Formalism for testing theories of gravity using lensing by compact objects: Static, spherically symmetric case

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 series expansions for the observables that are accurate to second order in the ratio $\ensuremath{\epsilon}={\ensuremath{\vartheta}}_{\ifmmode\bullet\else\textbullet\fi{}}/{\ensuremath{\vartheta}}_{E}$ of the angle subtended by the lens's gravitational radius to the weak-deflection Einstein radius, which scales with mass as $\ensuremath{\epsilon}\ensuremath{\propto}{M}_{\ifmmode\bullet\else\textbullet\fi{}}^{1/2}$. The positions, magnifications, and time delays of the individual images have corrections at both first and second order in $\ensuremath{\epsilon}$, as does the differential time delay between the two images. 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 general relativity in the weak-deflection limit. For the Reissner-Nordstr\"om 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 $10\text{ }\text{ }\ensuremath{\mu}\mathrm{arc}\text{ }\mathrm{s}$ (microarcseconds), 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.