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Relativity as measurement · A course with experiments

L1 · Chapter 1

Events, light signals and the delay in seeing

A flash 3 light-years away reaches you 3 years after it happens. Relativity starts from that delay. An event has its own place and moment, and a light signal reports it only when the light arrives. This chapter puts events and light signals on one diagram so the two times never blur together.

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An event is a place and a moment

A firecracker bursting, a clock ticking, a pulse of light leaving a lamp: each happens at one place and at one moment. Physicists call such a happening an event. We keep space to a single line, so an event needs only two numbers, its position and its time.

Draw time upward and position to the right. A clock that stays at x=0x=0 traces a vertical line as time passes; a clock that moves traces a slanted line. The line made of every event in an object’s life is its worldline.

Measure time in years and distance in light-years, the distance light covers in one year. Light then moves one unit of distance per unit of time, so the speed of light is c=1 ly/yrc=1\,\mathrm{ly/yr} and every light signal on the diagram is a line at 45 degrees. Einstein’s 1905 paper takes the same speed as a principle, whether the light is emitted by a body at rest or by a moving one.

Two light lines split the diagram

Pick one event and draw the two light lines through it. Events above both lines can receive a signal from it. Events below both lines could have sent it one. Events off to the left or right are out of reach: no signal travelling at or below the speed of light joins them to the chosen event. In three dimensions of space these lines become a cone, the light cone.

One number sorts any pair of events. Take their time difference Δt\Delta t and their separation Δx\Delta x and form the squared interval below. A positive value means the pair is timelike: a slower-than-light signal can join them. Zero means only light can. A negative value means the pair is spacelike and neither event can influence the other.

Chapter 3 shows that observers moving at different constant velocities all compute the same squared interval for the same pair of events, even though they disagree about the time difference. That shared number is why every observer sorts a pair of events into timelike, lightlike or spacelike in the same way.

s2=(Δt)2−(Δx)2s^2=(\Delta t)^2-(\Delta x)^2

Worked example

When does A see clock B tick?

Observer A stays at x=0x=0. Clock B leaves A at t=0t=0 at 0.6 of the speed of light. One of B’s ticks happens at t=5 yrt=5\,\mathrm{yr} and x=3 lyx=3\,\mathrm{ly}. B’s own clock reads 4 years at that event; Chapter 4 explains why it reads less than 5.

The flash from that tick travels back to A at light speed. It has 3 light-years to cover, so it arrives at t=5+3=8 yrt=5+3=8\,\mathrm{yr}. A sees the tick 3 years after it happens. Those 3 years are travel time for the light. They depend on where B was, not on how B’s clock runs.

The tick and its arrival at A are joined by light, so their squared interval is zero: 3 years apart in time and 3 light-years apart in space. Every pair of events linked by a light signal has a zero interval.

s2=(8−5)2−(0−3)2=0s^2=(8-5)^2-(0-3)^2=0

Seeing late is not the same as seeing slow

Put a clock at rest 5 light-years from A. Every tick reaches A exactly 5 years late, so the clock A sees reads 5 years behind A’s own clock. Its ticks still arrive one per year. A constant delay shifts the reading but leaves the rate alone.

A clock coming toward A behaves differently. Each tick starts from closer than the one before, so each delay is shorter and the ticks arrive bunched together: the clock looks fast. A clock moving away looks slow for the opposite reason. Such a seen rate mixes two things, the changing light delay and the clock’s own rate. To pull them apart we need a rule for when a distant event happens. That rule is the subject of Chapter 2.

An experiment to think through

Predict, change, explain

  1. For a clock resting 5 light-years away, predict whether the gold curve of what A sees runs parallel to the blue curve of B’s clock by A’s coordinates.
  2. In the Lab, choose “A distant resting clock only looks late”, then “An approaching clock looks fast”.
  3. Use the light delay to explain why one gold curve is only shifted while the other is steeper than the blue one.
Time one tick three ways

The clock and twin Labs set light speed to one light-year per year, as these chapters do. The collision Lab uses PDG 2024 masses; the particle collider Lab draws its events from a teaching generator.

Try explaining it yourself

1. An event happens at t=2, x=5t=2,\ x=5. Can it affect A at x=0x=0 at t=6t=6?

Read an explained answer

No. Light from 5 light-years away needs 5 years and reaches A at t=7t=7. The squared interval is negative, so the pair is spacelike.

s2=(6−2)2−(0−5)2=−9s^2=(6-2)^2-(0-5)^2=-9

2. A flash leaves x=−4 lyx=-4\,\mathrm{ly} at t=1 yrt=1\,\mathrm{yr}. When does A see it?

Read an explained answer

At t=5 yrt=5\,\mathrm{yr}. Light covers 4 light-years in 4 years from either side.

3. Two events are 3 years apart in time and 3 light-years apart in space. What kind of signal can connect them?

Read an explained answer

Only light. Their squared interval is zero, so a slower signal would arrive too late and nothing faster is available.

Sources to read alongside

  1. Max Planck Institute for Gravitational Physics, Einstein Online: Spacetime

    Plain-language account of events, light cones and the relativity of simultaneity. It does not treat proper time; the clock calculations in this course are our own.

  2. Einstein, On the Electrodynamics of Moving Bodies, Annalen der Physik 17, 891 (1905), English translation by Perrett and Jeffery (1923), Fourmilab edition

    Section 1 defines synchronous clocks by a light signal sent from A, reflected at B and returned to A, and states the two consistency assumptions. We read Sections 1 and the opening of Section 2 only.

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