A.5 Galilean and special relativity

UNIT SYLLABUS

HL 8 hours · HL only

Relativity begins with a deceptively simple question: how do measurements made by two observers in relative motion compare? Galileo's answer — velocities simply add, time is universal — works beautifully until speeds approach that of light. Einstein's two postulates (the laws of physics are identical in all inertial frames, and light travels at $c$ for every observer) force a radical revision: moving clocks run slow, moving lengths contract, and simultaneity itself depends on the observer. The Lorentz transformations encode all of this, spacetime diagrams let you see it, and muons created in the upper atmosphere — reaching the ground only because of time dilation — prove it happens.

Guiding Questions

? Why is the speed of light the same for all observers, and what follows from that?
? How do space and time become linked into spacetime in special relativity?

✓ What the IB expects you to master

Describe reference frames, and state Galilean relativity: Newton's laws hold in all inertial frames.
Use the Galilean transformations ( $x' = x - vt$ , $t' = t$ ) and velocity addition $u' = u - v$ .
State the two postulates of special relativity.
Apply the Lorentz transformations ( $x' = \gamma(x - vt)$ , $t' = \gamma(t - vx/c^2)$ with $\gamma = 1/\sqrt{1 - v^2/c^2}$ ) to the coordinates of events.
Use relativistic velocity addition $u' = \dfrac{u - v}{1 - uv/c^2}$ .
Explain proper time and proper length, and calculate time dilation $\Delta t = \gamma\Delta t_0$ and length contraction $L = L_0/\gamma$ .
Show that the spacetime interval $(\Delta s)^2 = (c\Delta t)^2 - (\Delta x)^2$ is invariant between frames.
Interpret spacetime diagrams: world lines, the relativity of simultaneity, and $\tan\theta = v/c$ for a moving particle's world line.
Explain how muon decay experiments provide evidence for time dilation and length contraction.

1 Key Formulas

Lorentz factor

\gamma = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}

Galilean transformation

x' = x - vt,\quad u' = u - v

Lorentz transformation (position)

x' = \gamma(x - vt)

Lorentz transformation (time)

t' = \gamma\left(t - \frac{vx}{c^{2}}\right)

Relativistic velocity addition

u' = \frac{u - v}{1 - \frac{uv}{c^{2}}}

Time dilation

\Delta t = \gamma\Delta t_{0}

Length contraction

L = \frac{L_{0}}{\gamma}

Spacetime interval

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

World line angle

\tan\theta = \frac{v}{c}

2 Exam Preparation & Topic Explanations

Keeping the frames straight

Most lost marks in relativity come from mixing up which observer measures the proper quantity. Ask: which frame sees both events at the same place (proper time)? In which frame is the object at rest (proper length)? Label those first and the formulas apply themselves.

Gamma is always $\ge 1$ : dilated times are longer, contracted lengths shorter — sanity-check every answer against that.

Pro Exam Strategy

Compute $\gamma$ once at the start; most numerical questions reuse it repeatedly.
Simultaneity is relative: events simultaneous in one frame are not in another — this resolves nearly every apparent "paradox".
The muon experiment is the guide's named evidence — know both frame descriptions cold.
On spacetime diagrams, steeper world lines are slower; light is always the 45° line.

3 MCQ Practice

Q1. Which quantity is agreed upon by all inertial observers?

The time interval between two events
The distance between two events
The spacetime interval between two events
The simultaneity of two events

Q2. A muon has a proper lifetime of $2.2\ \mu\text{s}$ and travels at $0.98c$ ( $\gamma \approx 5.0$ ). In the Earth frame its lifetime is approximately:

$0.44\ \mu\text{s}$
$2.2\ \mu\text{s}$
$11\ \mu\text{s}$
$22\ \mu\text{s}$

Q3. Spaceship A moves at $0.8c$ relative to Earth, and fires a probe forwards at $0.8c$ relative to itself. The probe's speed relative to Earth is:

$1.6c$
$c$
$0.98c$
$0.89c$

4 Short Answer Questions

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