# Two-Dimensional (+1 time) Gravity

While listening to one of those Science channel shows…they were talking about higher dimensions and gravity. It was said that 3 dimensional space (4 dimensional if include time) is the lowest number of dimensions that can have gravity.

Ok, I can see that 1 dimension couldn’t have gravity…but what is wrong with 2? If planets and stars were essentially circles then gravity would follow a inverse linear fall-off in effect…twice the distance away means half the gravity. Why is this impossible?

Oh, if I do get an answer if you could also explain why in a 4 dimensional univers (+ time) why planets (now hypersheres) can’t have stable orbits…I would appreciate it Twas another thing I heard years ago - posted on here but didn’t get a response.

Where’s Chronos or Stranger when you need him!

I don’t see why they would say that, looking at it from a general relativistic perspective, there’s no reason why you couldn’t describe gravity in terms of geodesics, just like you do in 3+1 dimensional spacetime.

For example BTZ black holes are solutions in 2+1 dimensional gravity.

The Wikipedia article does say that the 2+1 dimensional gravity has no Newtonian limit (i.e. it fails to behave like Newtonian gravity within certain limits), which may be what they’re referring to, but on the other hand, when you’re missing a whole dimension why should you expect it to?

Whether or not you have gravity in 2+1 dimensions depends somewhat on what kinds of things you want to call ‘gravity’. There’s certainly general relativity, for example (as on preview Pants already pointed out), so if you equate that with gravity, as a lot of people would, you can have gravity in 2+1 dimensions; it just would have properties that differ from the 3+1 dimensional case, but then again, that’s probably not too shocking. (As an aside, 2+1 dimensional gravity as a theory has some importance in quantum gravity, since as Witten has shown, it’s possible to quantize gravity using standard methods in 2+1 dimensions.)

As for unstable orbits in higher dimensions, well, the two body problem in > 3 spatial dimensions simply doesn’t admit stable solutions; everything either falls in or spirals out, so to speak. There’s a paper by Max Tegmark (PDF link) reviewing the various pathologies of different dimensionalities, if you’re interested.

I think this is what the OP’s science show was getting at. The short version of why this happens is because the dynamics of orbital motion, the angular momentum, introduce additional terms to the equation of motion. These are typically added to the gravitational potential and the total quantity is called the “effective potential”, written as a function of the orbit radius. If there is no radius such that the effective potential goes to zero, there will always be some force pushing the orbiting object to a different radius. Ergo, no stable orbits. Going to dimensions higher than 3+1 (or lower, too? I’m not sure), changes the term added by the angular momentum. It so happens that change is just enough to remove stable solutions.

Thanks for the link Half Wit!

So, no stable orbits in ANY dimensions >3 wow.

He does try to answer my OP to be saying that it is a WELL KNOWN FACT that no gravity in 2-D…but doesn’t explain

I don’t have a copy of MWT so I’m not sure exactly what Tegmark is referring to there, but he says there’s no gravitational force in 2-D. In GR gravity is not described as a force, it’s described as the curvature of spacetime. That said when you work within appropiate limits in GR gravity does behave very much like a force (so much so that you can pretty much describe at as one), however as per the wikipedia article on BTZ black holes, there exist no Newtonian limit in 2+1 gravity and so no limit in which to describe gravity as a force.

Maybe in 2D you can’t have a curvature in that 2D space that works like gravity, unlike in 3D where gravity is or can be described as curvature of space.

Just gussin.

In 2+1D general relativity, you end up getting curvature only where there’s mass: In vacuum, everything’s flat. You could measure global effects (it’d be a conical geometry, so pi would be less for any circle containing mass), but if you put two masses down near each other, they’d just sit there.