Gravity in 2+1D spacetime and inverse linear law

Question

In our 3+1D universe, gravity obeys the inverse square law. In a 4+1D universe, gravity would be expected to obey the inverse cube law et cetera.

In a 2+1D universe, one would similarly expect gravity to obey an inverse linear law. I've seen it claimed that this is not actually true, when you work it out according to general relativity there would actually be no gravity at all in such a universe. Is this true? If so, is there a layman's explanation of why? Is it related to the fact that gravitational potential wells would be infinitely deep?

Would electrostatic forces follow the same law as gravity?

If there existed infinitely long strings in our universe, would they have the same gravitational effect as point particles in a 2+1D universe?

Answer 1

The question asks simultaneously about both Newton gravity (NG) and Einstein gravity/general relativity (GR), which are two different theories.

1. For Newton gravity (NG) in 2+1D, the gravitational force is inversely proportional with distance. More generally, in $n$ spatial dimensions, then the gravitational force $F\propto r^{1-n}$. This is due to Gauss' law, because a Gauss surface is $n-1$ dimensional. The Coulomb force in electrostatics will have a similar radial dependence, since it too has to obey Gauss' law.

2. In $n\geq 3$ spatial dimensions Newton gravity (NG) in an asymptotic region can be derived from GR (assuming no cosmological constant $\Lambda$), using methods similar to my Phys.SE answer here.

3. Einstein gravity in 2+1D is a topological field theory. The linearized EFE is without propagating physical degrees of freedom.

   There are typically gravistatic effects from conical singularities. In particular there is no meaningful notion of a Newtonian asymptotic region due to the presence of a deficit angle. Perhaps surprisingly, the Newtonian $ln$ potential has no analog in GR!

   For negative cosmological constant $\Lambda<0$, there exist BTZ black holes in 2+1D.

4. Concerning cosmic strings in 3+1D, see e.g. this related Phys.SE post.

Answer 2

In Electrostatics you can write $F=qQ/(4\pi\epsilon_0 r^2)$ in 3D, or as $\vec\nabla \cdot \vec E = \rho/\epsilon_0$ and $\vec F=q\vec E.$ And the later generalize to 2d as a $1/r$ force.

In Newtonian Gravity you can write $F=mMG/r^2$ in 3D, or as $\vec\nabla \cdot \vec C = \rho 4\pi $ and $\vec F=m\vec C.$ And the later generalize to 2d as a $1/r$ force.

But you can object and say that is this isn't electromagnetism, after all there is more to electromagnetism than electrostatics. In n=4=3+1 spacetime dimensions you can imagine an n-vector potential $A$ and its outer derivative is the electromagnetic field, and the $n(n-1)/2$ components of the electromagnetic field break into two (n-1)-vector fields (relative to a frame) only in a 3+1 dimensional spacetime. You can still break it into a relative (n-1)-vector and a relative (n-1)-bivector which comes from a $(n-1)(n-2)/2$ dimensional space.

Similarly you can say that gravity is more that a $1/r^2$ force. What you have is curved spacetime. And you can have different types of curvature. You can have the kind of curvature in the vacuum around a star of mass M. You can have the kind of curvature in the vacuum around a star of mass m.

And you can sew them together along a spherical surface of area $4\pi R^2.$ And if you sew the outside of the $M$ solution to the inside part of the mass $m$ solution and $M>m$ then this is exactly how some energy of energy $(M-m)c^2$ can sew together two solutions. This is what energy does in general relativity, it takes two regions that can normally be how vacuum curves naturally and sew them together. And this is how curvature gets strong in general relativity, as matter collapses, the outside region that has the $M$ type curvature extends deeper towards the center of the forming star or planet. And that type of curvature gets stronger closer in. Thus the collapsing matter leaves stronger curvature outside it as the interface between the two types of curvature ends up in a new location.

This is similar to how in electromagnetism you can have a radial electric field outside a conductor transition to a zero field on the inside. Both are find vacuum solutions, and the charge on the surface allows you to sew them together.

But remember how we replaced the simplified electrostatic force with a full electromagnetic field that was the outer derivative of a scalar potential? And then this was a totally different beast in different dimensions?

In General Relativity the fundamental thing is curved spacetime. (Just like the electromagnetic field was the fundamental thing in electromagnetism.) And all that sources do are change which solutions can be sewn together. (Just like charges and current let electromagnetic fields diverge and evolve differently than they otherwise were able.)

So just like we had to change our vector potential to and all of a sudden the magnetic field was a scalar field, similarly you have totally different vacuum solutions to work with. The very things that the sources are to sew together are different. In 1+1 and 2+1 dimensions the vacuum spacetimes are flat so there are no longer things to sew together so can only have matter someplace to have curvature so you can't leave curvature behind as something collapses.