How can astronomers say, we know there are black holes at the centre of each galaxy?
What methods of indirect detection are there to know where and how big a black hole is?
There are three main feasible ways of detecting a black hole:
Gravitational lensing: the strong gravitational attraction of a black hole bends space time and the light coming from nearby stars (nearby in the sense of being in the same are in our sky) is bent inwards. There are a few well known distorsion types due to gravity, but mainly we can see galaxies, which are more or less elliptical, bent into pancake shapes.
Accretion disks and jets: as the black hole "sucks in" dust and other similar matter from nearby space, the matter is accelerated at relativistic velocities and it emits x-rays as it goes to die inside the event horizon.
Stars orbiting black holdes: if a star is orbiting a black hole, it will appear to be orbiting empty space (since we can't basically see a black hole directly).
Other ways, like Hawking radiation, are only theoretically possible for now -we could maybe be able to see old mini black holes "popping" but it's not really clear how that would happen exactly and none has been seen so far.
For the record, gravitational lensing has never been used to detect or even observe a known black hole. The above spectacular image of lensed galaxies is due to a galaxy cluster which makes the lens, and is roughly a million times more massive than the most massive black hole.
In all likelihood, we will detect black holes with gravitational waves (e.g., LIGO or LISA) before strong lensing. However microlensing may very soon be used to detect isolated BHs in our galaxy, along the line of site to individual stars in the galactic bulge.
For a stellar mass black hole the quick answer is there is no hard surface from which infalling material can crash on and splash or scatter back out. The event horizon is not a hard or material substance. So a black hole has some distinct signature differences from a neutron star, where material in an accretion disk strikes a hard surface. It is in this way one can identify an object as a black hole.
The Chandra X-Ray Observatory FAQ answers this question, about the supposed black hole at the center of the Milky Way. One of the steps is to input value(s) observed into GR's equations. Therefore any black hole observed depends on GR's validity, as that link confirms. If GR is invalid, whatever we think is a black hole could instead be an object with a physical surface, with an escape velocity at its surface that is very close to the speed of light.
The way the astronomers used to detect the black hole is by using x-ray imaging from one of the many types of telescope that orbit the earth.
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