Weitzenböck’s torsion, Fermi coordinates, and adapted frames

We study Weitzenböck’s torsion and discuss its properties. Specifically, we calculate the measured components of Weitzenböck’s torsion tensor for a frame field adapted to static observers in a Fermi normal coordinate system that we establish along the world line of an arbitrary accelerated observer in general relativity. A similar calculation is carried out in the standard Schwarzschild-like coordinates for static observers in the exterior Kerr spacetime; we then compare our results with the corresponding curvature components. Our work supports the contention that in the extended general relativistic framework involving both the Levi-Civita and Weitzenböck connections, curvature and torsion provide complementary representations of the gravitational field.

Figure 1

Schematic representation of an infinitesimal parallelogram. Here $(BD{)}^{\mu }=A^{\mu }-{\mathrm{\Gamma }}_{\alpha \beta }^{\mu }(P)B^{\alpha }{A}^{\beta }$, while $(AC{)}^{\mu }=B^{\mu }-{\mathrm{\Gamma }}_{\alpha \beta }^{\mu }(P)A^{\alpha }{B}^{\beta }$. Hence $(CD{)}^{\mu }=({\mathrm{\Gamma }}_{\alpha \beta }^{\mu }-{\mathrm{\Gamma }}_{\beta \alpha }^{\mu })A^{\alpha }{B}^{\beta }$.

Figure 2

Schematic representation of the nonclosure of the parallelogram when ${\mathrm{\Gamma }}_{(\alpha \beta )}^{\mu }(P)=0$.