The Principle of Relativity · Elementary Particle Physics

Assume space-time is featureless, that boosts compose like a group, and that causes precede effects. Those five structural facts leave exactly one family of transformations, with one number left to measure.

🎯 Why this matters

Everything this book tabulates — masses, lifetimes, cross-sections — is chosen to be a quantity these transformations leave alone. A number that changed when you boarded a moving train would be useless as a property of a particle.

Why this chapter exists

Everything in this book happens at speeds close to $c$ , and there are exactly two reasons particle physics needs enormous energies:

To make new particles. Mass is energy, so creating a particle of mass $m$ costs at least $m$ of collision energy. No energy, no new particles.
To see small things. Resolving power goes as inverse momentum — the point §1.8 makes properly. To look inside a proton you must hit it hard.

Chapter 1 assembles the toolkit for both: relativistic kinematics, units, cross-sections, the cast of particles, and the machines and detectors. This first section is the foundation under all of it, and Bettini does something unusual with it — so it is worth reading even if you think you know special relativity.

The principle, stated

The principle of relativity says that infinitely many reference frames exist in which the laws of physics take the same form, and that no experiment can pick one out as special. Those are the inertial frames , and the equations connecting the coordinates of two of them are the Lorentz transformations .

Fig. 1.1One event, two sets of coordinates. Everything in this section is about the function that maps one to the other.

📐 Physics you need first — what “covariant” actually demands

A transformation here is just a function that takes the four numbers

(t, x, y, z)

one observer assigns to an event and returns the four numbers a second, uniformly moving observer assigns to the same event. Nothing more exotic than a change of coordinates.

A law is covariant under that transformation when it has the same algebraic form in both coordinate systems — not when its numbers are unchanged. Maxwell’s equations are covariant; the numerical value of an electric field is not. Covariance is a constraint on the shape of your equations, which is exactly why it is so powerful: it rules out candidate laws before you ever test them.

The usual story, and why this book tells a different one

You were probably taught special relativity from Einstein’s two postulates, the second being that the speed of light is the same in every frame. That is historically how it happened, and it is not wrong. But it hides the logical structure and gives electromagnetism a starring role it does not deserve — the strong, weak and gravitational interactions are Lorentz-covariant too, and none of them cares about light.

So Bettini follows von Ignatowsky (1911): assume only structural facts about space-time, and see what transformations are possible. Light never appears.

The five assumptions — and that is the whole input

#	Assumption↕	What it buys you
1		No preferred place, no preferred direction — so the transformation can depend on intervals, not on positions
2	At least one inertial frame exists	Something to start from
3	The relativity principle holds	No frame is privileged, so the transformation from S′ to S must have the same form as S to S′
4	The transformations form a group
5	Causality

The derivation, one justification at a time

Work through it. Each step uses exactly one of the five assumptions, and the caption tells you which — the algebra is the easy part.

🪜 From five assumptions to the Lorentz transformations

Step 1 of 8 — Linearity(1.4)

x' = \gamma(V)\,(x - Vt), \qquad t' = \gamma(V)\left[\lambda(V)\,t - \mu(V)\,x\right]

Why you may do this: Inertia. A straight, uniform motion in S must still be straight and uniform in S′, and only a linear map does that to every trajectory. Homogeneity forces the dependence to be on (x − Vt) rather than on x and t separately.

Three unknown functions of V remain: γ, λ and μ. The rest of the derivation is killing them off.

This is Bettini pp. 4–7, restructured. Note that at no point did light, electromagnetism or any experiment enter.

Play with all three worlds below. The α < 0 one is worth a minute: watch what its velocity-addition law does as $V_1V_2 \to \kappa^2$ .

🧭 Three possible worlds — pick the sign of α

x' = \frac{x - Vt}{\left(1 - V^2/c^2\right)^{1/2}}, \qquad t' = \frac{t - Vx/c^2}{\left(1 - V^2/c^2\right)^{1/2}}

U = \frac{V_1 + V_2}{1 + V_1V_2/c^2}

V₁ (units of c or κ)0.90V₂0.90

γ(V₁) = 2.2942 · U = 0.9945— still below c, however hard you push

α > 0: Lorentz
α = 0: Galilei
α < 0: unbounded

Add V₁ to every possible V₂. Lorentz (blue) never leaves the band between ±c — the speed limit is not an extra postulate, it is what the composition law does. Galilei (slate) is a straight line through everything. The α < 0 world (rose) diverges when V₁V₂ = κ² and comes back with the opposite sign, which is the arithmetic face of its causality problem.

The causality test — can a boost reverse cause and effect?

Δx between the two events0.50Δt (cause first, so Δt > 0)1.00

Δt′ = γ(Δt − αVΔx) changes sign when V = 2.0000. That boost would need |V| > c, which this world forbids. The order of the two events is the same for everybody. Causality survives.

Current interval is time-like (|Δx/Δt| ≤ c) — slide Δx past 1.00 to cross over and watch the verdict change.

The result

\htmlClass{t-xp}{x'} = \htmlClass{t-g}{\gamma}\,(\htmlClass{t-x}{x} - \htmlClass{t-b}{\beta}\, \htmlClass{t-ct}{ct}), \qquad \htmlClass{t-ctp}{ct'} = \htmlClass{t-g}{\gamma}\,(\htmlClass{t-ct}{ct} - \htmlClass{t-b}{\beta}\,\htmlClass{t-x}{x}), \qquad y' = y,\; z' = z

(1.3)

The Lorentz transformation, written with β = V/c and γ = (1 − β²)^(−1/2). Note the symmetry between the two lines — x and ct enter each other's transformation identically.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

💡 What this really says — the speed limit was not assumed — it fell out of the group structure

Three things worth carrying forward.

The speed limit was not assumed. It fell out of demanding that boosts form a group and that causes precede effects. Nothing was said about light.

c is the invariant speed, not “the speed of light”. Anything massless travels at it; photons happen to be massless, so light travels at it. Bettini makes the point explicitly: if the photon had a small mass, the Lorentz transformations would be completely unchanged, and light would simply travel slightly slower than c — as the W and Z bosons, which are massive, already do.

Galilei is not wrong, it is a limit. α = 0 is the c → ∞ case, and it fails only the requirement that space-time be a single manifold. Every velocity you have ever measured directly is in the regime where it is indistinguishable from the truth.

⚙️ Engineer’s bridge

Read assumption 4 again: the transformations form a group. Written out, that is three requirements you already impose on any well-designed API:

identity — there is a do-nothing element (V = 0);
inverse — every operation can be undone (boost by −V);
closure — composing two operations gives another operation of the same type, never something outside the set.

Three of the seven steps above are nothing but cashing in one of those three axioms. And the payoff is the shape most engineers already know it in: the boost is a matrix multiply,

\begin{pmatrix} ct' \\ x' \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma\beta \\ -\gamma\beta & \gamma \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix},

with determinant exactly 1 — the transformation is volume-preserving in space-time, and composing two of them multiplies the matrices. The snippet below verifies both facts numerically.

Where the analogy breaks: unlike a rotation matrix, this one is not orthogonal. The quantity it preserves is $(ct)^2 - x^2$ , not $x^2 + y^2$ — one sign is flipped. That single minus sign is the entire difference between Euclidean geometry and the world, and it is what will define invariant mass in the next section.

🔢 Worked example — you cannot get there from here

A rocket leaves Earth at

0.9c

. It launches a probe forwards at

0.9c

relative to itself. How fast is the probe moving in Earth’s frame?

Galilei says $1.8c$ . The composition law says

U = \frac{0.9c + 0.9c}{1 + (0.9)(0.9)} = \frac{1.8c}{1.81} = 0.9945\,c.

Push harder — $0.99c$ on $0.99c$ — and you get $0.99995c$ . The denominator grows exactly fast enough to keep you inside the light cone, always.

Now the other direction. Two cars at $100\ \text{m/s}$ , closing:

U = \frac{200}{1 + (100)(100)/c^2}\ \text{m/s} = 199.999\,999\,999\,978\ \text{m/s},

short of the Galilean answer by $2.2 \times 10^{-11}$ m/s. That is why Newton lasted 200 years: the correction is 13 decimal places down.

Reproduce it

import numpy as np

add = lambda a, b: (a + b) / (1 + a*b)        # velocities in units of c
g   = lambda v: 1 / np.sqrt(1 - v*v)
L   = lambda v: np.array([[g(v), -g(v)*v],    # boost matrix acting on (ct, x)
                          [-g(v)*v, g(v)]])

print(f"0.9c + 0.9c              = {add(0.9, 0.9):.6f} c")
print(f"0.99c + 0.99c            = {add(0.99, 0.99):.8f} c")
c = 299792458.0
u = (100 + 100) / (1 + 100*100/c**2)
print(f"100 m/s + 100 m/s        = {u:.12f} m/s  (Galilei: 200)")
print(f"  shortfall              = {200 - u:.3e} m/s")

U = add(0.60, 0.75)                            # closure: two boosts make one
print(f"group closure: L(0.75) L(0.60) == L({U:.6f})?  "
      f"max|diff| = {np.abs(L(0.75) @ L(0.60) - L(U)).max():.2e}")
print(f"det L = {np.linalg.det(L(U)):.6f}  (boosts preserve space-time volume)")

r = np.random.default_rng(0).uniform(-0.999, 0.999, (2, 1_000_000))
print(f"never exceeds c: max U over 10^6 random pairs = {np.abs(add(*r)).max():.6f}")

prints

0.9c + 0.9c              = 0.994475 c
0.99c + 0.99c            = 0.99994950 c
100 m/s + 100 m/s        = 199.999999999978 m/s  (Galilei: 200)
shortfall              = 2.225e-11 m/s
group closure: L(0.75) L(0.60) == L(0.931034)?  max|diff| = 1.78e-15
det L = 1.000000  (boosts preserve space-time volume)
never exceeds c: max U over 10^6 random pairs = 0.999999

🔑 If you remember only three things

Five assumptions, and only one is physics. The other four are structure. Causality is the single claim about the world here, and it is what rules out the universe with no speed limit.
The toolkit is not electromagnetic. Strong, weak and gravitational processes are Lorentz-covariant too, which is why a derivation that never mentions light is the right foundation for a book about all four.
The method outlives the result. Write the most general map the symmetry allows, impose closure, then let one physical requirement choose among the survivors — that pattern returns whenever this book cannot derive something outright.

Where this goes next

§1.2–1.3 applies the same transformation to energy and momentum — with the same γ — and defines mass from the invariant it leaves behind.
§1.4–1.5 builds the kinematic invariants every collision in this book is analysed with.
Chapter 11 returns to the constant $c$ : gravitational waves travel at it too, which is the sharpest test that it is not a property of light.

✅ Check yourself — the principle of relativity

0/5 answered · 0 correct

1.Suppose tomorrow the photon were found to have a tiny non-zero mass. What would happen to the Lorentz transformations?
2.The derivation uses "the transformations form a group" three separate times. Which three facts does that assumption supply?
3.Why does causality kill the $\alpha < 0$ world but spare the Lorentz one?
4.In the widget, set $V_1 = V_2 = 0.9$ . What does the Lorentz law give, and what would Galilei give?
Hint: $U = (V_1+V_2)/(1 + V_1V_2)$ in units of $c$ .
5.The boost matrix acting on $(ct, x)$ has determinant 1 but is not orthogonal. What is the physical content of that distinction?