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The advent of Einstein's General Theory of Relativitygave physicists a mathematical language for describing the gravitational force in a manner consistent with the constant speed of light. Most of what we believe we know about black holes has come from abstract theoretical models in general relativity.
But in order to observe black holes in Nature we need to know how those abstract theoretical models translate to a Universe filled with other stuff.
Abstract theoretical black holes
In the abstract theoretical model of black holes, a black hole is studied as if it were the only thing in the Universe. Using that approximation, the math of general relativity becomes doable, and we can make predictions about black hole behavior that are useful in understanding the black holes we see. In addition, we learn a lot of things about black holes mathematically that we may never get a chance to witness directly through observation.In general relativity, the paths of light can be calculated for many different distributions of matter and energy using equations call thegeodesic equations. The geodesic equations give us the paths that would be followed by freelyfalling test particles. For example, a baseball after being hit by Sammy Sosa and before being caught by an eager fan would be a freely falling particle, travelling on a geodesic path through spacetime.
Light travels on geodesics paths through spacetime. When those geodesic paths cross the event horizon of a black hole, they never come back out. Interestingly, in a Universe where the energy density is never negative, this behavior of light leads mathematically to two very crucial properties of black holes:
 The surface area of the event horizon of a black hole can only increase, never decrease. This also means that although two black holes can join to make a bigger black hole, one black hole can never split in two.
 The pull of gravity at the event horizon is constant; it has the same value everywhere on the event horizon.
Observable astrophysical black holes
If a black hole traps all the light that crosses the event horizon, then how can we ever hope to observe one?In the abstract theoretical model of a black hole, it sits alone forever in the Universe letting us do math on it. In the Nature we observe, the Universe is filled with dust and gas in addition to stars, planets and galaxies. When dust and gas fall into a black hole, they can be sucked towards the event horizon so fast that the atoms are ionized and release bright light that escapes without crossing the event horizon.
So the way astronomers and astrophysicists detect black holes in astronomical observations is to look for light from ionized dust and gas being sucked into something so fast that it could only be a black hole, not a normal gravitating massive object like a star.
However, this bright light can be hard to see, because most black holes also attract giant clouds of interstellar dust that hide many of their features, as shown on the previous page. The suspected black hole shown in the photo above has a warped dust cloud around it, so that the bright light from the ionized gas can be seen.