Suppose two ideal clocks begin together, follow different journeys, and meet again. Comparing them at the reunion avoids a familiar difficulty: nobody needs to decide what a distant clock reads “right now.” Both readings are available at the same place. Can the clocks nevertheless show different elapsed times?
Relativity answers yes. The relevant quantity is proper time: the time accumulated along a particular worldline, or path through spacetime.
A clock along a path
In flat spacetime, choose an inertial laboratory frame. If an ideal clock moves at constant speed v relative to that frame for a coordinate duration Δt, its elapsed proper time is
Δτ = Δt √(1 − v²/c²).
The symbol c is the speed of light. The formula compares the duration assigned by this frame with the time accumulated by the moving clock. For changing velocity, calculate the contribution along each part of the path; in the continuous description, this becomes an integral.
The statement is about clock readings. It is not merely a statement about when light from a clock reaches a viewer. Signal travel time matters when reconstructing distant events, but it cannot explain away a difference that remains when the clocks are brought together.
A ten-year example
Let one clock remain at rest in the laboratory. A second travels outward at 0.6c for five laboratory years, reverses direction, and returns at the same speed for another five. We neglect gravity and idealize the turnaround as very brief.
The turnaround occurs three light-years away in the laboratory frame. The travelling clock accumulates 5 × √(1 − 0.6²) = 4 years on each leg, giving eight years in total. The laboratory clock accumulates ten. They reunite at one event with a two-year difference.
This construction is symmetric between the outward and return legs, but not between the two complete journeys. The laboratory clock follows one inertial path; the traveller changes velocity. There is no single inertial frame in which the traveller is at rest throughout both legs.
Compare the readings
An ideal round trip lasting ten laboratory years. Change the speed on both legs. Their laboratory durations remain five years each.
At 0.60c: laboratory clock 10.00 years; travelling clock 8.00 years; difference 2.00 years. Turnaround distance: 3.00 light-years.
Flat spacetime, ideal clocks, equal constant-speed legs and a negligible turnaround duration. At zero speed the two clocks remain together. These are calculated predictions for the stated model.
Notice what is held fixed. Increasing the speed here also moves the turnaround point farther away, because each leg still lasts five laboratory years. A different comparison could hold the destination fixed instead. A numerical demonstration should always state which quantities it varies.
What the turnaround does
The turnaround permits a reunion and changes the traveller's inertial frame. It is tempting to explain the entire clock difference by saying that acceleration “slows time.” That wording hides the calculation: the elapsed proper time depends on the whole path, including the long constant-speed portions.
A realistic journey has a finite acceleration interval. Its contribution must be included, rather than treated as a mysterious correction. The idealized two-leg result is the limit obtained when that interval is short compared with the rest of the trip under the stated assumptions.
Neither person needs to experience their own processes as unnaturally slow. The disagreement appears when comparing the amounts of proper time accumulated along different histories.
Clocks as evidence
Clock comparisons belong to experimental physics, not only to thought experiments. Chou and colleagues reported optical-clock measurements sensitive to relativistic effects at small laboratory speeds and height differences. Their experiment also addressed gravity, which our flat-spacetime example leaves out. The paper is a useful reminder that a measured frequency comparison, its corrections and its uncertainty belong to the evidential account.
For the Single Monad and Duality of Time programme, the clock law supplies a concrete benchmark. An underlying account of temporal generation must explain how ordinary clock comparisons arise in the appropriate regime. Calling a process “inner time” does not itself reproduce the eight-year prediction.
Sources and further reading
- Einstein (1905), On the Electrodynamics of Moving Bodies, English translation: the original special-relativistic framework.
- Chou, Hume, Rosenband and Wineland (2010), “Optical Clocks and Relativity,” Science 329, 1630–1633; published paper via NIST.
- Order is not duration: why an event sequence alone does not determine clock readings.
- Inner generation and outer time: the corresponding question within the research programme.