Lyapunov Exponents to Test General Relativity

Photon rings are key targets for near-future space-based very-long baseline interferometry missions. The ratio of flux measured between successive light-rings is characterized by the Lyapunov exponents of the corresponding nearly-bound null geodesics. Therefore, understanding Lyapunov exponents in this environment is of crucial importance to understanding black hole observations in general, and in particular, they may offer a route for constraining modified theories of gravity. While recent work has made significant progress in describing these geodesics for Kerr, a theory-agnostic description is complicated by the fact that Lyapunov exponents are time-parameterization dependent, which necessitates care when comparing these exponents in two different theories. In this work, we present a robust numerical framework for computing and comparing the Lyapunov exponents of null geodesics in Kerr with those in an arbitrary modified theory. We then present results obtained from calculating the Lyapunov exponents for null geodesics in two particular effective theories, scalar Gauss-Bonnet gravity and dynamical Chern-Simons gravity. Using this framework, we determine accuracy lower-bounds required before a very-long baseline interferometry observation can constrain these theories.