Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 21; Problem 21.6.
Because of the symmetries of the Riemann tensor, there are six independent, (possibly) non-zero components in 3 dimensions. We can take these components to be , , , , and . As before, we use the Einstein equation
with and consider a vacuum so that , meaning that . We’ll look at a local inertial frame (LIF), where the metric is . Then we get
Now we look at the 6 independent components of . Because , any component with either the first two indices or last two indices equal is zero, so we get
The last 3 equations show that 3 of the Riemann components are zero. The first 3 equations can be rewritten using the symmetries of the Riemann tensor:
Solving these equations gives
Thus all 6 components of the Riemann tensor are zero, showing that 3d spacetime must be flat and gravity cannot exist in 3 spacetime dimensions. (As usual, a tensor equation valid in a LIF is valid in all coordinate systems, so the conclusion is general.)