general relativity - Gravity in 2d space and inverse linear law

In our three-dimensional universe, gravity obeys the inverse square law. In a four-dimensional universe, gravity would be expected to obey the inverse cube law et cetera.

In a two-dimensional 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 2d universe?

Answer

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. 
   

   Einstein gravity in 2+1D is a topological field theory. The linearized EFE is without propagating physical degrees of freedom. That doesn't exclude the existence of e.g. gravi-static effects due to conical singularities.

   
   


3. 
   

   In 3+1D Newton gravity (NG) can be derived from GR in an appropriate limit, see e.g. methods used in this Phys.SE post. In 2+1D, GR has no Newtonian limit for zero cosmological constant $\Lambda=0$. For negative cosmological constant $\Lambda<0$, there exist BTZ black holes in 2+1D.