Formalism for testing theories of gravity using lensing by compact objects. II. Probing post-post-Newtonian metrics

We study gravitational lensing by compact objects in gravity theories that can be written in a post-post-Newtonian (PPN) framework: i.e., the metric is static and spherically symmetric, and can be written as a Taylor series in ${\mathsf{m}}_{•}/r$, where ${\mathsf{m}}_{•}$ is the gravitational radius of the compact object. Working invariantly, we compute corrections to standard weak-deflection lensing observables at first and second order in the perturbation parameter $\epsilon ={\vartheta }_{•}/{\vartheta }_E$, where ${\vartheta }_{•}$ is the angular gravitational radius and ${\vartheta }_E$ is the angular Einstein ring radius of the lens. We show that the first-order corrections to the total magnification and centroid position vanish universally for gravity theories that can be written in the PPN framework. This arises from some surprising, fundamental relations among the lensing observables in PPN gravity models. We derive these relations for the image positions, magnifications, and time delays. A deep consequence is that any violation of the universal relations would signal the need for a gravity model outside the PPN framework (provided that some basic assumptions hold). In practical terms, the relations will guide observational programs to test general relativity, modified gravity theories, and possibly the cosmic censorship conjecture. We use the new relations to identify lensing observables that are accessible to current or near-future technology, and to find combinations of observables that are most useful for probing the spacetime metric. We give explicit applications to the galactic black hole, microlensing, and the binary pulsar J0737-3039.