Schwarzschild's Black Hole
The Schwarzschild metric is for spherical, non-rotating, static mass distributions. A black hole is believed to be a possible end point in the life of a very massive star. All stars that are observed in our universe are observed to be rotating, and that rotation tends to cause their shape to deviate from spherical. So the Schwarzschild black hole is not an object that would tend to occur in nature. If we assume that a very advanced society is capable of constructing one could it be used for a time machine? Probably not. Consider a would-be black hole explorer, Chris. We decide to watch Chris enter the black hole from a fair distance. As Chris gets closer to the black hole the gravitational field becomes quite intense. We observe the clock that Chris holds by light signals that it emits and see that it appears to run slower and slower. Eventually, the gravitational differential from head toe becomes so great that it should literally pull Chris apart. That's if the black hole mass is small, say 1 solar mass. Paradoxically, supermassive black holes would have much less intense tidal forces near the event horizon. But we will assume that we are sufficiently advanced such that we can build a space suit capable of withstanding the tidal forces and keeping Chris alive through this adventure. Because of the time dilation effects at a black hole's surface light waves will appear to change color, they will be red-shifted. The light from Chris' clock will shift out of the visual wavelengths into the infrared and then into the radio and eventually out of our detection ability. We can no longer observe the adventure. It is just as well, because if we could detect Chris all the way to the event horizon we would have to watch forever (I don't think I'll live that long). Time dilates to infinity at the event horizon, as seen by an observer from an infinite distance. So to continue the narrative we must move to Chris' frame. Upon crossing the event horizon we will notice nothing special whatsoever. But we can no longer turn back. Now our folly is fully realized. We see the singularity and realize that if the black hole does deposit material somewhere else in spacetime it must pass through it and that will undoubtedly be fatal. This is where our story must end. If only it had been a rotating black hole.