From: Henry Baker <hbaker1@pipeline.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, July 24, 2009 12:15:14 PM Subject: [math-fun] dumb question about general relativity This question is about "classical" "infinitesimal" GR -- i.e., no quantum stuff like Hawking radiation & no Planck length. The laws of GR are symmetric w.r.t. time, so we have the following conundrum: If an object falls towards a black hole, from the perspective of an outside observer, time on the object gets slower & slower & in the limit it stops. I've been told that as a result of this time dilation, everything that "falls into" the black hole doesn't really fall in, but ends up getting painted onto its surface (from the perspective of the outside observer). Yes. There is a last time at which a signal emitted by the infalling body can propagate to infinity. The final cycle of that signal is red shifted infinitely, and exponentially fades out. But if the laws are truly symmetric w.r.t. time, then it should be possible for the black hole to "burp" out objects, as well. White holes do not contradict classical general relativity, but perhaps they cannot exist in quantum gravity due to the vacuum becoming unstable. The big bang is something like a white hole, but quantum gravity will be needed to explain it. Then also, black holes do exhibit a whiteness, in the form of Hawking radiation, but again, it's a quantum effect. I've read expositions of what happens to an observer _on_ the object falling into the black hole, and time supposedly doesn't stop. Also, if the black hole is large enough, then the tidal forces near the surface aren't large enough to destroy even a human observer, so one can ask the question about what such an observer would see. An observer on the infalling body observes nothing in particular upon passing the event horizon, but reaches the central singularity in a finite proper time, which is about 0.1 ms for a solar mass black hole. Tidal forces become infinite as the singularity is approached. While falling in, the observer continues to receive signals from the external universe. The backward light cone from the point at which the singularity is reached delineates the final time for which external signals are received. I am not sure about whether such signals are red or blue shifted. Only a finite history of the universe can be viewed before the observer reaches his end-of-time. All of this presupposes some sort of invertible transform in going from the perspective of the outside observer to the observer on the moving object. But clearly, such an invertible transform has a singularity at the "time" when the object gets painted onto the black hole (from the outside observer's perspective). Is there an analogy here with Taylor & Laurent series in the complex plane, where you can extend a function beyond its usual convergence by utilizing a different type of series? Many of these questions can be answered by staring at Penrose diagrams. In particular, it sure does look like the central singularity is some kind of natural boundary to spacetime. The event horizon, however, is no more than a coordinate singularity, and spacetime is smooth here. -- Gene