This question is apparently quite simple but I can't seem to find an answer to it, so I was hopping anyone could clarify me.
Are the Einstein field equations (EFE) only valid for a 3+1 dimensional space-time?
I've read somewhere, which I can't remember or find, that there were problems with the EFE in a 2+1 dimensions...Why would that be?
What about 1+1?
Answer
There is nothing "wrong" with the Einstein field equations in 2+1 as indicated by the comments, but they do have interesting, significantly restricted behavior in 2+1 dimensions.
For example, the Wikipedia page referred to by Olof in the comments says that in 2+1, every vacuum solution is locally either R2,1, AdS3, or dS3. Here's why. In d+1 with d≠1, the vacuum field equations (those with Tμν=0) can be manipulated to show that Rμν=Rd+1gμν
Notice that this behavior is in stark contrast to the vacuum behavior in 3+1. For example, take the vacuum region outside of a spherically symmetric massive body in 3+1 (like a black hole). This region is not flat, but in 2+1 with vanishing cosmological constant any vacuum region outside of a massive body would be. Pretty strange.
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