If torque is allowed to exist in the space part of the stress energy tensor $T_{\mu\nu}$ in the Einstein field equations $$ R_{\mu\nu}-\frac12 g_{\mu\nu} R = 8\pi T_{\mu\nu} $$ it would lead to $T_{\mu\nu}$ being asymmetric. That would require some tensor in the left hand side of the field equations to be asymmetric as well. The Ricci tensor $R_{\mu\nu}$ can hardly be generalized to become asymmetric. However the metric tensor $g_{\mu\nu}$ can become asymmetric by using a quasimetric where the symmetry criteria is dropped meaning that the distance from a to b is different from the distance from b to a. This can be further imagined as taking the gauge freedom in the Lorentz transformation to act on the speed of light, as is done in the Sagnac effect where a longer distance is experienced when measuring in the direction of a motion instead of measuring against a motion. The distance is measured symmetrically in the stationary reference frame.

Could such an extension of the field equations be made theoretically and tried on a real physical situation?

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    $\begingroup$ You may want to look into Finsler geometry, though not all Finsler structures arise from quasimetrics. $\endgroup$ – Michael Seifert Apr 18 at 1:40

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