Special relativity
Study H3 Special Relativity in prerequisite order: evidence and postulates, simultaneity, Lorentz transformations, relativistic effects, velocity addition, and energy–momentum.
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Before you begin
Special relativity describes space and time using inertial frames and the invariant vacuum speed of light. The lessons start with the Michelson–Morley result and Einstein’s two postulates, show that simultaneity depends on the frame, and then derive time dilation, length contraction and velocity addition from the Lorentz transformations. They continue with relativistic momentum and energy and the twin paradox, and finish by showing that all three effects come from the same pair of equations.
Be comfortable with: frames of reference (H3) and kinematics (H2).
Learning goals
- Discuss the result of the Michelson–Morley experiment and what it means for the ether theory
- State the two postulates of special relativity: the laws of physics and the speed of light in free space are the same in every inertial frame
- Explain why the Galilean transformation fails for a moving light source
- Discuss the relativity of simultaneity
- Explain proper time and proper length
- Use the Lorentz transformation to solve one-dimensional problems
- Derive the time dilation formula and the length contraction formula, making use of the Lorentz factor.
- Use time dilation and length contraction, for example for the lifetime of fast-moving muons
- Use relativistic velocity addition to find velocities in different inertial frames
- 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.
Syllabus statements covered
- 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 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
- use the relativistic energy–momentum relation E² = (pc)² + (mc²)² to solve problems, and show that it reduces, in the appropriate limits, to: (1) E = pc (for massless particles); or (2) E = mc² + ½mv² (for particles moving at low speeds v ≪ c).
Lessons
Work through them in order.
- Michelson–Morley experimentInterpret the null result and the failure of a detectable ether wind.
- Einstein’s postulates of special relativityState the two postulates precisely.
- Simultaneity and relativity of timeDistinguish observation delay from frame-dependent simultaneity.
- Lorentz transformationsTransform an event forward and back between inertial frames with consistent space and time coordinates.
- Comparing events between inertial framesApply Lorentz event differences using simultaneity and proper-measurement conditions.
- Time dilationIdentify proper time and derive the moving-clock relation.
- Length contractionIdentify proper length and enforce simultaneous endpoint measurements.
- Relativistic velocity additionCombine collinear velocities without exceeding c.
- Relativistic momentum and energyDistinguish energy quantities and calculate energy, momentum and invariant rest mass.
- Massless and low-speed energy limitsDerive the massless and low-speed limits and compare exact and Newtonian kinetic energy.
- The twin paradoxResolve the twin paradox by comparing proper time along different worldlines.Supporting
- Checking Lorentz measurement conditionsChoose the event pair and measurement condition before reconstructing Lorentz results.Supporting
Practise and check
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Topic reference
Quick reference
S' moves at + v relative to S, with parallel axes.
| Idea | Relation | Notes |
|---|---|---|
| Lorentz factor | β = v/c, γ = 1/square root of (1-β²) | γ ≥ 1 |
| Lorentz transformation | x' = γ(x-vt), t' = γ(t-vx/c²) | use the same two events in both frames |
| time dilation | Δ t = γΔτ | Δτ: ticks at one place in the clock’s rest frame |
| length contraction | L = L₀/γ | L: both ends recorded at one time in the measuring frame |
| velocity transformation | u' = (u-v)/(1-uv/c²) | signed velocities along x |
| energy–momentum | E² = (pc)² + (mc²)² | E is total energy, p the momentum magnitude |
Core knowledge to remember
- The laws of physics have the same form in every inertial frame, and the speed of light in a vacuum is c in every inertial frame.
- Events simultaneous in one frame are generally not simultaneous in a frame moving along their separation.
- Proper time is measured by one clock present at both events; proper length is measured in the object’s rest frame.
- Time dilation and length contraction follow from the Lorentz transformations once the zero condition (Δ x' = 0 or Δ t = 0) is imposed.
- For a photon, m = 0 and E = pc; for v≪ c, the kinetic energy reduces to (1/2)mv².
Common mistakes
- 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.