Black Hole Shadow Drift and Photon Ring Frequency Drift

The apparent angular size of the shadow of a black hole in an expanding Universe is redshift-dependent. Since cosmological redshifts change with time - known as the redshift drift - all redshift-dependent quantities acquire a time dependence, and a fortiori so do black hole shadows. We find a mathematical description of the black hole shadow drift and show that the amplitude of this effect is of order $10^{-16}$ per day for M87$^{\star}$. While this effect is small, we argue that its non-detection can be used to constrain the accretion rate around supermassive black holes, as well as a novel probe of the equivalence principle. If general relativity is assumed, we infer from the data obtained by the Event Horizon Telescope for M87$^{\star}$ a maximum accretion rate of $|\dot{M}/{M}| \leq 10^5 M_{\odot}$ per year. On the other hand, in the case of an effective gravitation coupling, we derive a constraint of $|\dot{G}/G| \leq 10^{-3}-10^{-4}$ per year. The effect of redshift drift on the visibility amplitude and frequency of the universal interferometric signatures of photon rings is also discussed, which we show to be very similar to the shadow drift. This is of particular interest for future experiments involving spectroscopic and interferometric techniques, which could make observations of photon rings and their frequency drifts viable.