M87* in space, time, and frequency.

Observing the dynamics of compact astrophysical objects provides insights into their inner workings, thereby probing physics under extreme conditions. The immediate vicinity of an active supermassive black hole with its event horizon, photon ring, accretion disk, and relativistic jets is a perfect place to study general relativity, magneto-hydrodynamics, and high energy plasma physics. The recent observations of the black hole shadow of M87* with Very Long Baseline Interferometry (VLBI) by the Event Horizon Telescope open the possibility to investigate its dynamical processes on time scales of days. In this regime, radio astronomical imaging algorithms are brought to their limits. Compared to regular radio interferometers, VLBI networks typically have fewer antennas. The resulting sparser Fourier sampling of the sky brightness distribution can only be partially compensated for by co-adding observations from different days, as the source changes. Here, we present an imaging algorithm that copes with the data scarcity and the source's temporal evolution, while simultaneously providing uncertainty quantification on all results. Our algorithm views the imaging task as a Bayesian inference problem of a time-varying brightness, exploits the correlation structure between time frames, and reconstructs an entire, $2+1+1$ dimensional time-variable and spectrally resolved image at once. The degree of correlation in the spatial and the temporal direction is inferred from the data and not assumed a priori. We apply the method to the EHT observation of M87* and validate our approach on synthetic data. The time- and frequency-resolved reconstruction of M87* confirms variable structures on the emission ring on a time scale of days. The reconstruction features extended and time-variable emission structures outside the ring itself.