How do we know black holes exist? The answer lies in Einstein's General Relativity theory and its attendant “field equations.” The field equations are a class of math problems that describe how gravity works in General Relativity. The equations deal with things like masses, spherical objects and space-time curvature.
An interesting feature of Einstein's field equations is that there are an infinite number of interchangeable solutions to them. Quantum physics god Wolfgang Pauli described the situation in his book Theory of Relativity : “The many possible solutions of the field equations are only formally different. Physically they are completely equivalent.” In other words, there is nothing in any of them which specifies that any one of them is the unique “right answer.” Each solution is a system for describing physical events, and therefore may not contradict another solution with respect to the fundamental nature of those physical events.
Does Einstein's Theory of Relativity predict the existence of black holes? The answer is commonly thought to be “yes,” although the real answer, according to this route of analysis, is “yes and no.” The solutions to Einstein's field equations must be said to be ambiguous on this point. What we face is the possibility that black holes might not exist.
The field equations have been lauded for years as “predicting” black holes. When this is said, what they really should be saying is (like the old peanut butter cups commercial), “You got your mathematical singularities in my description of physical events.”
You see, as Tom Van Flandern understands, “Singularities [or infinities] occur routinely in mathematics.” And they just as routinely do not translate into physical singularities or infinities.
For instance, Newton's Universal Law of Gravitation contains a singularity that is commonly dismissed as a physical impossibility and as a partial inadequacy of the theory on that point. The most well-known example of a mathematical singularity predicting something impossible is the “Ultraviolet Catastrophe.” Here there were these equations that indicated the energy of re-radiation of absorbed light should go to infinity. Some simple, good-old-fashioned hands-on lab work was all it took to clean those equations right up, avoiding the ludicrous infinite values and correctly describing the physical situation. It is essentially openly acknowledged in all cases - except in the case of black holes – that nature presents a physical constraint that stops singularities or infinities from forming.
Why should black holes get the only exemption? Some charge that General Relativity has been “taken over” by mathematicians, as opposed to the physicists who would presumably have had more sense about the whole thing. Einstein himself (a physicist) rejected the idea that, just because the math got a little hairy, his theory “predicted” black holes. In a paper written late in his career, Einstein asserted that the mathematical singularities (known as the Schwarzchild singularities) in his field equations “do not exist in cases which have physical reality… do not exist in physical reality… the question [is] answered by this paper in the negative, as to whether physical models are capable of exhibiting… a singularity.”
That being the case, we might as well also admit that the observational evidence for black holes is, well, minimalist. It all has to be inferred from things happening around the supposed black hole, since no information, light or matter of any kind can ever escape from a black hole to directly provide evidence of its existence. It is suggested by some in the alternative astronomical community that these supposed “black holes” inferred by observation are actually mere super-massive objects, which have not collapsed into a singularity just because some of the math does.
Recommended Reading: “Not-So-Cosmic Censorship and Black Holes” by Karl Brunstein in Meta Research Bulletin , vol. 8, #4, Dec. 15, 1999, and “Physics Has Its Principles” by Tom Van Flandern in vol. 9, #1, March 15, 2000. The Einstein quote is from Annals of Mathematics , vol. 40, #4, October 1939. See also Seeing Red by Halton Arp pp. 228-229.