Two identical clocks disagree although neither is broken. One flies past a laboratory: its synchronized clocks count 10 seconds while only 8 pass on the travelling clock. This happens at 60% of light speed in our hypothetical special-relativistic example.
A moving clock accumulates less proper time between two events on its path than the coordinate interval between those events in the laboratory’s inertial frame. This compares specific measurements; it does not mean time feels sluggish inside the cabin. Let us establish what the difference means.
First decide which clocks to compare
Imagine a laboratory far from massive bodies, where gravity can be neglected on the relevant scale and flat spacetime is a good approximation. Its inertial reference frame neither rotates nor accelerates: free objects move uniformly in straight lines. A spacecraft passes at constant velocity.
Choose two events: the travelling clock emits one signal and later another. Aboard the ship, both happen at the same clock. The interval it records is its proper time. In the laboratory frame, the ship has moved, so the events occur at different places. A local clock from the laboratory’s previously synchronized network can timestamp each event. Their difference is the laboratory, or coordinate, time interval. OpenStax: proper time and the event pair.
This is more than a clock video delayed by distance. Signal travel must be accounted for separately. When a clock approaches or recedes, the received flash frequency also depends on the Doppler effect. After correcting for light propagation, the relativistic interval difference remains.
Light takes a longer route without travelling faster
Build an imaginary clock with two mirrors: a pulse travels from the lower mirror to the upper one and returns. One tick is a complete return trip, starting and ending at the lower mirror. Mirror separation stays constant and the mirror axis is perpendicular to the ship’s motion.
Aboard the ship, light travels vertically. In the laboratory frame, the mirrors keep moving sideways: the pulse must reach the upper mirror’s new location, then the lower mirror’s new location. Its path has two diagonal legs.
In both inertial frames, the speed of light in vacuum is c. Consequently the longer light path in the laboratory frame requires a longer interval. Between two ticks of the travelling clock the laboratory counts more time. Compare the same pair of events in the figure, rather than two different beams. Einstein Online: the light clock.
One complete tick in two frames
Travelling with the clock
Proper interval: 8 s
In the laboratory frame
Laboratory interval: 10 s
Clock moving right: v = 0.6c
A — emission; B — reflection; C — return to the lower mirror.
Copper line: one pulse’s path. Short horizontal marks: mirrors; dashed lines connect their positions at different times.
The light clock makes the geometry visible, but the effect is no special mirror fault. Good atomic, mechanical and light clocks must agree in their common rest frame. Otherwise their disagreement could reveal a privileged state of uniform motion. A second aboard remains ordinary: local clocks and processes are compared with each other, rather than with an invisible “absolute time.”
Branch · With formulasOptional · safe to skip
Why 10 seconds correspond to 8
Let D be mirror separation, Δτ the proper interval of a full tick, Δt its laboratory interval and v the relative speed. Then D = cΔτ/2. In the laboratory frame, during half a tick light travels the hypotenuse cΔt/2 while the mirror moves vΔt/2. Pythagoras gives
(cΔt())^(2) = (2D())^(2) + (vΔt())^(2).
Rearranging:
Δτ = Δt√(1 − (v)^(2)/(c)^(2)).
At v = 0.6c the factor is √(1 − 0.36) = 0.8. If 10 laboratory seconds elapse between these events, the moving clock records 8 seconds. At v = 0 the intervals agree. As v approaches c, the factor decreases; a massive clock cannot be assigned speed c. The speed ratio is dimensionless and the result has units of time.
This formula applies to one clock moving uniformly between specified events in an inertial frame, with no gravitational correction. For changing velocity, add the proper times of small path segments; do not substitute an arbitrary “average speed.” OpenStax: geometry and equation.
How each can call the other slower
The spacecraft can also describe itself as stationary and determine that laboratory clocks passing it run slowly. This is reciprocity between inertial frames, rather than contradictory verdicts about a single clock.
The key is “simultaneous.” The frames pair readings of distant clocks at one moment differently. Swapping laboratory and ship in the time-dilation statement also changes which distant events count as simultaneous. We are not comparing the same clocks at two shared meetings and obtaining opposite final readings. Einstein Online: reciprocity and inertial comparison.
To meet again in this simple scenario, the spacecraft must change its motion. It follows a different path through spacetime: accumulated proper time depends on the entire path between departure and reunion. At reunion the clocks can be placed together and their difference read directly.
Branch · Travelling twinsOptional · safe to skip
The turn does not contain the entire age difference
In flat spacetime, let a station remain inertial while a spacecraft travels away at 0.6c and returns at the same speed. In the idealization of a very brief turn, the station counts 10 years across the two coasting legs and the ship 8. This is a new event pair: shared departure and reunion, rather than one tick in the previous diagram.
Acceleration during the turn breaks the symmetry of the paths, but the difference is not a mechanical “clock slowdown caused by a jolt.” Even as the turn’s duration becomes small, the difference accumulated over long uniform legs remains. A realistic smooth turn contributes its own proper time too. Einstein Online: acceleration and paths.
Testing clocks without an interstellar journey
In 2010, NIST researchers compared optical atomic clocks and altered the motion of the ion in one clock. The ion’s oscillatory motion at several metres per second produced a tiny frequency decrease consistent with relativity. The entire clock was not flown at that speed, and this was no slowed video: the experiment compared precise clock frequencies. NIST: the experiment.
Our 60% of light speed makes the difference obvious on paper. The effect does not switch on above a threshold: it is simply very small at low speeds. If our hypothetical ship reduces its speed to zero, its proper interval agrees with the laboratory interval. If a clock disagrees while resting beside its laboratory counterpart, that is a separate instrument error.
“Motion slows time” is therefore shorthand. Name the frame, the clock and the event pair. Light-speed invariance and spacetime geometry connect the intervals; a local clock does not experience a malfunction. Follow these distinctions into a long voyage in the article about interstellar travel.