Constraining possible variations of the fine structure constant in strong gravitational fields with the K$\alpha$ iron line

In extensions of general relativity and in theories aiming at unifying gravity with the forces of the Standard Model, the value of the "fundamental constants" is often determined by the vacuum expectation value of new fields, which may thus change in different backgrounds. Variations of fundamental constants with respect to the values measured today in laboratories on Earth are expected to be more evident on cosmological timescales and/or in strong gravitational fields. In this paper, I show that the analysis of the K$\alpha$ iron line observed in the X-ray spectrum of black holes can potentially be used to probe the fine structure constant $\alpha$ in gravitational potentials relative to Earth of $\Delta \phi \approx 0.1$. At present, systematic effects not fully under control prevent to get robust and stringent bounds on possible variations of the value of $\alpha$ with this technique, but the fact that current data can be fitted with models based on standard physics already rules out variations of the fine structure constant larger than some percent.