An Infalling Observer Does Not Freeze at the Horizon

The phrase “an infaller freezes at the horizon” conflates three different questions: how much time the infaller’s clock records, what value the static Schwarzschild coordinate $t$ assigns, and which light signals an exterior receiver detects. Only the second quantity diverges at the horizon. The infaller crosses in finite proper time, while the receiver gets an increasingly delayed and redshifted sequence of pre-crossing signals—not a locally frozen body.

Proper time and Schwarzschild time

Work in four-dimensional Schwarzschild spacetime, set $m=GM\gt0$, and write

$$f(r)=1-\frac{2m}{r}, \qquad ds^2=-f\,dt^2+f^{-1}dr^2+r^2d\Omega_2^2.$$

For an ingoing radial timelike geodesic with specific Killing energy $\mathcal E\gt0$,

$$\frac{dt}{d\tau}=\frac{\mathcal E}{f}, \qquad \frac{dr}{d\tau}=-\sqrt{\mathcal E^2-f}.$$

Thus $d\tau/dr\to-1/\mathcal E$ at $r=2m$: the remaining proper time is finite. For release from rest at infinity, $\mathcal E=1$ and

$$\Delta\tau(r_0\to2m) =\frac{2}{3\sqrt{2m}} \left[r_0^{3/2}-(2m)^{3/2}\right].$$

By contrast,

$$\frac{dt}{dr} =-\frac{\mathcal E}{f\sqrt{\mathcal E^2-f}} \sim-\frac{1}{f},$$

so $t$ grows logarithmically without bound as the worldline approaches the horizon from outside. This is a statement about the static chart, whose time coordinate is normalized by observers at infinity; it is not a statement about the infaller’s clock. The geodesic and chart analysis is given in Reall 2013, §§12.4–12.5, printed pp. 138–143, PDF.

The future horizon in regular coordinates

Define the tortoise coordinate and advanced time by

$$r_*=r+2m\ln\left|\frac{r}{2m}-1\right|, \qquad v=t+r_*.$$

The metric becomes

$$ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega_2^2.$$

Its $(v,r)$ block has determinant $-1$, including at $r=2m$. These ingoing Eddington–Finkelstein coordinates cross the future horizon smoothly; the pole in the diagonal Schwarzschild components was coordinate failure. Bičák gives the same horizon-penetrating metric and uses its null cones to describe collapse in Bičák 2000, §2.3, printed pp. 22–24, PDF. The complete chart comparison is in the Schwarzschild Coordinate Atlas.

What an exterior receiver detects

Reception is a separate null-signal problem. For outgoing radial light, $u=t-r_*$ is constant. An emission event approaching the future horizon can have finite $v$, but

$$u=v-2r_*\longrightarrow+\infty.$$

Consequently successive outward pulses reach a fixed exterior receiver at ever later receiver times. If each pulse has fixed frequency $\omega_e$ in the infalling emitter’s frame, a static receiver at $R\gt2m$ measures

$$\frac{\omega_R}{\omega_e} =\frac{f(r_e)}{\sqrt{f(R)} \left(\mathcal E+\sqrt{\mathcal E^2-f(r_e)}\right)} \longrightarrow0.$$

The signal therefore becomes delayed and redshifted. A real detector also has finite sensitivity, bandwidth, exposure, and observing time, so after a finite time it records no further pulses. An idealized receiver operated forever can associate arbitrarily late arrivals with emission arbitrarily close to the horizon, but that does not show matter held stationary there. Any observational claim must name the receiver worldline and the outgoing and returning—or one-way—null signals; horizons can prevent such protocols from being global, as explained by Perlick 2008, §§1–2, printed PDF pp. 1–4.

Local crossing check

The Schwarzschild Kretschmann scalar is

$$R_{abcd}R^{abcd}=\frac{48m^2}{r^6}, \qquad \left.R_{abcd}R^{abcd}\right|_{r=2m}=\frac{3}{4m^4}.$$

It is finite at the horizon and becomes small for a large black hole. No universal local curvature singularity or sudden invariant event marks the crossing in a small freely falling laboratory; the event horizon is a global causal boundary. Regularity does not mean that gravity vanishes: a finite laboratory can measure tidal relative acceleration, but orthonormal curvature components at $r=2m$ scale as $m^{-2}$. For a sufficiently large black hole those effects can be weak, so neither a fixed curvature threshold nor a universal detector signal defines the horizon. The useful remnant of the familiar slogan is narrower: Schwarzschild $t$ diverges, and an exterior receiver can obtain pre-crossing signals at arbitrarily late idealized times. For the full geometry and its invariant limits, continue to Schwarzschild and Reissner–Nordström and Invariants, Singularities, and Controlled Limits.

References

• Bičák, Jiří. “The Role of Exact Solutions of Einstein’s Equations in the Developments of General Relativity and Astrophysics: Selected Themes.” In Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics 540, 1–126. Berlin: Springer, 2000. Open PDF.
• Perlick, Volker. “On the Radar Method in General-Relativistic Spacetimes.” In Lasers, Clocks and Drag-Free Control: Exploration of Relativistic Gravity in Space, edited by Hansjörg Dittus, Claus Lämmerzahl, and Slava G. Turyshev, 131–152. Astrophysics and Space Science Library 349. Berlin: Springer, 2008. DOI; Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes. Cambridge: University of Cambridge, 2013. Open PDF.