Special Relativity
Study H3 Special Relativity in prerequisite order: evidence and postulates, simultaneity, Lorentz transformations, relativistic effects, velocity addition, and energy–momentum.
Learning goals
- discuss qualitatively the results of the Michelson–Morley interferometer experiment and its implications on the ether theory (knowledge of the details of the experiment is not required)
- state the postulates of the special theory of relativity, that in all inertial frames, the laws of physics are the same and the speed of light in free space is the same regardless of the motion of the light source or observer
- appreciate the failure of Galilean transformation equations when applied to a moving source of light
- discuss the concept of simultaneity
- show an understanding of the terms proper time and proper length
- apply the Lorentz transformation equations to solve one-dimensional problems
- Derive the time dilation formula and the length contraction formula, making use of the Lorentz factor.
- apply the time dilation formula and the length contraction formula in related situations (e.g. the lifetime of fast-moving muons) or to solve problems
- use the one-dimensional relativistic velocity addition formula to calculate velocities in different inertial frames or to solve problems
- Apply the relativistic energy–momentum relation E² = (pc)² + (mc²)² to solve problems, including selecting its limiting form.
- Show that E² = (pc)² + (mc²)² reduces to E = pc for massless particles and to E = mc² + ½mv² at low speeds.
Special relativity replaces absolute space and time with a single consistent framework built around inertial frames and the invariant speed of light. The safest route is conceptual first, mathematical second: establish the postulates and frame-dependent simultaneity before using Lorentz transformations to derive the familiar effects.
Choose your route
Start with the Michelson–Morley experiment, then follow the ordered lessons on this hub when you want a slower derivation or more examples on one subtopic.
Start Here
Prerequisites:
Study rule: define the two events and identify the frame before choosing a formula.
Deep-dive lessons
- Michelson–Morley Experiment — interpret the null result and the failure of a detectable ether wind.
- Einstein’s Principle of Relativity — state the two postulates precisely.
- Simultaneity and Relativity of Time — distinguish observation delay from frame-dependent simultaneity.
- Lorentz Transformation Equations — transform event coordinates and intervals between inertial frames.
- Time Dilation — identify proper time and derive the moving-clock relation.
- Length Contraction — identify proper length and enforce simultaneous endpoint measurements.
- Relativistic Addition of Velocities — combine collinear velocities without exceeding c.
- Relativistic Momentum and Energy — apply the energy–momentum relation and its limiting cases.
- The Twin Paradox — compare proper time along different worldlines.
- Confirming Time Dilation, Length Contraction and Simultaneity — consolidate the three results from the Lorentz difference equations.
The optional Einstein’s Significant 1905 Papers page supplies historical context; it is not part of the core derivation sequence.
Revision
Quick Reference
- β = v/c
- γ = 1/square root of (1-β²)
- Lorentz position transform: x' = γ(x-vt)
- Lorentz time transform: t' = γ(t-vx/c²)
- Time dilation: Δ t = γΔτ
- Length contraction: L = L₀/γ
- Velocity transformation: u' = (u-v)/(1-uv/c²)
- Energy–momentum relation: E² = (pc)² + (mc²)²
Problem-Solving Workflow
- Name frames S and S' and state which moves at + v.
- Define the two events and write their coordinates or coordinate differences.
- Identify the special condition: same place for proper time, rest frame for proper length, or same time for a length measurement.
- Choose the direct, inverse, or difference form of the Lorentz transformation.
- Keep full calculator values during working, round the final answer to the data’s significant figures, and check the v≪ c limit.
Top Exam Traps
- Treating “seen at the same time” as equivalent to “simultaneous in this frame”.
- Using time dilation or length contraction before identifying the proper quantity.
- Forgetting the c² in the Lorentz time transformation.
- Mixing primed and unprimed coordinates in one substitution.
- Using Galilean velocity addition when speeds are relativistic.
Practice
Practice (Past-Year + Self-Check)
- Transform one pair of events and explain the sign of both coordinate differences.
- Derive time dilation and length contraction by imposing the correct special conditions.
- Check a velocity-addition result against the limits v → 0 and u → c.
- Use E² = (pc)² + (mc²)² once for a massive particle and once for a photon.
Continue Learning
- Return to the H3 Physics Hub for the full syllabus map.
- Use the H3 Paper Map to plan mixed-topic practice.
Course and syllabus information
- Course
- GCE A-Level H3 Physics
- Edition
- GCE A-Level H3 Physics 2027