# General relativity without curvature?

Is there a reformulation of general relativity without curved space time, just with fields (like classical E&M)?

Edit: removed the part about E&M with curvature (multiple posts).

• @JohnDuffield It is incredibly easy to make solutions to Einstein's Field Equation that that have closed timelike curves. For instance consider $\{(a,A,x,y,z)\in\mathbb R^5:a^2+A^2=1\}$ with the subspace metric from $d\tau^2$=$da^2$+$dA^2$-$dx^2$-$dy^2$-$dz^2.$ If you just object to calling a CTC time travel that is understandable but I'm trying to spell out that GTG is a different theory than GR because GTG doesn't allow solutions like that and GR does. – Timaeus Aug 22 '15 at 18:25
GR can be recast into an equivalent but conceptually quite different form, using teleparallel gravity. This approach introduces the Weitzenboeck connection, which has no curvature, but has torsion. The presence of torsion indicates that gravity is not geometrized. Recall that in GR, we can always choose a locally inertial coordinate system such that the geodesic equation is simply $$\frac{d^2 x^{\mu}}{d\tau^2}=0,$$ since there is a reference frame in which all Christoffel symbols vanish - this is the weak equivalence principle. However, this implies that in the same frame, $${T^{\sigma}}_{\mu\nu}={\Gamma^{\sigma}}_{\mu\nu}-{\Gamma^{\sigma}}_{\nu\mu}=0$$ and therefore this tensor vanishes in all frames. In Teleparallel gravity this is false, so gravity is a force represented by the nonvanishing symbols. That is, it's a field. The approaches are equivalent. Check out the link at the beginning for more.