Reference

The math, in one place.

Every formula used in the interactive lessons, derived once, with the assumptions called out. Everything below uses natural units where convenient (set $c = 1$ ); the boxed final results are written with $c$ explicit.

1. Postulates

Principle of relativity: the laws of physics take the same form in every inertial frame.
Constancy of c: light propagates at the same speed $c$ in every inertial frame, independent of the motion of source or observer.

2. The Lorentz factor

For relative speed $v$ , write $β = v / c$ . Then

γ = \frac{1}{1 - β ^{2}}

$γ \to 1$ as $β \to 0$ , $γ \to \infty$ as $β \to 1$ . It is the multiplier everywhere SR shows up.

3. Lorentz transformations (boost along +x)

Frame $S^{'}$ moves at $+ v$ along x relative to $S$ . Coordinates of a single event in the two frames are related by:

x^{'} y^{'} = γ (x - v t), = y, t^{'} z^{'} = γ (t - \frac{v}{c ^{2}} x), = z .

The inverse — swap $v \to - v$ :

x = γ (x^{'} + v t^{'}), t = γ (t^{'} + \frac{v}{c ^{2}} x^{'}) .

Derivation sketch: insist (i) the transform is linear, (ii) it reduces to Galilean as $v \to 0$ , (iii) a light pulse $x = c t$ in $S$ still satisfies $x^{'} = c t^{'}$ in $S^{'}$ , and (iv) the inverse is symmetric under $v \to - v$ . These four constraints fix $γ$ uniquely.

4. Time dilation

A clock at rest in $S^{'}$ at $x^{'} = 0$ ticks off proper time $Δ τ$ . In $S$ , the elapsed time between two of its ticks is given by the inverse transform with $Δ x^{'} = 0$ :

Δ t = γ (Δ t^{'} + \frac{v}{c ^{2}} Δ x^{'}) = γ Δ τ .

$Δ t = γ Δ τ$

Each frame regards the other’s clock as the moving one — and therefore as the slow one. The situation is symmetric; there is no contradiction because each measurement is internal to a single frame.

5. Length contraction

A rod of proper length $L_{0}$ at rest in $S^{'}$ has endpoints with fixed primed coordinates $x_{1}^{'}, x_{2}^{'}$ . To measure its length in $S$ , sample both endpoints simultaneously ( $t_{1} = t_{2} = t$ ). Using $x^{'} = γ (x - v t)$ :

L_{0} = x_{2}^{'} - x_{1}^{'} = γ (x_{2} - x_{1}) = γ L .

L = L_{0} / γ

Lengths along the direction of relative motion contract. Lengths transverse to the motion are unchanged.

6. Velocity addition

An object moves at $u^{'}$ in $S^{'}$ , which is itself moving at $v$ along x relative to $S$ . The object’s velocity in $S$ is:

u = \frac{u ^{'} + v}{1 + u ^{'} v / c ^{2}}

Set $u^{'} = c$ : the formula returns $u = c$ exactly, for any $v$ . That is postulate 2 made algebraic.

Galilean limit: when $u^{'}, v ≪ c$ , the denominator $\to 1$ and you recover $u = u^{'} + v$ .

7. The spacetime interval

The combination

(Δ s)^{2} = c^{2} (Δ t)^{2} - (Δ x)^{2}

is invariant under any Lorentz boost. Frames disagree on $Δ t$ and $Δ x$ separately; they agree on this combination. Three flavours:

Timelike $(Δ s)^{2} > 0$ : a single particle can connect the events; there is a frame in which they happen at the same place.
Lightlike $(Δ s)^{2} = 0$ : only a light pulse connects them.
Spacelike $(Δ s)^{2} < 0$ : no signal can connect them; there is a frame in which they happen at the same time.

8. Relativity of simultaneity

Two events with lab-frame coords $(x_{1}, t)$ and $(x_{2}, t)$ — same time in $S$ — transform to:

t_{1}^{'} - t_{2}^{'} = - \frac{v}{c ^{2}} γ (x_{1} - x_{2}) .

Whenever $v \neq = 0$ and $x_{1} \neq = x_{2}$ , the two events are not simultaneous in $S^{'}$ . This single fact is the engine behind every "paradox" in SR — twins, ladders, trains, you name it. None of them are paradoxes; they all resolve once you keep careful track of which frame slices "now" through which set of events.