🧍 Astronaut’s Own Frame (Free-Falling into the Black Hole)

For the astronaut, their proper time \( \tau \) flows normally. Their wristwatch ticks at a steady rate. As they cross the event horizon, they do not feel any sudden change in time — the crossing happens in finite proper time.

Mathematically, for a radial free fall from rest at infinity in Schwarzschild spacetime:

\( \displaystyle \frac{d\tau}{dt} = \sqrt{1 – \frac{2GM}{rc^2}} \)

where \( t \) is Schwarzschild coordinate time (used by a distant observer), \( \tau \) is the astronaut’s proper time, \( r \) is the radial coordinate, and \( r_s = \dfrac{2GM}{c^2} \) is the Schwarzschild radius (event horizon).

As \( r \to r_s \), \( \dfrac{d\tau}{dt} \to 0 \), meaning the astronaut’s clock compared to the distant coordinate time slows to zero. But in their own frame, they smoothly cross the horizon in finite \( \tau \) — nothing special happens locally.

🚀 Hovering Observer Just Above the Event Horizon

Suppose there’s an observer hovering (using rockets) just above \( r = r_s + \epsilon \). For them:

• Their clock ticks slower than a distant clock due to gravitational time dilation.
• The closer they hover to the horizon, the stronger their proper acceleration must be to remain at fixed \( r \).

Their proper time relative to Schwarzschild coordinate time is:

\( \displaystyle d\tau_{\text{hover}} = \sqrt{1 – \frac{r_s}{r}} \, dt \)

As \( r \to r_s \), the factor \( \sqrt{1 – \dfrac{r_s}{r}} \to 0 \). So time essentially “freezes” at the horizon for this hovering observer relative to infinity. They would see the astronaut’s clock slow down drastically as the astronaut approaches the horizon.

🌌 Distant Observer (Far Away from the Black Hole)

• The astronaut’s descent appears to slow down more and more as they approach the horizon.
• Light signals from the astronaut become increasingly redshifted and arrive more infrequently, taking longer and longer to reach the distant observer.
• The distant observer never actually sees the astronaut cross the event horizon. Instead, the astronaut seems to asymptotically freeze just outside it, fading due to infinite redshift.

This comes directly from the Schwarzschild coordinate time for radial free fall:

\( \displaystyle t(r) \sim – \frac{r_s}{c} \,\ln \left| \frac{r}{r_s} – 1 \right| + \text{(finite terms)} \)

As \( r \to r_s \), the logarithmic term diverges, meaning it takes infinite Schwarzschild time \( t \) for the astronaut to reach the horizon from the distant observer’s perspective.