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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).

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The Gauge Theory of Gravity (GTG) by Lasenby, Doran and Gull has a background spacetime with fields on it.

It is basically derived from the same physical principles but as a background theory. It ends up not being the same theory, for instance it doesn't have the same isotropic solutions, and I think it does not allow time travel and such (unlike General Relativity).

There are others as well.

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  • $\begingroup$ Good stuff Timaeus. But GR doesn't allow time travel. That's just something promoted by the likes of Misner Thorne and Wheeler. See the screenshot from A World Without Time here. $\endgroup$ – John Duffield Aug 22 '15 at 11:46
  • $\begingroup$ @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. $\endgroup$ – Timaeus Aug 22 '15 at 18:25
  • $\begingroup$ Wheeler conflated a circle with a cycle. See Ben Crowell's answer here. There's no motion in spacetime, such as around a CTC. No problem with the GTG. $\endgroup$ – John Duffield Aug 23 '15 at 10:55
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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.

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