Pound-Rebka experiment and torsion in the Schwarzschild spacetime

Abstract

We develop some ideas discussed by E. Schucking [arXiv:0803.4128] concerning the geometry of the gravitational field. First, we address the concept according to which the gravitational acceleration is a manifestation of the space-time torsion, not of the curvature tensor. It is possible to show that there are situations in which the geodesic acceleration of a particle may acquire arbitrary values, whereas the curvature tensor approaches zero. We conclude that the space-time curvature does not affect the geodesic acceleration. Then we consider the Pound-Rebka experiment, which relates the time interval $\ensuremath{\Delta}{\ensuremath{\tau}}_{1}$ of two light signals emitted at a position ${r}_{1}$, to the time interval $\ensuremath{\Delta}{\ensuremath{\tau}}_{2}$ of the signals received at a position ${r}_{2}$, in a Schwarzschild type gravitational field. The experiment is determined by four space-time events. The infinitesimal vectors formed by these events do not form a parallelogram in the $(t,r)$ plane. The failure in the closure of the parallelogram implies that the space-time has torsion. We find the explicit form of the torsion tensor that explains the nonclosure of the parallelogram.