

Accreting matter around a black hole would form an accretion disk, however this should not be mistaken for the black hole itself.

enter image description here

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Do we know what determines whether or not one rotates? Would it be mass or size? – rougon May 1 '14 at 18:26
Formation: rotating black holes are usually formed from the collapse of a massive spinning star – polyphant May 1 '14 at 18:41
Also merging black holes could sum their angular momentum. – Py-ser May 2 '14 at 1:27
Specifically, all the things that black holes do (form, absorb things, merge, evaporate by Hawking radiation, gravitationally interact with other bodies, etc) conserve angular momentum. So in practice, a black hole rotates if the total of all the stuff that's fallen into it was rotating, because that's most of the angular momentum that the black hole has ever "seen" in its "life". – Steve Jessop Mar 3 at 19:34

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