Puzzles in Time Delay and Fermat Principle in Gravitational Lensing

TL;DR

This paper critically examines the standard time delay formula in gravitational lensing, identifies an error involving the geometric path difference, and emphasizes the need for clearer interpretation of measurements and reanalysis of data.

Contribution

The paper corrects a fundamental error in the standard time delay formula and clarifies the coordinate dependence and interpretation of time delays in gravitational lensing.

Findings

The geometric path difference term in the CSTD is an error.

Coordinate-dependent terms in time delay are irrelevant for image arrival time differences.

CSTD can generate correct lens equations but its physical significance is unclear.

Abstract

The current standard time delay formula (CSTD) in gravitational lensing and its claimed relation to the lens equation through Fermat's principle (least time principle) have been puzzling to the author for some time. We find that the so-called geometric path difference term of the CSTD is an error, and it causes a double counting of the correct time delay. We examined the deflection angle and the time delay of a photon trajectory in the Schwarzschild metric that allows exact perturbative calculations in the gravitational parameter $GM$ in two coordinate systems -- the standard Schwarzschild coordinate system and the isotropic Schwarzschild coordinate system. We identify a coordinate dependent term in the time delay which becomes irrelevant for the arrival time difference of two images. It deems necessary to sort out unambiguously what is what we measure. We calculate the second order corrections for the deflection angle and time delay. The CSTD does generate correct lens equations including multiple scattering lens equations under the variations and may be best understood as a generating function. It is presently unclear what the significance is. We call to reanalyze the existing strong lensing data with time delays.