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

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.

12 pages, added one figure in section 3; a paragraph in Introduction rewritten without changing the argument; corrected typos; one reference added for section 2; Eq. (84) rewritten in a more elegant form; slightly revised argument in section 9, results unchanged

References in corpus (4)

• General post-Minkowskian expansion of time transfer functions
• Gravitational bending of light by planetary multipoles and its measurement with microarcsecond astronomical interferometers
• Light propagation in the field of a moving axisymmetric body: theory and application to JUNO
• Light propagation in the gravitational field of N arbitrarily moving bodies in 1PN approximation for high-precision astrometry

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