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Four Metrics

A central idea in general relativity is that physics should not depend on the space-time coordinates in use \cite{einstein}. But the qualitative description of various phenomena can appear superficially quite different. Here we consider falling into a classical black hole using four distinct but equivalent metrics. First is the Schwarzchild case, with extreme time dilation at the horizon. Second, rescaling the dilation allows falling into the hole in finite proper time. Third, time and space are rescaled into a Penrose motivated picture where light trajectories all have unit slope. Fourth, a white hole variation of the second metric allows passage out through the horizon, with reentry forbidden.