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Yes, I am aware of observations of "massive objects", for example near the center of the galaxy. But is there any evidence that the object in question is a black hole, with all its commonly attributed properties? Or are black holes just an unconfirmed artifact of general relativity, i.e. something that is an abstract mathematical concept and not a real object?

I am not asking about means of detecting them, I am aware of those means. I'm interested if there's anything suggesting that the properties commonly attributed to black holes, considered a "fact", other than large mass, have anything to do with reality, or are completely abstract and deduced from math.

Qmechanic
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I don't think you're going to find an answer that is satisfactory to you, especially since you refuse to enumerate what "properties commonly attributed to black holes" are. There are extensive review articles on this, and numerous questions here. The Cygnus X-1 observations are pretty defninitive, and involve X-Ray signatures of a known object, rather than just constraining mass.

Really, it sounds like you'd only be satisfied by a tabletop black hole, which is far, far beyond the capacity of our current technology. I think most theoretical physicists would be happy to abandon general relativity for an alternative, if for no other reason to have another chance to resolve the quantum gravity problem, but for now, it is the best model we have, and we have to take it seriously, at least in the classical regime. The solar system tests of general relativity really are quite stringent, and have either ruled out or severely constrained most other theories. And most of the novel features of general relativity HAVE been tested by solar system tests, just not in as extreme environment as a black hole.

Zo the Relativist
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    The only truly missing piece is gravitational waves, and we have indirect evidence of that, and hopefully LIGO or LISA (hey, LISA Pathfinder went into the sky yesterday!) will directly detect those. – Zo the Relativist Dec 03 '15 at 15:28