IBDP Physics HL Cheat sheet - A.5 Galilean and special relativity (HL only) | IBDP Physics Cheatsheets

Reference frames and inertial observers

Reference frame = a coordinate system used by an observer to describe position, time and motion.
Inertial reference frame = a non-accelerating frame.
In all inertial frames, Newton’s laws have the same form: this is Galilean relativity.
In classical mechanics, time is absolute, so all observers agree on $t$ .
For frames moving at relative speed $v$ along the $x$ -axis, the Galilean transformation is $x' = x - vt$ and $t' = t$ .
The classical velocity addition result is $u' = u - v$ .
Galilean relativity fails at speeds close to $c$ because it assumes absolute time and does not preserve the speed of light.

This diagram shows how events are represented in space–time relative to an observer. It helps link the idea of a reference frame to the geometry of events, light signals and causality. Source

Postulates of special relativity

Postulate 1: The laws of physics are the same in all inertial frames.
Postulate 2: The speed of light in vacuum, $c$ , has the same value for all inertial observers, regardless of the motion of source or observer.
These postulates replace Galilean transformation with Lorentz transformation at high speeds.
Relativistic effects become important when $v$ is a significant fraction of $c$ .
The Lorentz factor is $\gamma = \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$ .
Key exam idea: as $v \to c$ , $\gamma$ increases rapidly.

HL only: Lorentz transformations and relativistic velocity addition

For motion along one dimension, the Lorentz transformations are $x' = \gamma(x-vt)$ and $t' = \gamma\left(t-\dfrac{vx}{c^2}\right)$ .
Unlike Galilean relativity, space and time both transform.
The term $-\dfrac{vx}{c^2}$ in the time transformation is why simultaneity is relative.
The relativistic velocity addition formula is $u' = \dfrac{u-v}{1-\dfrac{uv}{c^2}}$ .
This formula ensures no observer measures a speed greater than $c$ .
For $u, v \ll c$ , the relativistic formula reduces approximately to $u' \approx u-v$ , matching the classical limit.
In exam questions, check whether the speed is small compared with $c$ before deciding whether Galilean or relativistic treatment is appropriate.

This Minkowski diagram shows the world line of a photon on axes $x$ and $ct$ . Light always appears at $45^\circ$ on such diagrams because its speed is $c$ . Source

Time dilation and proper time

Proper time $\Delta t_0$ = the time interval measured in the frame where the two events occur at the same place.
Time dilation: a moving clock is measured to run slower.
Formula: $\Delta t = \gamma \Delta t_0$ .
Therefore $\Delta t \ge \Delta t_0$ .
The proper time is always the shortest measured time interval between the same two events.
Typical exam setup: identify which observer carries the clock or in which frame the clock is at rest.
Do not confuse measured longer interval with time actually changing in its own frame.

This diagram compares two inertial frames and visualizes time dilation geometrically. It is useful for seeing why each observer judges the other’s moving clock to run more slowly. Source

Length contraction and proper length

Proper length $L_0$ = the length measured in the frame where the object is at rest.
Length contraction occurs only parallel to the direction of motion.
Formula: $L = \dfrac{L_0}{\gamma}$ .
Therefore $L \le L_0$ .
The proper length is always the greatest measured length.
No contraction occurs in directions perpendicular to the motion.
In exam questions, contracted length is measured by an observer for whom the object is moving.

Relativity of simultaneity

Events that are simultaneous in one inertial frame are not necessarily simultaneous in another.
This follows directly from $t' = \gamma\left(t-\dfrac{vx}{c^2}\right)$ because time depends on position as well as time.
If two events have $\Delta t = 0$ but occur at different positions $\Delta x \ne 0$ , then generally $\Delta t' \ne 0$ .
This is a major difference from Galilean relativity, where $t' = t$ for all events.
Many paradox-style questions are resolved by identifying that different frames disagree about simultaneity.

Space–time interval and invariance

The space–time interval between two events is invariant: all inertial observers calculate the same value.
Formula: $(\Delta s)^2 = (c\Delta t)^2 - (\Delta x)^2$ .
Because it is invariant, it plays the role that ordinary distance plays in Euclidean geometry.
For the proper time case, events occur at the same place in one frame, so $\Delta x = 0$ in that frame.
Proper time can therefore be linked to the invariant interval.
A common exam skill is using the invariant interval to compare descriptions from different frames.

Space–time diagrams and world lines

In IB questions, axes are usually $x$ horizontally and $ct$ vertically.
A world line shows how an object’s position changes through space–time.
For an object at rest in a frame, the world line is vertical.
For a particle moving at constant velocity, the world line is a straight line tilted to the $ct$ axis.
For light, the world line is at $45^\circ$ because $x = ct$ .
The syllabus relation is $\tan\theta = \dfrac{v}{c}$ , where $\theta$ is the angle between the world line and the time axis.
A steeper line means a lower speed; a line closer to $45^\circ$ means a speed closer to $c$ .
Use diagrams to visualize time dilation, length contraction and relativity of simultaneity.

This diagram illustrates how different world lines and simultaneity lines are used in the twin paradox. It is especially helpful for interpreting space–time diagrams beyond simple straight-line motion. Source

Muon decay as evidence

Muon decay experiments provide evidence for time dilation and length contraction.
Muons created high in Earth’s atmosphere should decay before reaching the ground if classical physics were correct.
In Earth’s frame, the muon’s internal clock runs slowly, so its lifetime is dilated.
In the muon’s frame, the atmosphere is length contracted, so the distance to Earth is smaller.
Both descriptions agree on the same observed outcome.
Exam point: the same experiment supports both time dilation and length contraction, depending on the chosen frame.

This graph is not a muon experiment, but it is a clean real-world illustration that moving clocks can tick at different rates. It reinforces the measurable nature of time dilation in modern technology. Source

Galilean vs special relativity: compare fast

Galilean relativity: $t' = t$ , $x' = x-vt$ , $u' = u-v$ .
Special relativity: $x' = \gamma(x-vt)$ , $t' = \gamma\left(t-\dfrac{vx}{c^2}\right)$ , $u' = \dfrac{u-v}{1-\dfrac{uv}{c^2}}$ .
Galilean relativity works well when $v \ll c$ .
Special relativity is needed when speeds are close to $c$ .
Absolute time is classical; relative time is relativistic.
Simultaneity is universal in Galilean relativity but frame-dependent in special relativity.
The speed of light is not invariant in Galilean relativity, but is invariant in special relativity.

Checklist: can you do this?

State the two postulates of special relativity clearly and accurately.
Use Galilean and Lorentz transformation equations appropriately for a given situation.
Identify proper time and proper length before applying time dilation or length contraction.
Interpret a space–time diagram, including world lines, light lines, and the meaning of gradient/angle.
Explain muon decay evidence using either Earth’s frame or the muon’s frame.

Written by:

Dr Shubhi Khandelwal

Qualified Dentist and Expert Science Educator

Shubhi is a seasoned educational specialist with a sharp focus on IB, A-level, GCSE, AP, and MCAT sciences. With 6+ years of expertise, she excels in advanced curriculum guidance and creating precise educational resources, ensuring expert instruction and deep student comprehension of complex science concepts.