Special relativity · A-Level Physics · Courselo

Lesson 3 of 8 · Options and board-specific extensions

Special relativity

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Special relativity is the second half of the AQA Turning points in physics option (Paper 3 Section B), and time dilation and relativistic energy also appear in OCR B (H557) Papers 1 and 2. It has little content and a lot of marks: most questions are either a clean substitution into one of three data-sheet equations or a “describe and explain” answer about Michelson–Morley, muons or Bertozzi. You lose marks here through muddled frames of reference and vague descriptions, not through hard maths.

By the end you’ll be able to

State the two postulates of special relativity and explain the significance of the Michelson–Morley null result
Use t = t₀/√(1 − v²/c²) and the relativistic factor γ for time dilation, including muon decay evidence
Use l = l₀√(1 − v²/c²) for length contraction and identify proper time and proper length
Use E = γmc² and explain why no massive object can reach the speed of light

What the exam asks

Michelson–Morley: outline the interferometer, what an aether theory predicted, the null result and why it matters. Often a 4–6 mark extended answer.
Postulates: state both exactly, and define an inertial frame of reference.
Time dilation: $t = \dfrac{t_0}{\sqrt{1 - v^2/c^2}}$ , identify the , and explain the muon evidence with a calculation.

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Flashcards

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PromptCard 1 of 3

What is an inertial frame of reference?

Core ideas

Frames and postulates

An inertial frame of reference is one that is not accelerating: Newton’s first law holds in it. Einstein’s two postulates are:

The laws of physics have the same form in all inertial frames.
The speed of light in free space is invariant: every inertial observer measures $c$ , whatever the motion of the source or the observer.

Everything else follows from these two statements. Write them word-perfectly, because “light always travels at the same speed” (with no mention of free space or of the observer) usually earns nothing.

The Michelson–Morley experiment

Light from a source hits a semi-silvered glass plate (the beam splitter). One beam travels along an arm to a mirror and back. The other beam travels along a perpendicular arm of equal length to a second mirror and back. The two beams recombine and are viewed through a telescope, where they form interference fringes. A compensator plate makes sure both beams pass through the same thickness of glass.

The prediction came from the aether theory. If the Earth moved through a stationary aether at about $3 \times 10^4\ \text{m s}^{-1}$ , light would take slightly longer along the arm parallel to the motion than along the perpendicular arm. Rotating the whole apparatus through 90° swaps the roles of the arms, so the fringes should shift. With an effective path of 11 m, the expected shift was about 0.4 of a fringe, and the apparatus could detect a shift of about 0.01 of a fringe.

The result: no significant shift at any orientation, at different times of day, or in later repeats at different seasons. There was no detectable absolute motion through an aether. The speed of light was the same in both arms, whatever the Earth’s motion. This is the experimental basis for postulate 2.

Time dilation and proper time

The proper time $t_0$ is the time between two events measured by a clock that is at rest relative to the events, so both events happen at the same place in that frame. Every other inertial observer, moving at speed $v$ relative to that clock, measures a longer interval:

t = \frac{t_0}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \gamma t_0 \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \geq 1

“Moving clocks run slow.” At everyday speeds $\gamma$ differs from 1 by less than one part in $10^{9}$ , which is why we never notice it.

Muon evidence

Muons are produced by cosmic rays in the upper atmosphere and travel at more than $0.98c$ . Their half-life in their own rest frame is about 1.5 μs. Measurements at a mountain top and at sea level show that far more muons survive the descent than a non-relativistic calculation predicts. The Earth observer explains this with time dilation: the muons’ clocks run slow, so their half-life in the Earth frame is $\gamma t_{1/2}$ . The muon “observer” explains it with length contraction: the atmosphere is shorter in the muon’s frame. Both views predict the same number of survivors.

Length contraction

The proper length $l_0$ is measured in the frame where the object is at rest. An observer who sees it moving at $v$ measures a shorter length along the direction of motion only:

l = l_0\sqrt{1 - \frac{v^2}{c^2}} = \frac{l_0}{\gamma}

Mass, energy and Bertozzi

The total energy of a particle is $E = mc^2 = \gamma m_0c^2$ . Its rest energy is , so

E_k = mc^2 - m_0c^2 = (\gamma - 1)m_0c^2

For $v \ll c$ this reduces to $\tfrac{1}{2}m_0v^2$ . As , both and tend to infinity, so : it would need infinite energy.

Graph	Shape
mass $m$ against $v$	starts at $m_0$ when $v = 0$ , is almost flat at low speed, then rises steeply towards a vertical asymptote at

Bertozzi (1964) gave electrons known kinetic energies (0.5–15 MeV) using a Van de Graaff accelerator and a linear accelerator. He measured their speed by timing pulses over an 8.4 m flight path using an oscilloscope. He checked the kinetic energy independently by calorimetry: the electrons stopped in an aluminium target, and the rise in its temperature gave the energy per electron. The speed approached, but never exceeded, $c$ . Newtonian physics predicts $v^2 \propto E_k$ with no limit. This was direct evidence that kinetic energy varies with speed as relativity predicts.

Worked examples

Exam technique

Name the frame first. Before you substitute, write “proper time is measured by the muon”, or “proper length is measured on the launch pad”. That line often earns the method mark, and it stops you putting $t_0$ in the wrong place.
Sanity check: a dilated time must be longer than $t_0$ , a contracted length shorter than $l_0$ , and is never less than 1. If your answer breaks these rules, you have inverted the formula.

Common mistakes

Quick recap

An inertial frame is non-accelerating. The postulates are: the laws of physics take the same form in all inertial frames, and $c$ in free space is invariant.
The Michelson–Morley null result showed there is no absolute motion through an aether.
$t = \gamma t_0$ with $t_0$ the proper time. Moving clocks run slow, and the muon survival rate confirms it.

m_0c^2