Gravitational waves and the breaking of parallelograms in space-time

We show that plane-fronted gravitational waves induce the breaking of parallelograms in space-time, in the context of the teleparallel equivalent of general relativity. The breaking of parallelograms can be shown by considering a thought experiment that consists of a simple physical configuration, similar to the experimental setup that is expected to lead to the measurement of gravitational waves with the use of laser interferometers. An incident beam of light splits into two beams running along perpendicular arms, endowed with fixed mirrors at the extremes. The reflected light beams are detected at the same point of the splitting. Along each arm, the two light beams define two null vectors: the forward vector and the reflected vector. We show that the sum of these four vectors, the forward and reflected null vectors along the two arms, do form a parallelogram in flat space-time, but not in the presence of plane-fronted gravitational waves. The non-closure of the parallelogram is a manifestation of the torsion of the space-time, and in this context indicates the existence of gravitational waves.