# Why are orbits around black holes stable?

Black hole theory involves space (or space-time), itself, being sucked into the black-hole, with the event horizon marking the point at which space/space-time is moving faster than the speed of light. I find it really hard to picture how this could be happening while objects maintained a reasonable stable orbit around black holes. If we take the stars that orbit the super massive black hole at the centre of the Milky Way, the orbital dynamics are used to calculate the mass of the black hole, in the normal way. In other words not taking account of the fact space is rushing at some speed inwards toward the black-hole. I appreciate I'm missing some knowledge here. That's the motivation for asking the question.

• The premise is false. Black holes are not "sucking" anything, and there is no notion of "spacetime moving". There is no "rushing" happening. Spacetime is curved around a blackhole, but it is not in motion. The spacetime around a black hole is stable, it is just different than what how you're used to thinking of spacetime. The event horizon is not a point where "spacetime is moving quickly," it is the point at which the natural coordinates of space reach a singularity and it stops making sense to think of the space near a black hole as essentially spherical. Commented Mar 17, 2015 at 21:35
• It's not fair giving me another minus point when all I'm doing is repeating what the leading scientists in the field are saying. I'm not bothering any more if this is the attitude Commented Mar 17, 2015 at 22:28
• The minus points are a reflection of the question, not of you.
– BMS
Commented Mar 17, 2015 at 22:34
• @BMS - What's wrong with the question? She was asking about a statement made by Susskind, and as you can see from my answer, Susskind's description does correspond to a mathematically valid way of analyzing the behavior of particles in a black hole spacetime. Commented Mar 18, 2015 at 1:22
• Lucy, Susskind takes great pains to explain that his explanation of black holes is only an analogy (~3:45 to 4:20) and can only be pushed so far; this is true both for the talking about fish and the simplified technical description after it. It is a good analogy, but it is limited. It is true that "black hole theory can be thought of as space being sucked into the black-hole" but the straightforward claim you start with is incorrect and it is giving you the downvotes. If you edit your post accordingly and appropriately reference it to Susskind, your post will be much better received. Commented Apr 30, 2015 at 20:04

In the comments you mention Susskind's use of a metaphor involving water flow 7 minutes into this video, but this shouldn't be understood in terms of spacetime behaving fundamentally differently around a black hole as opposed to any other gravitating body. Rather, I suspect Susskind is just referring to the analysis of a black hole in a particular type of coordinate system, Gullstrand–Painlevé coordinates, outlined in conceptual terms on this page. As mentioned on the page, "The Gullstrand-Painlevé metric ... is just the Schwarzschild metric expressed in a different coordinate system" (i.e., different from the Schwarzschild coordinates often used to described the curved spacetime around a non-rotating black hole, with the curvature itself understood as coordinate-invariant). The page goes on to say:

Physically, the Gullstrand-Painlevé metric describes space falling into the Schwarzschild black hole at the Newtonian escape velocity. Outside the horizon, the infall velocity is less than the speed of light. At the horizon, the velocity equals the speed of light. And inside the horizon, the velocity exceeds the speed of light.

The author of the page also has a more technical paper elaborating on this coordinate system for pedagogical purposes, titled The river model of black holes. In section II the paper mentions some nice features of this coordinate system:

We demonstrate two features that are the essence of the river model for spherical black holes: first, that the river of space can be regarded as moving in Galilean fashion through a flat Galilean background space [eqs. (14) and (15)], and second, that as a freely- falling object moves through the flowing river of space, its 4-velocity, or more generally any 4-vector attached to the freely-falling object, can be regarded as evolving by a series of infinitesimal Lorentz boosts induced by the change in the velocity of the river from one place to the next [eq. (18)]. Because the river moves in a Galilean fashion, it can, and inside the horizon does, move faster than light through the background. However, objects moving in the river move according to the rules of special relativity, and so cannot move faster than light through the river.

Also note that in general relativity, according to Birkhoff's theorem the metric outside the surface of any spherically symmetric, non-rotating massive body, like a star or planet with no angular momentum, is just the spacetime geometry given by the Schwarzschild metric, no different than that of a non-rotating black hole of the same mass (the same sort of thing isn't true for rotating bodies vs. rotating black holes, see the last paragraph on p. 39 of this pdf). So, this presumably means that you could also describe the space outside a non-rotating star in the Gullstrand-Painlevé coordinate system, in terms of space acting like a fluid flowing inwards.

This needn't imply orbits are impossible though, since in this description an object also has a velocity relative to the local space, and its total velocity is the sum of the velocity of space and its velocity relative to space, just as the total velocity of an object in a fluid can be split up into the velocity of the local fluid plus its velocity relative to the local fluid. See my discussion in this answer about the "Newton's cannon" thought-experiment to get a better intuitive feel for how an object which is constantly experiencing a force (which could be from a fluid) pushing towards the center of a circle can nonetheless move in an orbit due to its own tangential velocity.

• Dear Hypnosift - Susskind isn't using a metaphor in his own opinion. He makes this clear by following his 'metaphor' with the emphasis "and that really is true". Commented Mar 19, 2015 at 2:05
• hypnosift, then you say "...is just the Schwarzschild metric expressed in a different coordinate system" apparently to mean it isn't legitimate physics, but an illustration. But is that really the implication? I thought that if two models are the same thing, translated through a coordinate system, then they are equivalent. Commented Mar 19, 2015 at 2:11
• Hi Hypnosift - I can see you conclude with the knowledge that space doesn't behave differently than around any body. In other words space can be seen as flowing toward any gravitating body. Which answers my confusion. Thanks for that :o) Commented Mar 19, 2015 at 2:36
• I didn't mean "the Schwarzschild metric expressed in a different coordinate system" to mean it was just an "illustration"--it's presumably totally rigorous mathematically. But when physicists have multiple mathematical ways of getting to exactly the same physical predictions, normally they don't see either as any more physically "correct" than the other, so although modeling a black hole in terms of space flowing inwards in a flat space is perfectly valid, it isn't any more "true" than a model in which space is curved but static. Commented Mar 19, 2015 at 2:48

I feel that in general there is a lot of confusion around what black holes are and what they do. in reality, it only gets very different from other objects such as stars very close to or inside the event horizon; in most circumstances it is simply a heavy object. therefore objects would simply orbit around it like any other massive object, as long as the orbital path does not lie to close to the event horizon.

• If inflowing space is equivalent to the description everyone is familiar with, that has to impact the path of a satellite (especially close-pass elliptical orbits which the orbits of many stars are, at the SMB at the centre of the Milky Way). Commented Mar 19, 2015 at 2:19
• The only alternative is either (a) Hypnosift is correct that it is a metaphor. I will write Susskind if that's what you say, and ask him to visit this question and confirm it is a metaphor. The other possibility is that the equivalence holds for all gravitation effects. So gravity on Earth is equivalent to space flowing toward earth. that must be it Commented Mar 19, 2015 at 2:30