You said "instantaneously". That usually would imply traveling outside of your lightcone. That really is undefined; an "instantaneous" transport from here can end up anywhere outside of your lightcone, which are all equally well(/poorly) defined in relativity. You can't distinguish those points in either time or space without creating a preferred reference frame.
This is the same reason FTL drives are also, fundamentally, time machines. It's really a very profound thing; there's no "yeah, but really we know there's an absolute time frame underneath it all"... there really, really isn't an absolute time frame, and you really, really can't say how old the black hole would be if you "instantaneously" transported there; it is just as valid to instantaneously appear when it is 31 years old as it would be to appear when it is 100,000,029 years old (taking of course the estimated age as its true age). Or, IMHO, just as invalid because FTL and instantaneous truly don't exist.
Yes, if you change your claim to be sublight and non-instantaneous we can compute how old it would be, but that's a profound goal post shift.
The Minkowski metric is a mindbender, the moreso once you realize that it is not some weird mathematical abstraction with no connection to reality, it is reality, or at least a much better approximation than the one we instinctively use. The "weird mathematical abstraction with no connection to reality" is the implicitly Euclidean/Galilean metric that exists in our heads. (Which can actually be demonstrated to contain fundamental contradictions if you poke them hard enough.)
I get it. There is a disconnect between my timeframe and that of a different acceleration frame. But they are related in a precise calculatable and empirically-confirmed way that lets me set a clock in one and know exactly what the clock in the other will read.
Knowing the age of something at a given time in its own reference point isn't contradictory, thus we can know the age of the Earth, etc. It's is contradictory to say that now, in our reference frame, the [supernova, black hole, star, etc.] is x years old. The only thing that makes sense is to judge how old the object appears to us, judging from the information (light) that's reaching us now from the given object.
It may help to read the preceding bit of the work.
It is probably also not necessarily clear which part of that page is talking about the fundamental contradictions of the metric if you haven't read the preceding chapters. The key sentence that makes the point I am referring to is "Before blithely dismissing this concern as non-sensical, it's worth noting that modern physics has concluded (along with Zeno) that the classical image of space and time was fundamentally wrong, and in fact motion would not be possible in a universe constructed according to the classical model.", where the "classical model" is one based on a Euclidean metric in which time is simply another spatial dimension that acts weird for no apparent-within-the-model reason, rather than being an actually special dimension as it is in the Minkowski metric. Also, "In all four of Zeno's arguments on motion, the implicit point is that if space and time are independent, then logical inconsistencies arise regardless of whether the physical world is continuous or discrete." where again, space and time being independent is a characteristic of the Euclidean/Galilean metric. (A little rough to be pointing you to the middle of that very-fine work, but it makes the point far better than I can.)
This is the same reason FTL drives are also, fundamentally, time machines. It's really a very profound thing; there's no "yeah, but really we know there's an absolute time frame underneath it all"... there really, really isn't an absolute time frame, and you really, really can't say how old the black hole would be if you "instantaneously" transported there; it is just as valid to instantaneously appear when it is 31 years old as it would be to appear when it is 100,000,029 years old (taking of course the estimated age as its true age). Or, IMHO, just as invalid because FTL and instantaneous truly don't exist.
Yes, if you change your claim to be sublight and non-instantaneous we can compute how old it would be, but that's a profound goal post shift.
The Minkowski metric is a mindbender, the moreso once you realize that it is not some weird mathematical abstraction with no connection to reality, it is reality, or at least a much better approximation than the one we instinctively use. The "weird mathematical abstraction with no connection to reality" is the implicitly Euclidean/Galilean metric that exists in our heads. (Which can actually be demonstrated to contain fundamental contradictions if you poke them hard enough.)