Spacetime, by Hand

Coming attractions

Three things that are true

None of these is an illusion. Each one rests on effects that experiments have measured many times over, and you'll compute every number yourself before the course is over.

• ≡8 and 10

  years of aging

  Rae flies to a star 3 light-years away and back at three fifths of light speed. When she steps off the rocket she has aged 8 years. Her sister Lin, who stayed home, has aged 10. Nobody's clock was broken.

  Chapter 11, Straight Lines Age the Most

• ≡1 in 5

  muons reach the ground

  Muons are made 10 km up. A muon lives about 2.2 millionths of a second, enough to fly only about 660 m even at light speed. Yet about 1 in 5 muons falling at 99.5% of light speed reach the ground (ignoring the air). The reason: a fast muon's clock runs slow. Detectors on a mountain and at sea level confirm it runs slow by just the amount relativity predicts.

  The Prologue, then Chapter 8

• ≡3.7

  billionths of a gram

  Heat a liter of water from room temperature to boiling and its mass goes up by about 3.7 billionths of a gram. Not because anything was added. The energy itself has mass.

  Chapter 14, Momenergy and E = mc²

All three numbers are teal, with the ≡ sign. On this site, teal means every observer agrees on it: Lin, Rae and anyone flying past all compute the same 8 and 10 years. Finding out why these particular numbers are teal is most of the course.

Who you'll meet

The cast, and the color code

Every picture in the course uses the same few characters and colors, so you can read a diagram at a glance. The colors always come with a second cue, such as a dash pattern, an arrow or a word, so they work for every pair of eyes.

• Lin Lab · solid

  Runs the lab: a lattice of rods with a synchronized clock at every corner. Her lines are blue and solid.

• Rae Rocket · dashed

  Rides a rocket past the lab. Lin says Rae is moving; Rae says the same about Lin. Her lines are vermilion and dashed.

• Light Gold · 45°

  Always drawn in gold at exactly 45°, often with an arrowhead to show which way it travels. Everyone who measures its speed gets the same answer.

• Sagredo In the margin

  The doubter, borrowed from Galileo's Dialogue. He asks the question you were about to ask, and he's often right to worry.

Two more colors to know. Teal, drawn as a double line or marked ≡, marks an invariant: if a number is the same for every observer, it is teal. Purple marks your code, the functions you write that end up running the labs.

Sixteen chapters, one worldline

The map

We draw everything in this course as events on a worldline, so here is the course drawn the same way. Each dot is a chapter. They fill in as you go, and your progress is kept only in this browser.

• Not started
• In progress
• Checkpoint passed
• Mastered: a perfect checkpoint, and every function written by you

After Chapter 15 the worldline keeps going: the last chapter ends with a look toward gravity, which is where general relativity takes over. Rusty on square roots or graphs? The Math Toolkit has short refreshers.

The rhythm of every chapter

How it works

Every chapter is one long scroll with the same six beats. The order matters. Guessing first, then looking, is the fastest way we know to make a strange idea stick.

1. Predict

   Before each experiment you commit to a guess and say how sure you are. Being wrong on the record is what makes the surprise stick.

2. Play

   A live lab: drag a rocket, freeze a light clock, flip to the other frame. Every lab asks you to measure something rather than only admire it.

3. Reason

   Then we work it out by hand, with friendly numbers like 3-4-5 triangles. The equation comes last, as shorthand for what you already did.

4. Ask Nature

   Real experiments with real numbers: muons on a mountain, atomic clocks flown around the world, the clocks inside GPS satellites.

5. Practice

   Problems in three tiers, from back-of-the-envelope to head-scratcher.

6. Build, then explain

   You write a short JavaScript function, about eight lines at most. It's tested in a sealed sandbox, then it runs the labs. Get it wrong and you'll watch a wrong universe. Then a checkpoint, and finally Sagredo, who wants it explained in your own words. If you can convince him, you've got it.

What you need

Who this is for

Anyone curious enough to wonder why fast clocks would run slow, and stubborn enough to want to check. It was written with high-school students in mind, but nobody is checking ID.

• You know some algebra, Pythagoras and a little trigonometry. That's all the math you need. There is no calculus anywhere on the main path.
• You've never programmed? That's fine. Each chapter teaches the few pieces of JavaScript it needs in a short “Code you need” box, and you'll run your first function in the prologue.
• Rusty on square roots, graphs or exponents? The Math Toolkit has short refreshers, and every chapter links to the one you need at the moment you need it.
• Teaching a class? Everything is free, needs no accounts, and keeps progress only in the learner's own browser. The formula card prints on a single sheet.

Credit where it's due

Built on giants

The order of ideas here follows Edwin F. Taylor and John Archibald Wheeler's Spacetime Physics: start from the one quantity every observer agrees on, and let everything else follow from it. Their second edition (1992) is free to read on Edwin Taylor's website.

The voice owes a debt to The Feynman Lectures on Physics, which you can read free online from Caltech: concrete things first, honesty about what's weird, and no fear of a little arithmetic.

We borrowed an order of ideas and a love of concrete examples, nothing more. Every sentence, figure, exercise and line of code on this site is original. The full list of inspirations and sources is on the About page.

Ready? The muons are already falling. Start with the prologue.