What's the Newtonian potential in 2+1 gravity?

Question

I understand that there are no propagating degrees of freedom (i.e. gravitational waves) in 2+1 dimensions. There are a couple of arguments to show this. One is to count degrees of freedom of general relativity and find that there are $D(D-3)/2$. Another one relates to the fact that the Weyl tensor vanishes identically, so vacuum solutions with $R_{\mu\nu}=0$ also have vanishing Riemann tensor.

I think this all means that the vacuum of 2+1 gravity is flat everywhere. My question is much simpler than that. Let's use Electromagnetism as an example

Coulomb potential in 2+1

I'm used to thinking about, for example, the Coulomb field in different dimensions by simply solving Gauss' Law

$$ \vec{\nabla} \cdot \vec{E}=\rho$$

where $\rho$ is the charge density (in some units where I don't have to care about vacuum polarizations and factors of pi). Integrating over a contour line $\mathcal{C}$ surrounding the charge density we get the flux of $\vec{E}$ in the direction $\hat{n}$ perpendicular to $\mathcal{C}$

$$ \oint \vec{E}\cdot \hat{n}dl=Q$$

as opposed to the surface integral that we get in spatial three dimensions. The isotropy of the electric field allows us to take it out of the integral to get

$$\vec{E}=\frac{Q}{2\pi r} \hat{n}$$

which means that the electric field decays like $1/r$ in 2+1 dimensions, as opposed to $1/r^2$ in our 3+1 world.

Newtonian gravitational force in 2+1

Since the gravitational force in the Newtonian framework also satisfies a Gauss' Law, I think that it should also decay as $1/r$. Equivalently, we could solve for the gravitational potential $\phi$ and find that it increases as $ln(r)$.

My question is: Is this fact true as a limit of 2+1 General Relativity? In other words, if we put two point masses in a 2+1 theory of General Relativity. Will they experience a force inversely proportional to their separation as long as we stay in the $v/c\ll 1$ limit?

I don't think this would contradict the statement that 2+1 gravity has no local degrees of freedom since Coulomb-like fields are static and attached to the matter degrees of freedom, as opposed to propagating waves which are radiative degrees of freedom independent of the matter fields.

Finally, if anyone has a review or article about masses interacting in flat 2+1 spacetime I'd love to see it!

Answer 1

The motion of point mass in GR with 2+1 dimensions was studied in this classic paper by Deser, Jackiw, and 't Hooft. In 2+1 dimensions empty space is always flat. Consequently, there is no attraction force between static particles (there is a gauge dependent interaction between moving point particles though).

On the lack of correspondence with the Newtonian limit they write:

Finally, we comment briefly on the discontinuity in the full theory between the Newtonian limit and Newtonian gravity, implicit in the fact that $g_{00} = -1$ in the static solution, so that $\ddot{x}^i = \Gamma^i_{00} = 0$ for slow test particles. This discontinuity was noted long ago [6], essentially on the basis of the absence of linearized $T_{00}-t_{00} $ interaction at $D = 3$. There is no paradox here, for Newtonian correspondence is not guaranteed a priori for Einstein theory. [...]