Formalism for Testing Theories of Gravity Using Lensing by Compact Objects. I: Static, Spherically Symmetric Case

TL;DR

This paper develops a formalism to analyze gravitational lensing by compact objects in various gravity theories, providing a way to test these theories through observable lensing effects, especially in the strong-deflection regime.

Contribution

It introduces a coordinate-independent, Taylor series-based formalism for calculating lensing corrections in static, spherically symmetric gravity theories, including higher-order effects.

Findings

First-order corrections to magnification and centroid vanish in GR-like theories.

Second-order corrections can distinguish modified gravity theories from GR.

Predicted measurable corrections for the Galactic black hole lensing with current and future instruments.

Abstract

We are developing a general, unified, and rigorous analytical framework for using gravitational lensing by compact objects to test different theories of gravity beyond the weak-deflection limit. In this paper we present the formalism for computing corrections to lensing observables for static, spherically symmetric gravity theories in which the corrections to the weak-deflection limit can be expanded as a Taylor series in one parameter, namely the gravitational radius of the lens object. We take care to derive coordinate-independent expressions and compute quantities that are directly observable. We compute first- and second-order corrections to the image positions, magnifications, and time delays. Interestingly, we find that the first-order corrections to the total magnification and centroid position vanish in all gravity theories that agree with general relativity in the weak-deflection limit, but they can remain nonzero in modified theories that disagree with GR in the weak-deflection limit. For the Reissner-Nordstrom metric and a related metric from heterotic string theory, our formalism reveals an intriguing connection between lensing observables and the condition for having a naked singularity, which could provide an observational method for testing the existence of such objects. We apply our formalism to the Galactic black hole and predict that the corrections to the image positions are at the level of 10 micro-arcseconds, while the correction to the time delay is a few hundredths of a second. These corrections would be measurable today if a pulsar were found to be lensed by the Galactic black hole; and they should be readily detectable with planned missions like MAXIM.