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). 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. 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. 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?