About this course

Why this exists

Special relativity has a reputation: either a pile of formulas to plug numbers into, or a parade of paradoxes that leave you more confused than when you started. Neither is the subject. The subject is a short list of facts about Nature, one strange minus sign, and a lot of honest arithmetic.

This course is for a curious high-schooler who knows algebra, Pythagoras and a little trigonometry, and wants more than a vague feeling of having understood. By the end you should be able to compute what happens, draw it on a spacetime diagram, explain it to a skeptical friend, and program it. The last part matters most to us. The labs on this site end up running on functions you wrote.

How the course is built

There are sixteen chapters in four parts, from a puzzle about muons to a spacetime simulator you assemble yourself. The order follows the idea that drives the whole subject: start with what every observer agrees on, the interval, and treat everything else as that one quantity seen from different angles.

Every chapter is one long page with the same rhythm:

1. Predict. You commit to a guess, with a confidence, before anything is revealed.
2. Play. A live lab asks you to measure something.
3. Reason. We work it out with friendly numbers and pictures; the equation comes last.
4. Ask Nature. A real experiment, with real numbers and a citation.
5. Practice. Exercises in three tiers, from back-of-the-envelope to head-scratcher.
6. Build, then explain it to Sagredo. You write a short JavaScript function. It runs against tests in a sealed sandbox, and once it passes it powers the chapter's labs. If it fails, you can still choose to run the lab with your code anyway and watch a wrong universe misbehave. Then comes a checkpoint, and a written explanation for our fictional doubter in the margin.

Programming is taught, not assumed. Each chapter introduces only the bits of JavaScript it needs, and no function you write is much longer than eight lines.

How to use it

• Go in order, at least the first time. Chapters lean on each other, and each opens with a short warm-up from earlier ones.
• Predict honestly. Nobody grades predictions. A wrong guess you wrote down teaches more than a right one you didn't.
• Keep a pencil nearby. The friendly numbers are chosen so the arithmetic works out by hand.
• Your progress lives in this browser. There are no accounts. To move to another device, download your progress from the privacy page and load it there.
• Print the formula card, and use the glossary whenever a word gets slippery.
• Teachers: everything is free to use in class. The text and figures are licensed so you can share and adapt them for non-commercial teaching; see Licenses.

Inspirations and sources

We stand on other people's work, and we'd like you to read them too.

Books and lectures

• Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics: Introduction to Special Relativity, 2nd edition (W. H. Freeman, 1992). The spine of this course: invariants first, the lattice of clocks, the parable of the surveyors, momenergy. A free copy is on Edwin Taylor's website.
• Richard P. Feynman, Robert B. Leighton and Matthew Sands, The Feynman Lectures on Physics, Volume I, chapters 15 to 17 (1963). The model for a voice that starts with something concrete and is honest about what's weird. Free online from Caltech.
• Hermann Bondi, Relativity and Common Sense (1964). The idea of building relativity from radar and the Doppler factor \(k\), which we use from Chapter 10 on.
• Galileo Galilei, Dialogue Concerning the Two Chief World Systems (1632). The ship's cabin of Chapter 2, and Sagredo, the reasonable doubter we borrowed for our margins.

How students learn this subject

• Rachel E. Scherr, Peter S. Shaffer and Stamatis Vokos, "Student understanding of time in special relativity: Simultaneity and reference frames", American Journal of Physics 69, S24 (2001).
• Rachel E. Scherr, Peter S. Shaffer and Stamatis Vokos, "The challenge of changing deeply held student beliefs about the relativity of simultaneity", American Journal of Physics 70, 1238 (2002).

These studies shaped our insistence on separating what a frame reckons from what a person sees, and on treating a frame as a lattice of clocks rather than a pair of eyes.

The experiments

In order of publication. Each is the primary source for the numbers a chapter quotes.

• H. Fizeau, "Sur les hypothèses relatives à l'éther lumineux", Comptes Rendus 33, 349–355 (1851). Light in moving water (Chapter 13).
• A. A. Michelson and E. W. Morley, "Influence of Motion of the Medium on the Velocity of Light", American Journal of Science 31, 377 (1886). The repeat of Fizeau's experiment, with a drag coefficient of 0.434 ± 0.02.
• A. A. Michelson and E. W. Morley, "On the Relative Motion of the Earth and the Luminiferous Ether", American Journal of Science (3rd series) 34, 333–345 (1887), doi:10.2475/ajs.s3-34.203.333.
• R. J. Kennedy and E. M. Thorndike, "Experimental Establishment of the Relativity of Time", Physical Review 42, 400–418 (1932), doi:10.1103/PhysRev.42.400.
• H. E. Ives and G. R. Stilwell, "An Experimental Study of the Rate of a Moving Atomic Clock", Journal of the Optical Society of America 28, 215–226 (1938), doi:10.1364/JOSA.28.000215.
• C. M. G. Lattes, G. P. S. Occhialini and C. F. Powell, "Observations on the Tracks of Slow Mesons in Photographic Emulsions", Nature 160, 453 and 486 (1947). The equal-length muon tracks from stopped pions (Chapter 14).
• O. Chamberlain, E. Segrè, C. Wiegand and T. Ypsilantis, "Observation of Antiprotons", Physical Review 100, 947 (1955), doi:10.1103/PhysRev.100.947. Segrè and Chamberlain shared the 1959 Nobel Prize in Physics for it.
• D. H. Frisch and J. H. Smith, "Measurement of the Relativistic Time Dilation Using μ-Mesons", American Journal of Physics 31, 342–355 (1963). Muons counted on Mount Washington and about 1907 m lower, at sea level in Cambridge, Massachusetts.
• T. Alväger, F. J. M. Farley, J. Kjellman and I. Wallin, "Test of the second postulate of special relativity in the GeV region", Physics Letters 12, 260–262 (1964), doi:10.1016/0031-9163(64)91095-9.
• W. Bertozzi, "Speed and Kinetic Energy of Relativistic Electrons", American Journal of Physics 32, 551–555 (1964), doi:10.1119/1.1970770. The experiment behind the film The Ultimate Speed; our Chapter 14 data are from its Table I.
• A. R. Whitney and others, "Quasars Revisited: Rapid Time Variations Observed Via Very-Long-Baseline Interferometry", Science 173, 225 (1971). One of the first two reports of apparent faster-than-light motion, here in the quasar 3C 279.
• J. C. Hafele and R. E. Keating, "Around-the-World Atomic Clocks: Predicted Relativistic Time Gains" and "… Observed Relativistic Time Gains", Science 177, 166–168 and 168–170 (1972), doi:10.1126/science.177.4044.166 and doi:10.1126/science.177.4044.168.
• J. Bailey and others, "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit", Nature 268, 301–305 (1977), doi:10.1038/268301a0.
• D. W. Allan, M. A. Weiss and N. Ashby, "Around-the-World Relativistic Sagnac Experiment", Science 228, 69–70 (1985), doi:10.1126/science.228.4695.69.
• N. Ashby, "Relativity in the Global Positioning System", Living Reviews in Relativity 6, 1 (2003), doi:10.12942/lrr-2003-1. The GPS clock budget and the Sagnac correction around the equator.
• The BRAHMS, PHOBOS, STAR and PHENIX Collaborations, the four "white papers" on the first three years of the Relativistic Heavy Ion Collider, Nuclear Physics A 757, 1–283 (2005). RHIC collided beams from 2000 until February 2026.
• C. W. Chou, D. B. Hume, T. Rosenband and D. J. Wineland, "Optical Clocks and Relativity", Science 329, 1630–1633 (2010), doi:10.1126/science.1192720.
• The OPERA Collaboration (T. Adam and others), "Measurement of the neutrino velocity with the OPERA detector in the CNGS beam", Journal of High Energy Physics 10 (2012) 093, arXiv:1109.4897. The corrected version traces the early arrival mainly to a badly connected optical fiber in the master-clock chain.
• T. W. Murphy, "Lunar laser ranging: the millimeter challenge", Reports on Progress in Physics 76, 076901 (2013), arXiv:1309.6294.
• M. L. Lister and others, the MOJAVE survey of jet motions, Astronomical Journal 152, 12 (2016). Most of the jets surveyed appear to move at under about 5 times light speed; the fastest appears to move at about 50.

Constants, units and astronomy

• 17th General Conference on Weights and Measures (CGPM), Resolution 1 (1983), the definition of the metre, which fixes the speed of light at exactly 299 792 458 m/s. The 3rd CGPM (1901), Resolution 2, standard gravity 9.80665 m/s².
• International Astronomical Union, Resolution B2 (2012), the astronomical unit of exactly 149 597 870 700 m, and Resolution B3 (2015), the nominal solar luminosity 3.828 × 10²⁶ W (A. Prša and others, Astronomical Journal 152, 41 (2016)). The light-year uses the IAU's Julian year of 365.25 days.
• F. van Leeuwen, "Validation of the new Hipparcos reduction", Astronomy & Astrophysics 474, 653 (2007), for the distance to Canopus (about 310 light-years).
• Particle Data Group (S. Navas and others), Review of Particle Physics, Physical Review D 110, 030001 (2024), for the muon lifetime, the muon and pion masses, and the range of muons in air. The proton, neutron, deuteron and electron masses are the CODATA 2018 values (E. Tiesinga and others, Reviews of Modern Physics 93, 025010 (2021)).
• Water's heat capacity and enthalpy, for the kettle in Chapter 14, from the NIST Chemistry WebBook.

Other history

• Smithsonian National Museum of American History, "Nanoseconds Associated with Grace Hopper", object NMAH 692464: the 30 cm wires she handed out from the late 1960s.

What's ours

Real people appear only through real quotations, word for word, with a citation. We never put invented words in anyone's mouth. Our "Why, really?" boxes are written in our own voice and credited to no one. Lin, Rae and Sagredo are fictional; Sagredo's name comes from Galileo's Dialogue, and his questions here are ours.

How we check the physics

• Every number is computed, not typed. Answers come from the same reference library of physics functions that you rebuild in the course, and an automated test suite checks that library against hand-verified values and against properties that must always hold, such as the interval staying the same under any boost.
• Every coding task is tested centrally, so the tests your code faces have been physics-reviewed in one place.
• Historical data is checked against the original papers. Measurements from real experiments are tagged in our source, and each one has been checked against the primary source listed above.
• Popular myths are named and corrected. We keep a list of things popular explanations often get wrong, from "moving clocks look slow" to "mass increases with speed", and each one gets a clear correction in the chapter where it comes up.

Found a mistake?

Please tell us. A wrong number in a physics course is a bug, and we treat it like one. Write to feedback@indicogni.com with the page, the spot, and what you think it should say.

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They keep this free. There's no paywall, no account and no premium tier, and there never will be. Hosting, review and the long hours of writing and checking have to come from somewhere, and a few ads are how we pay for them.

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Licenses

• Text and figures: Creative Commons Attribution-NonCommercial-ShareAlike 4.0. Share and adapt them for non-commercial purposes, with credit, under the same license.
• Site code: the MIT License.
• Your code: the functions you write are yours.
• Math typesetting: KaTeX, MIT License, copyright Khan Academy and other contributors, served from this site.
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