Reference
Formula Card
The whole course on one sheet. Print it, fold it, keep it next to your pencil.
Every equation here is one you'll derive yourself in the course; nothing on this card is meant to be memorized before you've earned it. Colors keep their meaning: blue for the lab, vermilion for the rocket, teal for anything every observer agrees on.
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Units: light speed is 1
We measure time and distance in the same unit, so light covers 1 unit of distance per unit of time and every speed is a fraction of light's.
| 1 m of time | 3.3356 ns (light crosses 1 m) |
|---|---|
| 1 ns | 0.2998 m of light travel, about a foot |
| 1 s | 299 792 458 m of time (exact) |
| 1 ly | 9.4607 × 1015 m; with years, light speed is 1 ly/y |
| Sun to Earth (1 au) | 499.005 s of light travel for 1 au, the average distance (491 to 507 s over the year) |
| 1 g | 1.0323 ly/y² |
| Particles | mass, energy and momentum all in MeV |
Speed \(v\) has no units: \(v = 0.6\) means 60% of light speed. Any clock has \(-1 < v < 1\).
The color code
- Lab (Lin): blue, solid; unprimed \(t, x\)
- Rocket (Rae): vermilion, dashed; primed \(t', x'\)
- Light: gold, with an arrow, always at 45°
- Invariant: teal, double line, marked ≡
- Your code: purple
If a number is the same for every observer, it is teal.
Key equations
- The interval, same in every free-float frame
- \[\inv{\tau^2} = \lab{\dt^2 - \dx^2} = \rkt{\dtp^2 - \dxp^2}\]\(\tau^2 > 0\) timelike · \(= 0\) lightlike · \(< 0\) spacelike. In 3D, also subtract \(\Delta y^2\) and \(\Delta z^2\).
- Gamma, the stretch factor
- \[\g = \frac{1}{\sqrt{1 - v^2}}\]In code:
1 / Math.sqrt(1 - v * v). (Near \(v = 1\) the limit is how precisely you know \(v\); see Chapter 5.) - Lorentz transformation (rocket moves at \(+v\) along \(x\))
- \[\begin{gathered}\rkt{t'} = \g\,(\lab{t} - v\,\lab{x}) \\ \rkt{x'} = \g\,(\lab{x} - v\,\lab{t})\end{gathered}\]Inverse: replace \(v\) with \(-v\). Example: \((5, 3)\) at \(v = 0.6\) becomes \((4, 0)\).
- Time stretching
- \[\lab{\dt} = \g\,\inv{\tau}\]\(\tau\) is read by the one clock present at both events.
- Lorentz contraction (along the motion only)
- \[L = \frac{L_0}{\g}\]\(L_0\) is the proper length, measured in the object's own frame.
- The rocket's "now", drawn on the lab diagram
- \[\lab{t} = v\,\lab{x} + \frac{\rkt{t'}}{\g}\]A lab-simultaneous pair \(\Delta x\) apart has \(\rkt{\dtp} = -\g v\,\lab{\dx}\).
- Front clock lags
- \[\text{lag} = vL\]Clocks synchronized on a rocket of proper length \(L\), in lab reckoning: the front one is behind.
- Velocity addition
- \[w = \frac{u + v}{1 + uv}\]Rapidities add: \(\th_w = \th_u + \th_v\). Doppler factors multiply: \(k_w = k_u k_v\).
- Doppler factor
- \[\begin{gathered}k = \sqrt{\frac{1+v}{1-v}} \\ f_\text{received} = k\,f_\text{emitted}\end{gathered}\]Approaching: \(k\). Receding: \(1/k\). Also \(\g = (k + 1/k)/2\).
- Aberration
- \[\cos\theta_\text{lab} = \frac{\cos\theta' + v}{1 + v\cos\theta'}\]Here \(\theta\) is an angle to the direction of motion, not the rapidity.
- Momenergy
- \[\begin{gathered}E = \g m, \qquad p = \g m v \\ E^2 - p^2 = \inv{m^2}\end{gathered}\]Kinetic energy \(K = E - m = (\g - 1)m\). Light: \(E = |p|\), \(m = 0\).
- Maximal aging
- \[\inv{\tau_\text{total}} = \sum_i \sqrt{\Delta t_i^2 - \Delta x_i^2}\]Between two timelike-separated events, the straight (free-float) worldline has the greatest wristwatch time.
- Rapidity
- \[\th = \artanh v = \tfrac12 \ln\frac{1+v}{1-v}\]\(v = \tanh\th\), \(\g = \cosh\th\), \(\g v = \sinh\th\), \(k = e^{\th}\). Also \(\g v = (k - 1/k)/2\).
- Constant proper acceleration \(a\) (bonus)
- \[\begin{gathered}\th = a\tau, \qquad t = \frac{\sinh(a\tau)}{a} \\ x = \frac{\cosh(a\tau) - 1}{a}\end{gathered}\]
Friendly numbers
| \(v\) | \(\g\) | \(\g v\) | \(k\) | \(\th\) | triangle |
|---|---|---|---|---|---|
| 3/5 = 0.6 | 5/4 | 3/4 | 2 | ln 2 = 0.6931 | 3-4-5 |
| 4/5 = 0.8 | 5/3 | 4/3 | 3 | ln 3 = 1.0986 | 3-4-5 |
| 5/13 | 13/12 | 5/12 | 3/2 | ln 1.5 | 5-12-13 |
| 12/13 | 13/5 | 12/5 | 5 | ln 5 | 5-12-13 |
| 15/17 | 17/8 | 15/8 | 4 | ln 4 | 8-15-17 |
| 24/25 | 25/7 | 24/7 | 7 | ln 7 | 7-24-25 |
| √3/2 ≈ 0.866 | 2 | √3 | 2 + √3 | ln(2 + √3) | 30-60-90 |
Combinations: 0.6 ⊕ 0.6 = 15/17 (\(k = 4\)); 0.6 ⊕ 0.6 ⊕ 0.6 = 63/65 (\(k = 8\)); 0.6 ⊕ 0.8 = 35/37 (\(k = 6\)); 0.5 ⊕ 0.5 = 0.8. Small speeds: \(\g \approx 1 + v^2/2\).
Symbols
| Quantity | Symbol | Code |
|---|---|---|
| speed, as a fraction of light's | \(v\) | v |
| gamma, the stretch factor | \(\g = 1/\sqrt{1 - v^2}\) | gamma(v) |
| rapidity | \(\th = \artanh v\) | rapidity(v) |
| Doppler factor (approach; never below 1) | \(k = \sqrt{(1+v)/(1-v)}\) | none |
| frequency factor: \(k\) approaching, \(1/k\) receding | \(D = \sqrt{(1+v)/(1-v)}\), signed \(v\) | dopplerFactor(v) |
| squared interval | \(\tau^2 = \dt^2 - \dx^2\) | intervalSquared(a, b) |
| wristwatch (proper) time | \(\tau\) | properTime(a, b) |
| proper distance | \(\sigma = \sqrt{\dx^2 - \dt^2}\) | none |
| energy, momentum, mass | \(E, p, m\) | momenergy(m, v) |
Identities: \(\g = \cosh\th\), \(\g v = \sinh\th\), \(v = \tanh\th\), \(k = e^{\th}\), \(\g = (k + 1/k)/2\).
Drawing spacetime
- Time points up, \(x\) points right, on equal scales, so light runs at exactly 45°.
- The rocket's \(t'\) axis is the line \(x = vt\); its \(x'\) axis is \(t = vx\).
- Ticks on moving axes come from \(t^2 - x^2 = \pm n^2\). Never space them equally.
- Events are dots with a letter. A clock's worldline is steeper than 45°.
- The dashed line through P is the rocket's "now" through P, with slope \(v\). Teal numbers mark \(t^2 - x^2 = 1, 4, 9, 16\).
Four questions that prevent most mistakes
- Which clock is present at both events? Its reading is the proper time \(\tau\); every other frame gets \(\g\tau\).
- Whose "now"? Lengths and "at the same time" belong to a frame. Name it.
- Reckon or see? Lattice clocks reckon with no delay. Eyes and cameras see late news.
- Multiply or divide by \(\g\)? Check the friendly case: at \(v = 0.6\), the moving clock's 4 is the lab's 5.
Definitions of every term are in the glossary. Want a refresher on the algebra behind these? Try the Math Toolkit.