The Infalling Perspective
For the astronaut falling into the black hole, time passes normally. Their wristwatch measures <strong>proper time</strong>, denoted by \(\tau\). According to the Equivalence Principle, locally, physics always behaves as if they are in free fall in flat space. They do not feel time slowing down. They cross the event horizon in a finite amount of proper time and continue toward the singularity.
The Distant Observer's View
However, a distant observer using Schwarzschild coordinates sees something different. Due to extreme gravitational time dilation, light signals from the astronaut take longer and longer to escape. To the distant observer, the astronaut appears to slow down as they approach the horizon, becoming increasingly redshifted and dimmer, seemingly freezing at the edge forever. This is a coordinate effect, not a physical barrier for the infaller.
Resolving the Paradox
Both perspectives are correct within their own reference frames. The astronaut dies at the singularity in finite proper time. The distant observer never sees them cross, but also never sees them bounce back. The information about the crossing is lost behind the horizon.