Recent astrophysical observations of electromagnetic and gravitational emission from close to black-hole horizons offer novel tests of General Relativity (GR) in the strong-gravity regime. Constraining potential deviations requires reliable predictions not just in but also beyond GR. To set the stage, I discuss different assumptions about physics beyond GR and resulting expectations about where to look for deviations.
In the first part of my talk, I will focus on black-hole uniqueness and how to leverage VLBI observations of the photon ring to constrain the underlying spacetime. In particular, I will discuss how to systematically parameterize potential deviations and demonstrate a systematic lensing-band framework to obtain robust constraints.
In the second part of my talk, I will focus on the dynamical strong-gravity regime and how to leverage observed binary waveforms to constrain the effective field theory (EFT) of gravity. In particular, I will summarise recent progress on well-posedness that place numerical relativity in the EFT of gravity on the same solid mathematical footing as in GR.