File:NoMassToEventHorizon.JPG

English: Here are schematics of some rolled-up x-cτ plots for paths through the (open) center of spherical objects of various mass, from no mass (upper left) to black hole (lower right), inspired by Lewis Carroll Epstein's sketches^[1] 12-4 through 12-7. Free-float trajectories are straight-lines on the "locally-flat" x-cτ surface. Simple gravitational potential models were used to generate the radial profiles, over an axial distance of ±5 radii.

The rate of passage of proper-time τ (represented by angular position around the tube azimuth) for an observer at fixed "far-radius" r from the gravitational object's center (represented by distance from the center of each image-panel along the tube's axis), per unit far-time t (represented by distance in any direction on the curved surface), decreases as the tube-diameter increases near a gravitational object's center. Even further, the gravitational Lorentz-factor dt/dτ goes to infinity at the black-hole's event horizon in the bottom right panel so that events happening quickly to a plunging traveler seem to take forever to a far-time observer.