# Schwarzschild Solution, The constant of integration for 2+1 case

Lets say I want to find the spherical symmetric solution to EFE $$G=2T$$ in $$d+1$$ dimensions. The symmetries and EFE imply $$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2$$ With $$f(r)$$ satisfying $$f^\prime(r)\propto r^{1-d}$$ Therefore $$f(r)=A\int^r\frac{dx}{x^{d-1}}$$ Comparison with the Newtonian limit, allows us to $$call$$ the constant of proportionality $$A$$ such that $$\Delta f(r)=2\Delta\Phi_N$$ One can easily check that changing the constant of integration is NOT a matter of coordinate (therefore gauge) transformation and different constants give rise to different worlds. A single choice may however be singled out by assuming flat ($$f=1$$) at spatial infinity.

Question: What about $$d=2$$? In this case the (different) answers are $$f_a(r)=\frac{M}{\pi}\log(r/a)$$ None of which yields asymptotic flatness!

PS: My constants may differ from yours (Cf. My EFE) but that does not change anything. The problem is still there

• Maybe a (2+1) world can not exist? – K. Sadri Jun 4 '19 at 9:44
• Maybe no mass monopole exists in 2+1. – K. Sadri Jun 4 '19 at 10:17

Gravity in 2+1 dimensions does not have propagating degrees of freedom, so vacuum Einstein field equations $$Ric=0$$ simply mean that the metric is locally flat. As a result, there is no Schwarzschild solution. Instead, a point particle corresponds to a conical singularity of a spatial slice of spacetime. Also, if one considers a negative cosmological constant (EFE then ensure that the spacetime is locally AdS₃), then there is a black hole solution, the BTZ black hole.