Local gravitational physics of the Hubble expansion: Einstein’s equivalence principle in cosmology

Abstract

We study physical consequences of the Hubble expansion of Friedmann-Lemaıtre-Robertson-Walker (FLRW) manifold on measurement of space, time and light propagation in the local inertial frame. We use the results of this study to analyse the Solar System radar ranging and Doppler tracking experiments and time synchronization. FLRW manifold is covered by the coordinates (t, yi), where t is the cosmic time coinciding with the proper time of the Hubble observers and identified with the barycentric coordinate time (TCB) used in ephemeris astronomy. We introduce the local inertial coordinates xα = (x0, xi) in the vicinity of a world line of a Hubble observer with the help of a special conformal transformation that respects the local equivalence between the tangent and FLRW manifold. The local inertial metric is Minkowski flat and is materialized by the congruence of time-like geodesics of static observers being at rest with respect to the local spatial coordinates xi. The static observers are equipped with the ideal clocks measuring their own proper time which is synchronized with the cosmic time t measured by the Hubble observer. We consider the geodesic motion of test particles and notice that the local coordinate time x0 = x0(t) taken as a parameter along the world line of the particle, is a function of Hubble’s observer time t. This function changes smoothly from x0 = t for a particle at rest (observer’s clock), to x0 = t + (1/2)Ht2 for photons, where H is the Hubble constant. Thus, the motion of a test particle is non-uniform when its world line is parametrized by the cosmic time t. NASA JPL Orbit Determination Program operates under the assumption that the spacetime is asymptotically flat which presumes that the motion of light (after the Shapiro delay is excluded) is uniform with respect to the time t but it does not comply with the non-uniform motion of light on cosmological manifold. For this reason, the motion of light in the Solar System analysed with the Orbit Determination Program appears as having a systematic blue shift of frequency, of radio waves circulating in the Earth-spacecraft radio link. The magnitude of the anomalous blue shift of frequency is proportional to the Hubble constant H that may open an access to the measurement of this fundamental cosmological parameter in the Solar System radiowave experiments.