A clock near a black hole does not cease to experience its own local duration. Comparisons between clocks depend on their paths, their positions and the signals used to relate them. The distinction between a supported clock and a falling clock is essential.
First choose clocks that remain at fixed radii
Consider the exterior Schwarzschild geometry of a nonrotating, uncharged black hole. Define its horizon radius rₛ = 2GM/c² and the dimensionless radius x = r/rₛ. For a static clock held at x > 1, proper time relates to Schwarzschild coordinate time by dτ = L(x)dt, with L(x) = √(1 − 1/x).
The coordinate t is normalized to the static time standard at infinity. At a finite radius it is not the reading of the local clock. For two static clocks A and B compared over the same coordinate interval, dτA/dτB = L(xA)/L(xB).
A regular signal sent from A to B has received-to-emitted frequency ratio fB/fA equal to that same expression. The frequency at each end is measured by its own local clock. The stationary geometry makes the propagation delay between fixed positions constant; it does not make that delay vanish.
Move the supported clocks
STATIC SCHWARZSCHILD CLOCKS · RADII IN rₛ
Static clock rate L(x) relative to coordinate time. Green: emitter A. Gold: receiver B. Infinity is a reference limit outside the plotted range.
A signal from 2rₛ to 10rₛ arrives redshifted: its received frequency is approximately 74.54% of its emitted frequency.
Check the comparison in both directions
At A = 2rₛ and B = 10rₛ, the ratio is √(0.5/0.9) = √5/3, approximately 0.74536. A locally emitted frequency of 1,000 Hz is therefore received at approximately 745.36 Hz in this ideal stationary arrangement.
Reverse emitter and receiver and the ratio becomes approximately 1.34164: the signal is blueshifted. Choose equal radii and it becomes one. These checks prevent the misleading claim that every signal near a black hole must be redshifted. The comparison depends on both endpoints and their motion.
The infinity option represents a limiting normalization, not a detector reached in finite travel time at an actual infinitely distant position. Every numerical receiver radius denotes a finite supported observer outside the horizon.
Holding position is not free fall
The required proper acceleration of a static observer satisfies a rₛ/c² = 1/[2x²√(1 − 1/x)]. It diverges as x approaches one. No material observer can remain static exactly on the horizon in this geometry.
A falling clock follows a different trajectory and does not obey the fixed-radius comparison used above. A regular infalling observer can cross the horizon after finite proper time. The divergence of Schwarzschild coordinate time at crossing does not establish that local experience stops there.
Signals from a falling emitter involve its velocity as well as gravitational effects. Rotation, charge, accretion and changing geometry introduce further complications. The demonstration isolates the static exterior comparison so that its meaning can be checked exactly.
A reconstruction must recover the metric comparison
The book's discussion of gravitation asks for comparison fields, universal matter coupling and dynamics. For SMM/DTT, these clock ratios are concrete benchmarks for an effective geometry. An analogy between slower completion and stronger gravity is insufficient unless it produces consistent clock and signal predictions.
Sources and further reading
- David Tong, Black Holes, for Schwarzschild geometry and regular horizon coordinates. The radius-controlled calculation is an original teaching example.
- Continue with gravity and clock rates, proper time and black holes in cosmic entropy accounting.