Universal time delay in static spherically symmetric spacetimes for null and timelike signals

TL;DR

This paper introduces a perturbative method to compute travel times of signals in static spherically symmetric spacetimes, revealing a universal form of time delay depending on mass and post-Newtonian parameters, verified numerically with M87 black hole.

Contribution

A novel perturbative approach for calculating signal travel times in static spherically symmetric spacetimes, establishing a universal time delay formula in the weak field limit.

Findings

Time delay formula depends on mass and post-Newtonian parameter γ.

Universal form of time delay applies to all SSS spacetimes at leading order.

Numerical verification using M87 black hole confirms analytical results.

Abstract

A perturbative method to compute the total travel time of both null and lightlike rays in arbitrary static spherically symmetric spacetimes in the weak field limit is proposed. The resultant total time takes a quasi-series form of the impact parameter. The coefficient of this series at a certain order $n$ is shown to be determined by the asymptotic expansion of the metric functions to the order $n+1$. To the leading order(s), the time delay, as well as the difference between the time delays of two kinds of relativistic signals, is then shown to take a universal form for all SSS spacetimes. This universal form depends on the mass $M$ and a post-Newtonian parameter $\gamma$ of the spacetime. The analytical result is numerically verified using the central black hole of M87 as the gravitational lensing center.