Foundations — Simultaneity — relativity of simultaneity

This page assumes nothing. If the parent note Simultaneity used a symbol, we build it here from the ground up, in the order that lets each idea rest on the previous one.

1. An event — the atom of everything

The picture: on a map, a place is a dot with an address like "3 metres east." But an event needs a fourth number: when. So we tag each event with a position and a time.

2. Position $x$ — the "where" number

The picture: a ruler laid along the ground. The origin is the "0" mark. $x$ is just the reading on the ruler under the event. Positive $x$ = right, negative $x$ = left.

3. Separation $\Delta x$ — the gap between two wheres

The picture: two dots on the ruler. $\Delta x$ is the length of the stretch between them.

4. Time $t$ and the reading of a clock

The picture: a stopwatch frozen at the instant of the pop. Whatever it shows is $t$.

5. A reference frame $S$ — whose ruler and whose clocks?

The picture: imagine the platform tiled with metre-marks and a synchronised clock bolted at each mark. That entire grid is the frame $S$. The train carries its own grid of rulers and clocks — that is $S^{\prime }$.

6. Relative speed $v$ — how fast the frames slide past each other

The picture: the train's grid sliding steadily rightward over the platform's grid, $v$ metres to the right every second.

7. The speed of light $c$ — the universe's fixed number

The picture: a pulse of light racing down the $x$-axis. Alice clocks it at $c$. Bob, chasing it in the train, also clocks it at $c$ — not $c$ minus his own speed. That refusal to add up normally is what forces space and time to bend.

8. The ratio $v/c$ and the factor $\gamma$

How to read this fraction, piece by piece:

• $v^2/c^2$ is the ratio $v/c$ squared — always between $0$ and $1$.
• $1-v^2/c^2$ is therefore a number between $0$ and $1$ (smaller as you speed up).
• ​ (the "square root," read "root") asks: what number times itself gives this? It shrinks the bracket toward $0$ as $v\to c$.
• Dividing $1$ by that small root makes $\gamma$ large. So faster motion $\Rightarrow$ bigger $\gamma$.

9. Putting the symbols together — the transformation

Now every symbol in the parent's key equation has a meaning:

This machinery is developed fully in Lorentz Transformation; here we only needed to know what each letter is. The picture that turns all of this into geometry — worldlines, "now"-slices tilting — lives in Spacetime Diagrams, and the limit on which orderings may flip is Causality and the Light Cone.

Prerequisite map

Read it upward: event splits into where and when; those grow into separations and frames; the fixed c plus v build gamma; all of it feeds the simultaneity gap at the heart of the topic.

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, reread that section before the parent note.