Below Decks

Warm-up

Three quick ones from the prologue and the surveyors, to wake up the parts of your brain this chapter needs.

The puzzle

Shut yourself up with a friend in the main cabin below decks of a large ship. This isn’t a new idea: Galileo suggested it in 1632, and he meant it as an experiment. Bring some flies, a bowl of fish, and a bottle that drips into a jar below. Watch everything while the ship sits in harbor. Then let the ship sail off, smoothly, with no rocking, and watch again.

Two ship words you’ll need. The front of a ship is the bow, so the cabin’s front wall is the fore wall. The back of a ship is the stern, so the back wall is the aft wall.

Here’s the first test. A ball hangs from the ceiling right above a chalk mark on the floor. The ship sails smoothly at a steady 10 m/s. You let go.

Experiment

Galileo only had flies and fish. We have flashlights. Light has a speed: about 30 centimeters every nanosecond. Here’s the question that will end up costing us our common-sense idea of time: speed relative to what? If light moved at its speed relative to the water, or to some invisible stuff filling space, then inside a moving cabin you’d clock light slower one way and faster the other, and you’d know you were moving. Galileo’s promise would be broken.

Why would light need “something” to move through? Light is a wave, and every other wave anyone knew moves through some stuff: sound through air, ripples through water. So for most of the 1800s physicists were sure light moved through an invisible stuff filling space. They called it the ether.

The lab below lets you test both possibilities. In the ether universe, light moves at speed 1 relative to the ether, and the ship moves through the ether the way it moves through still air. The other universe is ours. Before you flash the lamp, one more prediction.

Here is the cabin, sealed tight. You can run three experiments in it, in either universe, and set the ship’s speed. Everything else stays the same.

Then play. Is there any experiment in this cabin, in either universe, that tells you the ship is moving? Which one?

What just happened?

Speed relative to what?

The principle of relativity

Here is Galileo’s principle in one sentence. In a sealed room that moves at a steady speed in a straight line, every experiment comes out exactly as it would at rest. A room like that, with its rulers and clocks, is a frame of reference, and the sentence is called the principle of relativity. We’ll also call it Galileo’s principle.

Two words carry the weight: steady and straight. When the ship turns, the coffee sloshes. When it brakes, you lurch forward. You can feel changes in motion. You can’t feel motion.

Adding speeds, the everyday way

Rae, who always rides whatever is moving (today, the ship), walks toward the bow at 2 m/s on a ship doing 10 m/s. From the dock she moves at 12 m/s. The ball she throws forward at 15 m/s, measured on board, does 25 from the dock. That’s the everyday adding rule:

\[ \lab{w} = \lab{u} + \rkt{v'} \]

Here \(\lab{u}\) is the ship’s speed and \(\lab{w}\) the ball’s speed, both measured from the dock (blue), and \(\rkt{v'}\) is the ball’s speed measured on board (vermilion; the prime marks an on-board measurement). It has worked for every ball, arrow and train anyone has ever timed. Light is the exception.

Sound plays by this rule too, with a twist: it moves at 340 m/s relative to the air. Inside the sealed cabin the air sails along with you, so a shout crosses the cabin equally fast both ways. No clue there. But stand on the open deck in a wind.

Light, in an ether universe

Physicists of the 1800s were sure the ether was real (the deep dive below says why). If it were, the light in a cabin moving through the ether at speed u would behave like your shout in the wind: slower toward the bow, faster toward the stern.

Let’s do the ether arithmetic once, slowly, with friendly numbers. The cabin is 6 m long, so each wall is 3 m from the lamp. Clocks are in meters of time, so light in still ether covers 1 m per meter of time.

1. The ship moves through the ether at \(u = 0.5\). Light heading for the fore wall moves at 1 through the ether, but the wall runs away at 0.5. Relative to the cabin, the light gains at \(1 - 0.5 = 0.5\). This is the everyday adding rule \(w = u + v'\), solved for \(v'\). Here \(w = 1\) is the light’s speed measured from the ether and \(u = 0.5\) is the cabin’s, so \(v' = w - u = 0.5\) is the light’s speed measured from the cabin.
2. Time to cover 3 m at speed 0.5: \(3 \div 0.5 = 6\). The fore clock reads 6. Time is distance divided by speed.
3. The aft wall rushes toward the light, so the closing speed is \(1 + 0.5 = 1.5\), and the time is \(3 \div 1.5 = 2\). The aft clock reads 2. Same rule, wall moving the other way.
4. Now run it backwards, as the crew would. Divide the difference of the readings by their sum: \(\dfrac{6 - 2}{6 + 2} = 0.5\). That’s \(u\). The readings are \(3/(1-u)\) and \(3/(1+u)\). Over the common bottom \((1-u)(1+u) = 1 - u^2\), their difference is \(\dfrac{3 \cdot 2u}{1-u^2}\) and their sum is \(\dfrac{3 \cdot 2}{1-u^2}\). Divide one by the other: the \(3\), the \(2\) and the \(1-u^2\) all cancel, and what’s left is \(u\).

So in an ether universe a crew with one lamp and two clocks could measure their speed, and Galileo’s principle would hold for balls and fail for light. That universe makes sense. It just isn’t ours.

What Nature says

Every test so far, in labs and in particle accelerators, gives the same answer. You’ll meet two of them in a moment. Here it is, in words this course uses from now on:

\[ \begin{gathered} \text{speed of light} = \inv{1} \\ \text{in every free-float frame} \end{gathered} \]

A free-float frame is a room, with its rulers and clocks, in which anything you let go stays put or coasts in a straight line at a steady speed. Our sailing cabin counts for sideways motion: a ball rolled across its floor coasts straight. Up and down, it doesn’t count, because a dropped ball still falls. The falling room below shows which rooms count fully.

It doesn’t matter how fast the lamp moves, or the lab. This is not an assumption we picked because it looked pretty. It is a measured fact, as solid as the ball landing on the chalk mark. Teal is the color this course uses for anything every observer agrees on: an invariant. In 1905 Einstein made it one of his two starting points, alongside Galileo’s principle. Many of the most direct tests came later, and every one has backed him. Everything else in this course, from slow clocks to \(E = mc^2\), follows from those two. The first casualty is the prologue’s rule 3: if two observers moving past each other both time the same flash at 1, their clocks can’t be ticking at the same rate. Chapter 5 shows exactly how they differ.

Free-float frames: the falling room

Which rooms count as “steady”? Our ship is good enough for anything that happens sideways. But drop a ball and it falls: gravity picks out a direction, “down”. Now picture a room very high above the ground that falls freely, like an elevator whose cable has snapped (a thought experiment; nobody gets hurt). Everything in it falls together: the floor, you, and the ball you let go. So the ball stays beside your hand, and your feet no longer press on the floor: the gentlest push would send you drifting. Inside that room, no experiment detects gravity at all.

Such a room is a free-float frame. (Textbooks also say “inertial frame”.) The principle of relativity is really a statement about free-float frames: the laws of physics, and the speed of light, are the same in all of them.

Why do the balls drift together? Each one falls toward the center of the Earth, along its own line, and two lines aimed at one point aren’t parallel. This drift is called a tidal effect: gravity pulls a little differently at different places. (The Moon’s version of the same effect raises the ocean tides.) For balls 10 m apart the drift is 7.7 micrometers after 1 s and about 28 mm after a minute. The rule is \(\tfrac12 (g\,d/R)\,t^2\), where \(g = 9.8\un{m/s^2}\), \(d = 10\un{m}\) is the distance between the balls, and \(R = 6371\un{km}\) is Earth’s radius. A spacecraft in orbit is a falling room that never lands.

So is the falling room free-float? For two balls 10 m apart, to 1 mm accuracy, yes, for about 11 seconds. To 1 micrometer, for under half a second. Physicists say free-float frames are local: every claim that a room is free-float comes with a stated accuracy, over a stated region and time.

So far, two facts: balls can’t tell you your speed, and in our universe neither can light. Let’s see the evidence for the second one.

Ask Nature

If there is an ether, Earth can’t be at rest in it all year: it circles the Sun at about 30 km/s, which is \(u \approx 10^{-4}\). In 1887, in Cleveland, Albert Michelson and Edward Morley set out to feel that wind.

Their trick was a race. Split one beam of light in two. Send one half along the wind and back, the other across it and back, then bring them together. In an ether universe the along trip takes longer, by about \(L u^2\) (Build it, below, computes it exactly). With arms of 11 m that’s \(11 \times (10^{-4})^2 = 1.1 \times 10^{-7}\) m of time, less than a millionth of a nanosecond. No clock can time that.

But light can. In \(1.1 \times 10^{-7}\) m of time, light goes 110 nanometers, so the slow half arrives 110 nm behind the other. Yellow sodium light has waves 589 nm long: the lag is about a fifth of a wave. When the halves meet, their crests and troughs make bright and dark stripes, called fringes. Turn the apparatus by 90° and the arms swap jobs, so the stripes should slide by twice that: about 0.37 of a fringe. That is easy to see.

A skeptic could still argue that light keeps its speed relative to the lamp that sent it, like a bullet from a gun. Then a moving lamp would send faster light. In 1964 a team at CERN tested exactly that, with the fastest lamps available.

Practice

Three tiers: ◇ back of the envelope, ◆ pencil and paper, ★ head-scratcher. Speeds are fractions of light speed unless a unit says otherwise.

For the formulas in one place, print the formula card; the small-number shortcuts live in the math toolkit.

Build it

Time to build the ether universe’s stopwatch in code. Your function will predict Michelson’s fringe slide, and once it passes its tests, the interferometer above runs on your numbers, with Nature’s data point sitting right beside them.

function rectangle(w, h) {
  if (w < 0 || h < 0) throw new RangeError('sides must not be negative');
  return { area: w * h, perimeter: 2 * (w + h) };
}

Here’s the physics, with friendly numbers: arm \(L = 1\), wind \(u = 0.6\).

• Along the wind. Out at \(1 - 0.6 = 0.4\): that takes \(1 \div 0.4 = 2.5\). Back at \(1 + 0.6 = 1.6\): \(1 \div 1.6 = 0.625\). Total \(3.125\).
• Across the wind. The light has to aim upstream to beat the drift, the way a swimmer crosses a river. Of its speed 1, it spends 0.6 cancelling the wind, which leaves \(\sqrt{1 - 0.6^2} = 0.8\) for going across. Each way takes \(1 \div 0.8 = 1.25\), so the total is \(2.5\).

For a general arm \(L\) and wind \(u\), the along trip is \(L/(1-u) + L/(1+u) = 2L/(1-u^2)\), and the across trip is \(2L/\sqrt{1-u^2}\). If \(|u| \ge 1\), the light heading upstream could never catch its mirror, so the trip would never end: your function should refuse.

The starter code’s comments repeat these steps: compute the along trip, then the across trip, and refuse \(|u| \ge 1\) with a RangeError.

When your function passes, scroll back up to the interferometer: the chip in its toolbar says it’s running on your code, and the purple bar on the gauge is your prediction. Nature’s bar doesn’t move. That’s the point.

Checkpoint

Five scored items, then one to explain in your own words. Feedback waits until you submit, and you can retake it as often as you like.

Explain it to Sagredo

Going further

Reading: Galileo’s own cabin, on the Second Day of his Dialogue Concerning the Two Chief World Systems (1632), is still a pleasure to read. Taylor and Wheeler’s Spacetime Physics (2nd edition, 1992) gives the principle of relativity a whole chapter.