The Shapiro time delay and the equivalence principle

TL;DR

This paper provides an accessible derivation of the Shapiro time delay from Einstein's equivalence principle, compares relativistic derivations, and discusses practical applications in solar system and pulsar observations.

Contribution

It offers a pedagogical derivation of the Shapiro delay from basic principles and compares different relativistic approaches, aiding understanding of general relativity.

Findings

Derivation of the Newtonian approximation accessible to students

Comparison of relativistic derivations from the Schwarzschild metric

Application of the delay formula to solar system and pulsar data

Abstract

The gravitational time delay of light, also called the Shapiro time delay, is one of the four classical tests of Einstein's theory of general relativity. This article derives the Newtonian version of the Shapiro time delay from Einstein's principle of equivalence and the Newtonian description of gravity, in a manner that is accessible to undergraduate students and advanced high-school students. The derivation can be used as a pedagogical tool, similar to the way that simplified derivations of the gravitational deflection of light are used in teaching about general relativity without making use of the more advanced mathematical concepts. Next, we compare different general-relativistic derivations of the Shapiro time delay from the Schwarzschild metric, which leads to an instructive example for the challenges of formulating the post-Newtonian limit of Einstein's theory. The article also describes simple applications of the time delay formula to observations within our solar system, as well as to binary pulsars.