The equivalence principle for a plane gravitational wave in torsion based and non-metricity based teleparallel equivalents of general relativity

Abstract: We study the energy-momentum characteristics of the plane +''-polarised gravitational wave solution of general relativity in the Teleparallel Equivalent of General Relativity (TEGR) and the Symmetric Teleparallel Equivalent of General Relativity (STEGR) using the previously constructed Noether currents. The current components describe locally measured by observer energy-momentum if the displacement vector $\xi$ is equal to the observer's 4-velocity. To determine the non-dynamical connection in these theories we use the unifiedturning of'' gravity principle. For a constructive analysis of the values of Noether currents and superpotentials in TEGR and STEGR, we use the concept of ``gauges''. The gauge changing can affect the Noether current values. We study under what conditions the Noether current for the freely falling observer is zero. When they are established, zero result can be interpreted as a correspondence to the equivalence principle, and it is a novelty for gravitational waves in TEGR and STEGR. We highlight two important cases with positive and zero energy, which reproduce the results of previous works with a different approach to determine gravitational energy-momentum in TEGR, and give their interpretation.