Foundations — Postulates of SR

This page assumes you have seen none of the notation on the parent note. We build every symbol from the ground up, in the order they depend on each other. Nothing is used before it is drawn.

0. The two postulates — the rules everything rests on

Since the whole page keeps pointing back to them, let us state the two rules of Special Relativity up front, in plain words.

We will lean on Postulate 2 especially: it is the reason the photon in our light-clock (§6) is forced to travel at exactly $c$ even when the clock is sliding sideways.

1. An "event" and a "frame of reference" — the ruler-and-clock you carry

Before any physics, we need the two most basic words.

Look at the figure: the black grid belongs to a person standing still; the red grid belongs to someone gliding to the right. The same event (the dot) sits at different grid coordinates for each — that is the entire drama of relativity in one picture.

2. "Inertial" — the special, well-behaved frames

Not every frame is fair game. Postulate 1 only speaks about inertial frames.

Before the picture-test, we must name the "mysterious tug".

The picture-test: put a ball on a frictionless table.

• On a smoothly cruising train → ball stays put. Inertial. ✓
• On a train that suddenly brakes → ball rolls forward with no one touching it. That phantom shove is a pseudo-force; the braking frame is non-inertial. ✗
• On a spinning merry-go-round → the ball curves sideways on its own (the Coriolis pseudo-force), and feels flung outward. A rotating frame is therefore also non-inertial, even at constant spin rate — because turning is a kind of acceleration. ✗

Why the topic needs it: Postulate 1 is only claimed for inertial frames. Accelerating and rotating frames need General Relativity — a different, later story.

3. Velocity $v$ — the number "how fast, which way"

The red arrow is the velocity of the moving frame. Throughout the parent note, $v$ is always the relative speed between two observers — never the speed of light, which gets its own letter (next).

4. The speed of light $c$ — the universe's speed limit

The single most important fact — the thing that makes relativity relativity — is that every inertial observer measures this same $c$ (that is exactly Postulate 2), even if they are chasing the light beam. Contrast this with $v$, which is different for different observers.

5. $v^2/c^2$ — comparing your speed to light's

Now we combine two symbols we own. The parent note constantly writes $v^2/c^2$. Let's earn every piece.

• $v/c$ = "what fraction of light-speed am I going?" A rocket at $v=0.6c$ has $v/c=0.6$, i.e. 60% of light-speed.
• Squaring ($v^2/c^2$) removes the sign: whether you move left or right, $v^2$ is positive. Direction shouldn't change how much time slows, so squaring is exactly the right tool — it answers "how fast" while ignoring "which way".

6. The square root $\sqrt{1-v^2/c^2}$ — built step by step from a triangle

The parent's key expression is $\sqrt{1-v^2/c^2}$. Where does this exact shape come from? From Pythagoras applied to a device called the light-clock. First, what is that device?

Now watch the same clock from a frame in which it glides right at speed $v$. Consider one half-tick (light going from bottom mirror up to top mirror). Two frames watch it:

• In the clock's own rest frame the light just goes straight up a distance $L_0$, taking a proper half-tick time we will label $\frac{1}{2}\Delta t_0$.
• In the frame where the clock moves, the top mirror slides forward while the light climbs, so the light traces a diagonal. Call this observer's half-tick time $\frac{1}{2}\Delta t$ (this is the moving-frame interval — the very $\Delta t$ we define in §8).

The diagonal (red) is the hypotenuse of a right triangle whose three sides, all measured in the moving frame, are:

• vertical leg $=L_0$ (the mirror gap — unchanged, because motion is sideways, not up-down),
• horizontal leg $=v\cdot \frac{1}{2}\Delta t$ (how far the mirror slid during this half-tick),
• hypotenuse $=c\cdot \frac{1}{2}\Delta t$ (the light's actual path — it must go at $c$, by Postulate 2, even though its path is longer).

Step 1 — apply Pythagoras (leg² + leg² = hypotenuse²). Writing $\tau \equiv \frac{1}{2}\Delta t$ to keep the algebra clean: $L_0^2+(v\tau {)}^2=(c\tau {)}^2$ What we did: related the three sides. Why: the right triangle is exactly what "moving light-clock" draws, and Pythagoras is the one tool linking legs to hypotenuse.

Step 2 — gather the ${\tau }^2$ terms on one side: $c^2{\tau }^2-v^2{\tau }^2=L_0^2\Rightarrow {\tau }^2(c^2-v^2)=L_0^2$

Step 3 — solve for $\tau$ by dividing then taking a square root: $\tau =\frac{L_0}{\sqrt{c^2-v^2}}$

Step 4 — pull a $c$ out of the root so the fraction $v^2/c^2$ appears. Write $c^2-v^2=c^2\left(1-\frac{v^2}{c^2}\right)$, and $\sqrt{c^2(\cdots )}=c\sqrt{\cdots }$: $\tau =\frac{L_0}{c\sqrt{1-v^2/c^2}}$

There it is: the exact shape $\sqrt{1-v^2/c^2}$ emerges from the triangle, not by decree. In the clock's own rest frame the half-tick is just $L_0/c$ (no sideways slide), so the moving observer's half-tick $\tau$ is longer by the factor $1/\sqrt{1-v^2/c^2}$. Doubling both to get full ticks turns this into the parent note's $\Delta t=\gamma \Delta t_0$ once we name that factor (§7).

Note $1-v^2/c^2$ is always between $0$ and $1$ (since $v^2/c^2$ sits between $0$ and $1$), so its square root is also between $0$ and $1$.

7. The Lorentz factor $\gamma$ — the master stretch number

Since the bottom (denominator) is a number between $0$ and $1$, dividing $1$ by it gives something $\ge 1$. So $\gamma$ is always at least $1$, and grows without bound as $v\to c$.

Read the curve: at low $v$ it hugs $1$ (Newton's world), then rockets skyward near $v=c$. Every relativistic effect — slower clocks, shorter rulers — is measured by how far above 1 $\gamma$ has climbed. A few landmark values:

• at $v=0$: $v^2/c^2=0$, so $\gamma =1$ (nothing changes).
• at $v=0.6c$: $v^2/c^2=0.36$, so $\gamma =1/\sqrt{0.64}=1.25$.
• at $v=0.9c$: $v^2/c^2=0.81$, so $\gamma \approx 2.29$.
• as $v\to c$: $v^2/c^2\to 1$, so $\gamma \to \infty$ (blows up).

8. Times $\Delta t_0$ and $\Delta t$ — "whose clock ticked?"

The symbol $\Delta$ (Greek "delta") means "a change in" or "an interval of". So $\Delta t$ = "an amount of elapsed time" — the gap between two events' timestamps.

The parent's headline result is $\Delta t=\gamma \Delta t_0$. Because $\gamma \ge 1$, the moving observer always measures a longer interval — the moving clock is seen to tick slow. That is Time Dilation.

9. Length $L_0$ and $L$ — proper length vs. contracted length

A length is a distance in metres — like the mirror gap in the light-clock. But just like time, a length depends on who measures it, so we keep two symbols, exactly matching the $L_0$ we already used in §6.

This same $\gamma$ that stretched time now shrinks length.

How these foundations feed the topic

Read the chain top-down. Frames (built from events) tell us who measures. The two speeds $c$ and $v$ form the ratio $v^2/c^2$. Pythagoras on the moving light-clock shapes that ratio into $\sqrt{1-v^2/c^2}$, whose reciprocal is the Lorentz factor $\gamma$. Finally $\gamma$, together with the two postulates ("same laws" forces "$c$ is constant"), produces time dilation $\Delta t=\gamma \Delta t_0$ and length contraction $L=L_0/\gamma$.

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