A.5 Galilean and special relativity (HL)

This topic compares the everyday, low‑speed world of Galilean relativity with the high‑speed domain of special relativity, where time, length and simultaneity become frame‑dependent in a precise, quantitative way.

Reference frames and Galilean transformations

An inertial frame is one in which a free particle moves with constant velocity; accelerating or rotating frames are non‑inertial and require fictitious forces to explain motion.
In Galilean relativity, the laws of mechanics are the same in all inertial frames and time is universal: $x' = x - vt,\ t' = t$ .
Velocities simply add: $u' = u - v$ , which works well for speeds much smaller than \$c\$.

Postulates of special relativity and Lorentz factor

Special relativity is built on two postulates: the laws of physics have the same form in all inertial frames, and the speed of light in vacuum has the same value \$c\$ for all inertial observers.
To satisfy these postulates, coordinates transform via the Lorentz transformations with factor $\gamma = 1/\sqrt{1 - v^{2}/c^{2}}$ , rather than the simpler Galilean rules.
At low speeds \$v \\ll c\$, \$\\gamma \\approx 1\$ and the Lorentz transformations reduce to Galilean ones, recovering Newtonian mechanics as an approximation.

Time dilation, length contraction and simultaneity

The proper time \$\\Delta \\tau\$ is measured in the frame where two events occur at the same place; other observers measure a longer interval $\Delta t = \gamma\,\Delta \tau$ (time dilation).
The proper length \$L_0\$ is measured in the object’s rest frame; moving observers measure a contracted length $L = L_{0}/\gamma$ along the direction of motion.
Events that are simultaneous in one frame need not be simultaneous in another; relativity of simultaneity is encoded in the time transformation $t' = \gamma(t - vx/c^{2})$ .

Space–time diagrams and world lines

In a space–time diagram, time (often ct) is plotted vertically and position horizontally; the world line of a particle shows its history through space and time.
The space–time interval $\Delta s^{2} = c^{2}\Delta t^{2} - \Delta x^{2} - \Delta y^{2} - \Delta z^{2}$ has the same value in all inertial frames, even though \$\\Delta t\$ and \$\\Delta x\$ individually change.
The angle of a world line with respect to the time axis is related to speed: steeper lines correspond to lower speeds, while light rays have 45° lines (in units where ct and x share the same scale).

Important formulas

x' = x - vt

Galilean transformation for position along the direction of motion.

t' = t

In Galilean relativity, time is the same in all inertial frames.

u' = u - v

Galilean velocity addition in 1D.

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

Lorentz factor in special relativity.

x' = \gamma (x - vt)

Lorentz transformation for position along the motion.

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

Lorentz transformation for time.

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

Relativistic velocity addition in 1D.

\Delta s^{2} = c^{2}\Delta t^{2} - \Delta x^{2} - \Delta y^{2} - \Delta z^{2}

Invariant space–time interval.

\Delta t = \gamma\,\Delta \tau

Time dilation: $\Delta \tau$ is the proper time.

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

Length contraction along the direction of motion.

Practice problems

Comparing Galilean and Lorentz transformations

A spacecraft moves at speed v relative to Earth along the x‑axis. Write down the Galilean transformation for position and time, and the corresponding Lorentz transformation. Explain when the Galilean version is a good approximation.

Galilean velocity addition

In a train moving at 15\ \mathrm{m\,s^{-1}} relative to the ground, a passenger walks forward at 2.0\ \mathrm{m\,s^{-1}} relative to the train. Use Galilean addition to find the passenger’s speed relative to the ground and state the assumption behind this rule.

Time dilation for a fast spacecraft clock

A clock on a spacecraft moving at v = 0.80c relative to Earth measures a proper time interval of 1.0 h between two ticks. How much time passes according to an observer on Earth?

Length contraction of a fast spaceship

The proper length of a spaceship measured in its own rest frame is L_{0} = 120\ \mathrm{m}. It passes an observer on a planet at speed 0.60c. What length does the planet observer measure along the direction of motion?

Relativistic velocity addition

A probe is launched from a spaceship that moves at 0.70c relative to Earth. The probe moves forward at 0.50c relative to the spaceship. Find the probe’s speed relative to Earth using the relativistic velocity‑addition formula.

Space–time interval and simultaneity

Two events occur in frame S with $\Delta x = 3.0\times10^{8}\ \mathrm{m}$ and $\Delta t = 2.0\ \mathrm{s}$. Compute the space–time interval $\Delta s^{2}$ and state whether the separation is time‑like or space‑like. Comment on whether all inertial observers agree on the simultaneity of these events.

Muon decay evidence for time dilation

Muons created high in the atmosphere have a mean lifetime at rest of about 2.2\ \mu\text{s}. They travel towards Earth at 0.98c. Show qualitatively why many muons can still reach detectors at the surface, and write the time‑dilation relation that explains this.

Clarity tip: For relativity questions, always specify the frame, mark which time or length is “proper”, and decide early whether to use Lorentz transformations, invariants (like \$\\Delta s^2\$) or time‑dilation/length‑contraction shortcuts.

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