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Special relativity · Measuring time

Light Signals and Moving Clocks

Why does seeing a clock tick differ from dating it or reading the clock itself? Light needs time to reach you, so you see every tick late. Echoes let you subtract that travel time and assign a coordinate time, which depends on how the clocks were synchronized. The moving clock's own reading is a third quantity: it counts time along its own path, and a round trip leaves it behind.

Drag a clock's path through spacetime and compare four times for one tick: when its light arrives, its coordinate time, its radar time from echoes and the clock's own reading.

Interactive modelLight Signals and Moving Clocks
Coordinate time in this frame5 yr5\,\mathrm{yr}
A's radar time5 yr5\,\mathrm{yr}
A sees it at8 yr8\,\mathrm{yr}
B's own clock4 yr4\,\mathrm{yr}

Blue uses A's synchronized coordinates; gold is what A sees when the light arrives.

More numbers and the saved record
A's clock when B's record ends10 yr10\,\mathrm{yr}
B's clock when its record ends8 yr8\,\mathrm{yr}
Ticks received per year of A, near this tick0.50.5
Time-dilation factor of this leg1.251.25

The JSON record lists every tick of B with its event coordinates, A's send and receive times and the radar time recomputed from that pair.

Physics tutorial

One tick, four times

BackgroundA distant clock flashes once a year. Observer A writes down when each flash arrives, sends light pulses that the clock echoes, and later compares notes with the clock itself.

Why it mattersPhysics needs times that do not depend on where the observer happens to stand. Light travel time, synchronization and the clock's own path each change a different number.

Start with the essentials

Focus question
Why does seeing a clock tick differ from dating it or reading the clock itself?
One-sentence intuition
Arrival time is a fact about A's clock, coordinate time is a convention built from echoes, and proper time is a fact about the moving clock's path.

Core mathematical model

Radar coordinates from one echo

t=12 (t1+t2),x=12 c (t2−t1)t=\tfrac12\,(t_1+t_2),\qquad x=\tfrac12\,c\,(t_2-t_1)

A sends light at its time t1t_1, the light meets the event, and the echo returns at t2t_2. For an observer at rest this reproduces Einstein's synchronization.

The spacetime interval

Δs2=c2Δt2−Δx2\Delta s^2=c^2\Delta t^2-\Delta x^2

Every inertial frame gives the same value. It is zero along a light ray and positive along the path of a clock.

Proper time on straight legs

τ=∑iΔti1−vi2/c2\tau=\sum_i \Delta t_i\sqrt{1-v_i^2/c^2}

Each leg contributes the square root of its interval divided by c. Out and back at six tenths of light speed for ten years of A's time gives eight years on B.

How fast the ticks arrive

ΔtreceivedΔτ=1+vr/c1−vr/c\frac{\Delta t_{\text{received}}}{\Delta\tau}=\sqrt{\frac{1+v_r/c}{1-v_r/c}}

Here vrv_r is the velocity away from A. Receding ticks arrive spread out and approaching ticks bunched together; this is the Doppler effect, not a change in the clock's rate.

Common difficulties

Seeing is not dating

Typical misconceptionThe time at which I see a distant event is the time at which it happened.

Better mental modelThe flash needs time to arrive. Coordinate time subtracts that delay using an agreed synchronization, here radar echoes.

Bunched ticks are not a fast clock

Typical misconceptionAn approaching clock really runs fast because its ticks arrive more often.

Better mental modelAfter the light delay is removed, a moving clock runs slow in A's frame whether it approaches or recedes. The bunching comes from the shrinking light path.

What the model leaves out

Typical misconceptionThe diagram shows how the clock would look through a telescope.

Better mental modelOne space dimension, with light speed set to one light-year per year. Observer A rests at the origin and uses Einstein-synchronized clocks. Clock B moves at a constant speed on each of two legs, so its turn is instantaneous. Flashes and echoes travel on straight light lines. The model has no gravity, no forces during the turn and no optics of real images; the drawing frame only relabels the same events.

Run the experiment

  1. 01

    Compare the four readouts

    Start with A round trip leaves fewer ticks and inspect the fourth tick, at the turning point.

    What to observe: A dates the turn at five years by coordinates and by radar, sees it at eight years, and B's own clock reads four.
  2. 02

    Separate delay from dilation

    Choose A distant resting clock only looks late.

    What to observe: Every tick is seen five years late, yet coordinate time and the clock's reading agree. Delay alone does not slow a clock.
  3. 03

    Change the frame, not the events

    Move the frame speed slider or choose Same events, moving frame.

    What to observe: The coordinate time of the tick changes, while the arrival time on A and the reading on B stay fixed.

Related experiments and articles

Relativity as measurement · Reading path

Relativity as measurement: clocks, light signals and four-momentum

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleEvents, light signals and the delay in seeing
  2. ArticleSetting distant clocks with a light echo
  3. ArticleWhy moving observers disagree about “at the same time”
  4. ArticleProper time is what a moving clock records
  5. LabTwin Clocks and Radar TimeCount the signals each twin receives
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