Weitzenb\"ock's Torsion, Fermi Coordinates and Adapted Frames

TL;DR

This paper investigates Weitzenb"ock's torsion in general relativity, calculating its components in various coordinate systems and comparing it with curvature to show their complementary roles in describing gravity.

Contribution

It provides explicit calculations of Weitzenb"ock's torsion components in Fermi and Schwarzschild-like coordinates, highlighting their relation to curvature in extended GR frameworks.

Findings

Torsion and curvature offer complementary descriptions of gravity.

Explicit torsion components are derived for static observers in different spacetimes.

Results support the view that torsion and curvature jointly characterize gravitational fields.

Abstract

We study Weitzenb\"ock's torsion and discuss its properties. Specifically, we calculate the measured components of Weitzenb\"ock'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 Weitzenb\"ock connections, curvature and torsion provide complementary representations of the gravitational field.