How does gravity act and propagate in a 2+1D universe?—Newtonian versus general relativity

Question

In a hypothetical 2+1D universe:

• if we apply the Newtonian concept of gravity, we might expect that the gravitational force between two mass points with a distance of $r$ would diminish linearly with distance proportional $1/r$, however,

• if we would apply Einstein's general relativity theory, we might expect that in 2+1D spacetime, the Einstein tensor vanishes identically because of the mathematical characteristics of the Ricci curvature tensor and the Ricci scalar in 2+1D.

The two concepts, appear to me somewhat paradoxical. My question is:

• whether my interpretations of Newton's and Einstein's laws for gravity for 2+1D are incorrect? or

• if they are correct, then how can the two concepts be harmonized,

to answering the question of the existence and propagation of gravity in a 2+1D universe.

Answer 1

In (2+1)D, it is a geometrical fact that the Weyl tensor vanishes. The Einstein tensor and the Ricci tensor give then all of the independent components of the Riemann tensor, and they are fully specified algebraically by the Einstein field equations. In particular, a consequence of this is that in (2+1)D gravity does not propagate: there is only curvature in points in spacetime which contain matter. Hence, the Earth would not orbit the Sun in a (2+1)D universe. The Einstein tensor can still be non-vanishing, and a common example is the BTZ black hole.

As far as I know, the Newtonian scenario is very different, and gravity indeed propagates (although I do not know from memory the specific expression for the gravitational force). Notice this means that GR does not reduce to Newtonian gravity in (2+1)D. To my knowledge, these two points of view cannot be reconciled in a straightforward manner. Perhaps something can be achieved with Newton–Cartan theory, but I'm not sufficiently familiar with its properties in (2+1)D to make useful comments about it.

In (1+1)D the Einstein tensor vanishes identically, and thus the Einstein equations reduce to stating that the stress tensor must too vanish identically. Hence, you can't have matter in these spacetimes (unless the stress tensor of the matter cancels exactly).