Time transfer functions in Schwarzschild-like metrics in the weak-field limit: A unified description of Shapiro and lensing effects

TL;DR

This paper derives exact closed-form expressions for time transfer functions in weak-field Schwarzschild-like metrics, unifying gravitational lensing and Shapiro delay effects, including novel higher-order enhanced terms.

Contribution

It provides a unified, exact formulation of time transfer functions in weak gravitational fields, revealing two regimes and introducing new higher-order enhanced terms beyond the third order.

Findings

Derived two regimes of light propagation in weak fields.

Obtained general expressions for angular separation and time delay in lensing.

Identified and calculated new higher-order enhanced terms in Shapiro delay.

Abstract

We present a complete analysis of the light rays within the linearized, weak-field approximation of a Schwarzschild-like metric describing the gravitational field of an isolated, spherically symmetric body. We prove in this context the existence of two time transfer functions and we obtain these functions in an exact closed-form. We are led to distinguish two regimes. In the first regime, the two time transfer functions correspond to rays which are confined in regions of spacetime where the weak-field approximation is valid. Such a regime occurs in gravitational lensing configurations with double images of a given source. We find the general expressions of the angular separation and the difference in light travel time between the two images. In the second regime, there exists only one time transfer function corresponding to a light ray remaining in a region of weak field. Performing a Taylor expansion of this function with respect to the gravitational constant, we obtain the Shapiro time delay completed by a series of so-called "enhanced terms". The enhanced terms beyond the third order are new.