Special Relativity · Section 4

A moving clock loses time.
Not what a camera sees: what the clock reads.

Choose a worldline. Compare the traveling clock's own proper time with the platform clocks it moves past, and read both clocks' faces at the reunion event.

Read Section 4 of Einstein’s 1905 paper →

Choose a scenario

Each button applies a named preset. Typed entry is available in the run summary below; visual and equivalent paths produce identical commands (this bead's action contract).

Worldline presets

What the clocks read

Clock speed (fraction of c): 0.600000
Worldline: out-and-back
Coordinate duration: 10 s

R0. The moving clock's proper time versus the platform's coordinate time.
Proper time (τ)	8 s
Coordinate time (t)	10 s
τ / t	0.800000

R1. Per-second loss: the stable exact form beside the printed second-order approximation ½β², never substituted for one another.
Exact loss per second	0.2000000000
Printed second-order form (½β²)	0.1800000000

R2. Reunion comparison at the reunion event: both clocks' readings, with the exact lag beside the printed approximation ½ t β², labeled as an approximation.
Exact lag at reunion	2 s
Printed approximation (½ t β²)	1.80000 s (approximation, not exact)

R3. Reciprocal description: each inertial frame reports the same dilation factor for the other clock's rate. Comparing separated readings needs a stated simultaneity convention; only the reunion comparison above is frame-independent.
Dilation factor (γ)	1.25000

The light clock (supplemental illustration, offered after the measurement definitions: it illustrates dilation, it does not define it).
Proper tick (2L₀/c)	2 s
Coordinate tick	2.50000 s

A clock losing exactly one second per day moves at β ≈ 0.00481124.

Model: an ideal clock whose rate depends only on its instantaneous speed. Not modeled: gravitational time dilation, real clock mechanisms under acceleration, rotating-frame synchronization, the geoid's actual shape, atomic-clock physics, and clock noise.

The Physical Argument

Two different questions, kept apart

“What does the clock read at a shared event?” and “What does a camera watching the clock across a growing distance see, including the travel time of the light itself?” are different physical questions. This instrument answers only the first: simultaneous-coordinate readings and reunion comparisons, never the optical appearance of a receding or approaching clock.

1-\sqrt{1-v^2/V^2}\approx\tfrac12\,v^2/V^2

Section 4 states the loss per second of coordinate time to magnitudes of fourth and higher order as

\tfrac12\,v^2/V^2

. The lab above shows this printed approximation beside the exact, numerically stable form

\beta^2/(1+\sqrt{1-\beta^2})

; the two agree to many digits at everyday speeds and separate visibly as speed grows.

The reunion is the honest comparison

Two separated clocks can only be compared by adopting a simultaneity convention; two clocks brought back together read whatever they read, with no convention involved. This instrument computes the frame-independent reunion comparison for closed worldlines (out-and-back, or a constant-speed circle) rather than a comparison of distant, unsynchronized readings.

An ideal clock, not a mechanism

The model is an ideal clock whose rate depends only on its instantaneous speed; not a model of any particular mechanism, and not a claim about how real atomic clocks behave under acceleration. Two worldlines with the same speed profile but different turning accelerations report exactly the same proper time.

Ideal model, host calculation. The full event-geometry evaluator (worldline proper time as a general integral, reunion comparisons, and reciprocal-rate redescription across boosted frames) is being built separately; until it lands, this instrument computes piecewise-constant-speed proper time directly from the Lorentz factor, which is exact for every scenario above.