The Clock Made of Light

Warm-up

Three quick questions from earlier chapters. Each one is a tool you’ll pick up again today.

The puzzle

Here is the simplest clock I can think of. Two mirrors face each other, 2 meters apart. A pulse of light bounces between them. Up and back is one tick: 4 meters of time, about 13 nanoseconds.

Bolt the clock to Rae’s rocket and send her past Lin’s lab at \(v = 3/5\). For Rae nothing has changed: the light goes straight up and straight down, 4 per tick. Lin doesn’t watch anything. Her lattice clocks note when a tick starts and when it ends, each one right where it happens.

Experiment

Here are both stories at once. The panel headed Rae’s frame shows her clock at rest. The panel headed Lin’s frame shows Lin’s own clock standing still and Rae’s sliding by. (On a wide screen they sit side by side. On a phone, Rae’s is on top.) The two panels run on different clocks: Rae’s panel shows Rae’s own time, Lin’s shows Lin’s lattice time. So 6 under one and 6 under the other are not the same moment. Compare counts within one panel.

Then play freely. Push the speed toward 0.99 and watch Lin’s count of Rae’s ticks fall far behind. Now set 0.10 with the slider and press Play. A tenth of light speed is still faster than anything people have built. Can you see any difference in the tick lengths? At 0.10 the difference is too small to see: Lin measures 4.02 instead of 4, and the readout rounds that to 4.0. Look at the ratio (γ) readout instead: 1.005.

The counters only show whole ticks. At 0.10, after Lin’s 20 ticks, Rae’s clock is about a tenth of a tick behind, so for a moment one counter can roll over before the other. That one-tick gap comes from rounding to whole ticks. The real difference is tiny.

What just happened?

A good worry, but the wrong culprit. Lin never looks at anything. A lattice clock sits at each event, one where the tick starts and one where it ends, and each writes down its own reading on the spot. No signal has to travel anywhere. The 5 is reckoned, not seen: worked out from two clocks’ readings, not seen with anyone’s eyes. (What Lin would actually see with her eyes is a different and stranger question. Chapter 12 answers it.)

So the surprise isn’t an illusion of looking. It’s a fact about time.

The triangle inside every clock

Half a tick, with friendly numbers

Let’s do Lin’s arithmetic at the puzzle’s speed, \(v = 3/5\). Call Lin’s measure of one whole tick \(\lab{t}\). We don’t know it yet, so let’s hunt for it one step at a time.

1. Rae’s view: the light climbs 2 and comes back 2, so her tick, on her own clock, is \(\inv{\tau} = 4\). (\(\tau\) is the Greek letter tau. Here it is the time read by a clock that rides along from one event to the other: Rae’s wristwatch time for one tick.) Half a tick is one climb of 2. Half a tick is the simplest piece: one straight trip from the bottom mirror to the top.
2. Lin’s view: half a tick lasts \(\lab{t}/2\). In that time the clock slides sideways by speed times time, \(0.6 \times \lab{t}/2\). Distance is speed times time, in any frame.
3. The light’s path is the slant of a right triangle. Its legs are the climb, 2, and the slide, \(0.6 \times \lab{t}/2\). Light moves at 1, so the slant’s length is the time it takes: \(\lab{t}/2\). Speed 1 means one meter per meter of time. For light, distance and time are the same number.
4. Now Pythagoras finds \(\lab{t}/2\): slant squared = climb squared + slide squared. One short line at a time:
   1. \((\lab{t}/2)^2 = 2^2 + (0.6 \times \lab{t}/2)^2\)
   2. \((\lab{t}/2)^2 = 4 + 0.36\,(\lab{t}/2)^2\)
   3. \(0.64\,(\lab{t}/2)^2 = 4\)
   4. \((\lab{t}/2)^2 = 6.25\)
   5. \(\lab{t}/2 = 2.5\)
   Check: the slide is \(0.6 \times 2.5 = 1.5\), and \(1.5^2 + 2^2 = 6.25 = 2.5^2.\) Line 2 squares each part. Line 3 takes \(0.36\,(\lab{t}/2)^2\) from both sides, and line 4 divides by 0.64. The answer is the 3-4-5 triangle cut in half: 1.5, 2, 2.5.
5. So Lin measures one tick as \(\lab{t} = 5\), while Rae’s clock says \(\inv{\tau} = 4\). The ratio is \(5/4 = 1.25\). Every tick repeats the same triangle, so every tick has the same ratio. It belongs to the speed, not to the tick.

Are the mirrors still 2 apart?

Sagredo is right to ask: it’s a length on a moving rocket. Later (Chapter 9) we’ll find that moving things are measured shorter along their motion. If lengths can change at all, why should the height, across the motion, stay put?

Here’s a test that needs no clocks at all. Lin and Rae each hold up a meter stick, at right angles to the motion, with a wet paintbrush at the 1 m mark. As the sticks pass, each brush paints the other stick. Paint is a fact. Afterward, both of them can walk over and look where it is.

One more thing makes the test sharp. Lin says Rae’s stick is the moving one, and Rae says Lin’s is. Neither frame is special (that’s the principle of relativity), so whatever moving sticks do in Lin’s story, they must also do in Rae’s story, with the roles swapped.

Any speed at all

Now trade the friendly numbers for letters. Call Rae’s tick \(\inv{\tau}\) and Lin’s measure of it \(\lab{t}\). Half a tick is a right triangle. Its upright leg is the mirror gap. We can write the gap with \(\inv{\tau}\), in two short steps. Rae’s half tick is one climb. Light covers 1 m per 1 m of time, so a 2 m climb takes 2 m of time: half tick = 2 = gap. In letters: gap = \(\inv{\tau}/2\). The meter sticks just showed that Lin agrees on the gap, so the upright leg is \(\inv{\tau}/2\) in Lin’s picture too. The sideways leg is the slide, \(v\lab{t}/2\), and the slant is \(\lab{t}/2\). Pythagoras says

\[ \left(\tfrac{\lab{t}}{2}\right)^2 = \left(\tfrac{\inv{\tau}}{2}\right)^2 + \left(\tfrac{v\,\lab{t}}{2}\right)^2 . \]

Multiply both sides by 4. Write \(\lab{x} = v\lab{t}\) for the distance the clock slides in one tick, as Lin measures it. Then move \(\lab{x}^2\) to the other side: Need a refresher?

\[ \inv{\tau^2} = \lab{t}^2 - \lab{x}^2 . \]

Check it at 3/5: \(5^2 - 3^2 = 25 - 9 = 16 = 4^2\). Lin’s 5 and 3 belong to her frame. But the combination \(t^2 - x^2\) is Rae’s own clock reading, squared, and anyone can compute it. That’s why \(\inv{\tau}\) is teal: in this course, teal means invariant, the same for everyone. Hold on to this line. In Chapter 6 it becomes the hero of the whole course.

Here it is on Lin’s spacetime diagram. Rae passes Lin at the event O, where both clocks read 0. Lin’s own light clock, standing in her lab, ticks at the events L1, L2 and L3, at t = 4, 8 and 12. Rae’s ticks are R1, R2 and R3, where Lin’s lattice records them. R1, at (5, 3), sits on the same teal curve as L1: same t² − x², same reading on the clock that was there. Each teal curve holds every event where a clock that left O, at any steady speed, reads the same time. A faster clock runs slower in Lin’s frame, so it reaches 4 on its own clock at a later lab time and farther out. Joining those events for every speed traces the curve \(t^2 - x^2 = 16\), which bends away from the light lines and never touches them.

Now press Show t² − x² in the lab (it opens by itself once you get the exercise right). Drag the speed again: t and x change for every one of Rae’s ticks, but t² − x² holds still at 16, 64 and 144.

The stretch factor

Solve the same equation for Lin’s tick instead. Put \(\lab{x} = v\lab{t}\) back in: \(\inv{\tau}^2 = \lab{t}^2 - v^2\lab{t}^2 = \lab{t}^2(1 - v^2)\). Take square roots: Need a refresher?

\[ \lab{t} = \g\,\inv{\tau}, \qquad \g = \frac{1}{\sqrt{1 - v^2}} . \]

Let’s call \(\g\) (the Greek letter gamma) the stretch factor, after what it does: it’s how many meters of Lin’s lattice time pass for each meter of time on Rae’s clock. Textbooks call it the Lorentz factor. At 3/5 it is \(1/\sqrt{1 - 0.36} = 1/0.8 = 5/4\), the ratio we found by hand.

Here’s a small gift from Pythagoras. Every right triangle with whole-number sides gives a speed where \(\g\) comes out as a neat fraction. Take 3-4-5: the speed is \(3/5\), \(\sqrt{1 - v^2}\) is \(4/5\), and \(\g\) is \(5/4\). In words: the speed is one short side over the long side, \(\sqrt{1 - v^2}\) is the other short side over the long side, and \(\g\) is that flipped over. Need a refresher?

The dial below draws the same half-tick triangle, shrunk so its slant is 1. At 3/5 that’s slide 3/5, climb 4/5, slant 1: the 3-4-5 triangle again, only smaller. During one meter of Lin’s time the light runs 1 along the slant while the clock slides \(v\), so the light climbs only \(\sqrt{1 - v^2}\). That climb is how far Rae’s clock advances, so in Lin’s frame her clock runs at \(1/\g\) of Lin’s rate.

Three things to notice. At everyday speeds \(\g\) is 1 to many decimal places: even at a tenth of light speed it is only 1.005. It is the same for \(v\) and \(-v\), because only \(v^2\) appears: direction doesn’t matter. And as \(v\) creeps toward 1 it grows without limit. At \(v = 1\) the formula divides by zero. No clock can ride a light beam.

One clock, or two?

Look again at who measured what. Rae’s tick starts at one event (the light leaves the bottom mirror) and ends at another (it comes back). Rae’s clock is present at both. Lin’s lattice needs two different clocks, 3 meters apart, one at each event.

The time read by the one clock that is present at both events is the wristwatch time the clockheads have carried since the Prologue. Physicists call it proper time and write it \(\inv{\tau}\), the Greek letter tau. A frame in which the clock moves needs two lattice clocks, one at each event, and it measures more: \(\lab{t} = \g\,\inv{\tau}\), never less, because \(\g\) is never less than 1. Let’s call that time stretching. Physicists call it time dilation.

Sagredo’s watch

All right: put your watch on the rocket beside the light clock. Suppose the watch kept its old pace while the light clock slowed. Then Rae, sealed in her cabin, could compare the two and discover that she’s moving. The principle of relativity from Chapter 2 says she can’t. So the watch slows too. So does her heartbeat, and so does the decay of muons in a vial. You forced a wristwatch to misbehave in the Experiment, and the warning in the Try to break it box caught it at once. Nature never lets that warning light up. It isn’t the clock that’s slow. It’s the time.

Remember rule 3 of the Prologue’s simulator, “every clock ticks at the same rate, moving or not”? It has now failed. A muon’s decay is a clock too. For a muon falling at 0.995, the Earth’s clocks should measure its life, from birth to decay, as about 10 times as long as by its own clock: about 22 μs instead of 2.2 μs, on average. (2.2 μs is the muon’s average lifetime. Its half-life, the time in which half of a batch decays, is a different, shorter measure: about 1.5 μs.) That could be how so many reach the ground. Chapter 8 checks the numbers against Nature.

That’s the whole idea in one triangle. Next: is it true? Let’s ask Nature.

Ask Nature

A light clock is a thought experiment. Do real clocks agree? At particle speeds, dramatically: Chapter 8 is full of fast particles whose lifetimes, measured by lab clocks, come out \(\g\) times their lifetime at rest, the best of them matching the formula to about a tenth of a percent. The harder test is at everyday speeds, where \(\g - 1\) is tiny.

How small is the effect at 10 m/s? Work it out.

An effect that small, and these clocks still measure it.

Practice

Three tiers: ◇ back of the envelope, ◆ pencil and paper, ★ head-scratchers. You can type fractions such as 5/3 and roots such as sqrt(3)/2.

◇ Back of the envelope

◆ Pencil and paper

★ Head-scratchers

Keep the formula card nearby: the stretch factor, the friendly speeds and the triangle are all on it.

Build it

Time to teach the computer the stretch factor. You’ll write gamma(v). It runs in a sealed sandbox, and once it passes its tests, two labs on this page run on your code: the light clock up in the Experiment, and the stretch-factor dial in “The stretch factor”.

function safeHalf(x) {
  if (Math.abs(x) > 100) {
    throw new RangeError('too big');
  }
  return x / 2;
}

Once it passes, scroll back up: the light clock in the Experiment and the stretch-factor dial in “The stretch factor” both carry a chip saying they run on your gamma().

Checkpoint

Six items. Feedback waits until you submit. To pass, get at least four of the five graded items right, including the three marked as key ideas.

Explain it to Sagredo

Going further

Two ★★ problems that look ahead, and some reading.

Reading

• Edwin Taylor and John Wheeler, Spacetime Physics, 2nd edition (1992), the source of this course’s invariant-first route. A free PDF is at eftaylor.com.
• Richard Feynman, Robert Leighton and Matthew Sands, The Feynman Lectures on Physics, Volume I, Chapter 15, which tells the light-clock story its own way. Free at feynmanlectures.caltech.edu.