Time Delay in Robertson-McVittie Spacetime and its Application to Increase of Astronomical Unit

TL;DR

This paper examines how light delay in Robertson-McVittie spacetime affects astronomical observations, specifically testing if cosmological expansion explains the observed increase in the astronomical unit, and finds it does not.

Contribution

It derives correction terms for light delay in Robertson-McVittie spacetime and applies them to the astronomical unit increase, showing cosmological effects are insufficient to explain the observation.

Findings

Cosmological expansion does not account for the AU increase.

Derived correction terms for light delay in Robertson-McVittie spacetime.

Applied light delay corrections to astronomical observations.

Abstract

We investigated the light propagation by means of the Robertson-McVittie solution which is considered to be the spacetime around the gravitating body embedded in the FLRW (Friedmann-Lema{\^i}tre-Robertson-Walker) background metric. We concentrated on the time delay and derived the correction terms with respect to the Shapiro's formula. To relate with the actual observation and its reduction process, we also took account of the time transformations; coordinate time to proper one, and conversely, proper time to coordinate one. We applied these results to the problem of increase of astronomical unit reported by Krasinsky and Brumberg (2004). However, we found the influence of the cosmological expansion on the light propagation does not give an explanation of observed value, $d{\rm AU}/dt = 15 \pm 4$ [m/century] in the framework of Robertson-McVittie metric.