Die Transformationsgleichungen , welche die Raumzeitkoordinaten (z, y, z, t) eines ruhenden Systems mit denen in einem bewegten System ( z ' , y' , z', t') verknlipfen, dessen Geschwindigkeit q nach Richtung und Grole konstant ist, haben in der heutigen Physik eine so gro6e Wichtigkeit erlangt, da6 es sich wohl lohnt, genau zu prufen, welche Voraussetzungen physikalischer oder anderer Natur eigentlich notwendig sind, urn die Gestalt dieser Gleichungen abzuleiten. Nach der Relativitatstheorie sind sie durch die Lorentztransformation gegeben. Diese lautet bekanntlich, wenn wir mit c die Lichtgeschwindigkeit im Vakuum bezeichnen nnd die Koordinatensysteme so wiihlen, da6 5ur Zeit Null das bewegte mit dem ruhenden zusammenfUlt und sich dann in der z-Richtung weiterbewegt :Als Brenzfall fur c = 00 ist in diesen Gleichungen bekanntlich die Galileitransformation enthalten :(2) t = t , z'=-q t + z.

The immediate vicinity of an active supermassive black hole—with its event horizon, photon ring, accretion disk and relativistic jets—is an appropriate place to study physics under extreme conditions, particularly general relativity and magnetohydrodynamics. Observing the dynamics of such compact astrophysical objects provides insights into their inner workings, and the recent observations of M87* by the Event Horizon Telescope1–6 using very-long-baseline interferometry techniques allows us to investigate the dynamical processes of M87* on timescales of days. Compared with most radio interferometers, very-long-baseline interferometry networks typically have fewer antennas and low signal-to-noise ratios. Furthermore, the source is variable, prohibiting integration over time to improve signal-to-noise ratio. Here, we present an imaging algorithm7,8 that copes with the data scarcity and temporal evolution, while providing an uncertainty quantification. Our algorithm views the imaging task as a Bayesian inference problem of a time-varying brightness, exploits the correlation structure in time and reconstructs (2 + 1 + 1)-dimensional time-variable and spectrally resolved images. We apply this method to the Event Horizon Telescope observations of M87*9 and validate our approach on synthetic data. The time- and frequency-resolved reconstruction of M87* confirms variable structures on the emission ring and indicates extended and time-variable emission structures outside the ring itself.

The data reduction procedure for radio interferometers can be viewed as a combined calibration and imaging problem. We present an algorithm that unifies cross-calibration, self-calibration, and imaging. Because it is a Bayesian method, this algorithm not only calculates an estimate of the sky brightness distribution, but also provides an estimate of the joint uncertainty which entails both the uncertainty of the calibration and that of the actual observation. The algorithm is formulated in the language of information field theory and uses Metric Gaussian Variational Inference (MGVI) as the underlying statistical method. So far only direction-independent antenna-based calibration is considered. This restriction may be released in future work. An implementation of the algorithm is contributed as well.