# Events and distant clocks

An event happens at one place at one instant. Clocks at different places can only be compared by a rule, and §1 of the relativity paper gives one: send light out and back, and call the clocks synchronous if the trip out takes as long as the trip back.

Written for this edition, not translated from Einstein.

## What does it mean to say that two distant events happen at the same time?

An event is something that happens at one place at one instant: a spark, a collision, a flash leaving a lamp. To give it a time, you read a clock at that place. Reading a distant clock by looking at it would mix in the time the light takes to reach you.

A reference frame is a set of rigid measuring rods and clocks at rest, with a clock at every place, so that each event gets three coordinates of position and one of time from the rods and the clock beside it. Before that works, the clocks must agree, and for clocks far apart agreeing needs a rule.

§1 of the relativity paper gives the rule. Send a flash of light from clock A to clock B and back. Note A's reading when it leaves, B's reading when it arrives, and A's reading when it returns. The clocks are called synchronous if the trip out takes as long, by the clocks, as the trip back:

$$
t_B - t_A = t'_A - t_B
$$

t B minus t A equals t A prime minus t B: the trip out takes as long as the trip back.

The paper calls this a definition. It cannot be checked by measurement, because checking it would need clocks that are already synchronous. §1 adds two assumptions: that the rule is consistent, so that if B agrees with A then A agrees with B, and two clocks that agree with a third agree with each other; and that light's speed measured out and back is a universal constant, which the paper writes V.

## Worked example: Setting a distant clock with one light signal

1. Clock A reads 0 s as a flash leaves it for clock B.
2. The flash is reflected at B and arrives back at A when A reads 10 s.
3. By the rule the two trips take equal times, so the reflection happened at 5 s on A's time.
4. If B's clock read 5 s at the reflection, it is synchronous with A. If it read 7 s, it is 2 s fast, and setting it back 2 s makes the two agree.

$$
t_B = \frac{1}{2}\left(t_A + t'_A\right)
$$

The time at B equals one half of the sum of A's reading when the flash left and A's reading when it came back.

The rule needs no distance. If B is known to be 1.5 × 10⁹ metres away, the same round trip also gives light's speed: 2 × 1.5 × 10⁹ m divided by 10 s is 3 × 10⁸ metres per second.

## Where this lesson stops

An event is something that happens at one place and one time. Whether two distant events are simultaneous depends on a rule for setting clocks, which §1 of the relativity paper supplies.

## This lesson builds on

- [Functions and graphs](/foundations/functions-graphs/)