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After the revelation of the first black hole images, it seems there is a bias towards its south side. Is it because of measuring it from earth or is it something more fundamental in the understanding of the gravitational field?

This is because of the tilt of the accretion disk with respect of our line of sight. The accretion disk is "in front" of the black hole in the southern part of the image.

You can compare with this image from the 5th paper of the ApJ Letters "First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring", which shows one of the best-fit theoretical simulations alongside the observed image:

There is a number of other effects at play, though. You see only a narrow frequency band, so once the light is shifted by the Doppler effect to a different wavelength in either direction, or when the plasma has the wrong temperature, you stop seeing it in the image. An additional general-relativistic effect is a "space-time push" the rotation of the black hole gives to co-rotating photons, which causes a slight west-east asymmetry in the image as well.

The reason is almost entirely due to Doppler beaming and boosting of radiation arising in matter travelling at relativistic speeds. This in turn is almost entirely controlled by the orientation of the black hole spin. The black hole sweeps up material and magnetic fields almost irrespective of the orientation of any accretion disk.

The pictures below from the fifth event horizon telescope paper makes things clear.

The black arrow indicates the direction of black hole spin. The blue arrow indicates the initial rotation of the accretion flow. The jet of M87 is more or less East-West (projected onto the page), but the right hand side is pointing towards the Earth. It is assumed that the spin vector of the black hole is aligned (or anti-aligned) with this.

The two left hand plots show agreement with the observations. What they have in common is that the black hole spin vector is into the page (anti-aligned with the jet). Gas is forced to rotate in the same way and results in projected relativistic motion towards us south of the black hole and away from us north of the black hole. Doppler boosting and beaming does the rest.

I was going to ask a similiar question, so my (niave and completely amateur) answer is based on the paper recently published in The Astrophysical Journal Letters First M87 Event Horizon Telescope Results.

Image from BBC News and EHT Collaboration

The best-known effect is that a black hole, surrounded by an optically thin luminous plasma, should exhibit a "silhouette" or "shadow" morphology: a dim central region delineated by the lensed photon orbit (Falcke et al. 2000). The apparent size of the photon orbit, described not long after Schwarzschild's initial solution was published (Hilbert 1917; von Laue 1921), defines a bright ring or crescent shape that was calculated for arbitrary spin by Bardeen (1973), first imaged through simulations by Luminet 1979, and subsequently studied extensively (Chandrasekhar 1983; Takahashi 2004; Broderick & Loeb 2006). The size and shape of the resulting shadow depends primarily on the mass of the black hole, and only very weakly on its spin and the observing orientation.

I found this animated GIF to explain this really well:

https://upload.wikimedia.org/wikipedia/commons/d/d6/BlackHole_Lensing.gif

Given this, it looks like it's just relative location of the black hole to the light sources behind/around it.