Space-time diagrams

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**ct** (speed of light × time) up the **vertical** axis, position **x** along the **horizontal** axis.

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The **path an object traces** on a space-time diagram — its position at every instant.

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A single **point** — a definite **place at a definite time**.

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A **vertical** line — x stays fixed while ct keeps climbing.

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At **45°**, because light travels $x = ct$, so equal steps in x and ct.

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**More tilted toward the x-axis** — the faster it goes, the further it leans (but never past 45°).

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$v = c\,\dfrac{\Delta x}{\Delta(ct)}$ — the more horizontal the line, the faster the object.

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$(\Delta s)^2 = (c\Delta t)^2 - (\Delta x)^2$ — the same in every inertial frame.

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It is **invariant**: all inertial observers measure the **same Δs**, even though Δt and Δx differ.

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$(c\Delta t)^2 = 2.25\times10^6$, $(\Delta x)^2 = 8.1\times10^5$, so $(\Delta s)^2 = 1.44\times10^6$ and **Δs = 1200 m**.

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**No** — events simultaneous in one frame need not be in another; the line of 'now' **tilts** for a moving observer.

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The **space-time interval** Δs, the cause-and-effect order of events, and that **light travels at 45°** (speed c).