Galilean and special relativity (HL)

A coordinate grid and clock you measure motion **against**. All motion is **relative** to a chosen frame.

A frame moving at **constant velocity** (no acceleration). Newton's first law holds in it.

Inertial: a train cruising in a straight line at steady speed. Non-inertial: a car going round a bend.

$u' = u - v$ — the object's velocity in the moving frame equals its ground velocity minus the frame's velocity.

$x' = x - vt$ — position in the moving frame, where v is the frame's speed.

Same direction ⇒ **subtract** the speeds; opposite directions ⇒ the speeds **add**.

Same direction ⇒ add: $12 + 1.5 = 13.5$ m s⁻¹.

$u' = u - v = 20 - 30 = -10$ m s⁻¹, i.e. 10 m s⁻¹ westward.

The **laws of mechanics are the same in every inertial frame** — no experiment can detect uniform motion.

**No.** All inertial frames are equivalent; 'at rest' only ever means 'relative to something'.

Near the **speed of light** — light travels at the same speed in every frame, so simple addition fails (→ special relativity).

Opposite directions ⇒ add: $25 + 30 = 55$ m s⁻¹.

**t′ = t**: one clock serves every frame, so time is absolute.

The frames differ by a constant velocity, so they measure the **same acceleration**, and F = ma gives the same force.

The **laws of physics are the same in all inertial (non-accelerating) reference frames**.

The **speed of light in a vacuum is the same for all inertial observers**, regardless of the motion of the source or observer.

$c = 3.00 \times 10^{8}$ m s⁻¹ — the same for every inertial observer.

A frame moving at **constant velocity** — no acceleration (no speeding up, slowing down, or turning).

Exactly **c**, not 1.5c — by postulate 2 light's speed never adds on the source's speed.

Classically speeds add; **relativistically light always measures c** for everyone, so they do not add.

**c** — the postulates make it impossible for anything with mass to reach or exceed the speed of light.

Whether two events happen **'at the same time' depends on the observer's motion** — observers in relative motion can disagree.

Because **c is the speed limit**, so any relative speed must stay **below c**; velocities do not add the everyday way near c.

**No** — lengths and time intervals depend on the observer's motion; only the speed of light c is the same for all.

Because **both measure light at the same speed c**, they are forced to disagree about **when** events happen.

'The speed of light in a vacuum is the same for **all inertial observers, regardless of the motion of the source or observer**.'

When the speed is a **tiny fraction of c** (for example a proton at 0.0015c): Newtonian physics is then accurate.

$\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}$ — and it is **always ≥ 1**.

The time between two events measured by a **single clock present at both** — the **shortest** possible time.

The length of an object measured **in its own rest frame** — the **longest** possible length.

$\Delta t = \gamma\,\Delta t_0$. Since γ ≥ 1, **moving clocks run slow**.

$L = \dfrac{L_0}{\gamma}$. Since γ ≥ 1, **moving objects contract** along the motion.

**Multiply** the proper time by γ ($\Delta t = \gamma\,\Delta t_0$) — the time gets bigger.

**Divide** the proper length by γ ($L = L_0/\gamma$) — the length gets smaller.

$\gamma = \dfrac{1}{\sqrt{1 - 0.80^2}} = \dfrac{1}{\sqrt{0.36}} = 1.67$.

Only the dimension **along the direction of motion**; width and height are unchanged.

$u' = \dfrac{u - v}{1 - uv/c^2}$ — it always keeps the result **below c**.

$\dfrac{1.00c}{1 + 0.25} = 0.80c$, **not** 1.0c.

Time **stretches** (Δt = γΔt₀, multiply); length **shrinks** (L = L₀/γ, divide). Both use the same γ.

**x′ = γ(x − vt)** and **t′ = γ(t − vx/c²)**; for two events use Δx and Δt.

Δt′ = γ(Δt − vΔx/c²) with Δt = 0: the sign of Δt′ tells you. Same t, larger x along the motion gives the **earlier** t′.

Far more reach the ground than 2.2 μs allows: in the Earth frame their clocks run slow (**time dilation**); in their frame the atmosphere is contracted (**length contraction**).

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

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

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

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

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

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

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

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

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

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

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

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

θ is the angle between a world line and the **ct axis**; a bigger angle means a faster object, and 45° means c.

Through the origin, tilted up from the x axis by the **same angle θ** as ct′ is tilted from the ct axis (the mirror image of ct′ in the light line).

Take the gap between the world lines of its two ends along a **horizontal** line (the same ground time).