Year 12 Physics Module 7 ⏱ ~25 min 5 MC · 2 Short Answer Lesson 8 of 14

Special Relativity, Inertial Frames and Postulates

You'll be able to identify inertial vs non-inertial frames.

On 30 June 1905, Albert Einstein, then a patent clerk at the Swiss Patent Office in Bern, published "On the Electrodynamics of Moving Bodies," building on the 1887 Michelson–Morley null result (11 m optical path, expected 0.4 fringe shifts, measured less than 0.01). His two postulates, that physics is the same in all inertial frames, and that $c = 2.998 \times 10^8$ m/s is constant for all observers, overturned two centuries of Newtonian absolute space and time.

Today's hook: In 1887, Albert Michelson and Edward Morley at Case Western Reserve University, Cleveland, used an 11 m optical path interferometer and expected to detect Earth's motion through the aether as fringe shifts of 0.4. They measured less than 0.01. In June 1905 Albert Einstein at the Swiss Patent Office Bern used this null result to argue that $c$ must be the same for all observers, so if you shine a flashlight from a train moving at 100 km/h, the platform observer still measures the light at exactly $c$. How can that be?
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Orient and predict

Predict how a torch beam behaves on a moving spaceship, then set your goals and vocabulary for special relativity.

Warm up first

Three quick questions from earlier lessons. Pulling old material back to mind before you learn something new makes the new material stick better, so this is not busywork.

Worksheets

Practise this lesson

Four printable worksheets that build from the foundations up to exam-style questions, start at whatever level suits you.

Build Foundations & guided practice Apply Application & data response Master Mastery tasks Exam Exam-style questions
Before you read, predict

You are on a train moving at 100 km/h. You shine a flashlight forward, and the light travels at $c$ relative to you.

  1. According to classical (Galilean) physics, how fast does the light travel relative to a person standing on the platform?
  2. If the speed of light is truly constant for all observers, what would the platform observer measure?
  3. What must happen to time and space measurements if both observers measure the same speed of light?

Write your predictions before reading on, you will revisit them at the end.

Warm-up, in Galilean relativity, if you throw a ball at 10 m/s forward on a train moving at 20 m/s, a platform observer sees the ball at:

Learning Intentions
goals

Know, Inertial Frames

  • Frames moving at constant velocity
  • No preferred inertial frame exists
  • Galilean relativity vs Einsteinian relativity

Understand, Einstein's Postulates

  • Laws of physics are identical in all inertial frames
  • Speed of light in vacuum is constant for all observers
  • Michelson-Morley null result and its significance

Can Do, Apply Relativistic Concepts

  • Identify inertial vs non-inertial frames
  • Apply the postulates to simple scenarios
  • Calculate the Lorentz factor $\gamma$
Scan these before reading
vocab
Inertial frameA reference frame in which Newton's first law holds; a frame moving at constant velocity (no acceleration).
Special relativityEinstein's theory describing physics in inertial frames, based on the two postulates of relativity.
Luminiferous aetherA hypothetical light-carrying medium; the Michelson-Morley null result was inconsistent with the expected stationary-aether wind.
Lorentz factor$\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}$; quantifies how much relativistic effects scale with velocity.
Null resultAn experimental outcome that detects no effect, often as significant as a positive result, as in Michelson-Morley.
Cross-lesson links: L02 introduced the Michelson–Morley 1887 null result as evidence that EM waves need no medium. L11 revisits the same experiment as the motivation for Einstein's June 1905 Swiss Patent Office Bern paper, the constancy of $c$ for all inertial observers. L12 applies the first concrete consequence: time dilation ($t = \gamma t_0$). L16 applies Einstein's other 1905 insight, the photon model of the photoelectric effect.

Core Content

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Weigh the aether evidence and the two postulates

Follow Galilean relativity to the Michelson-Morley null result, then state Einstein’s two postulates.

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From Galileo to Einstein
+5 XP

When common sense fails

Sit inside a smoothly gliding train with the blinds drawn. No experiment you can perform, dropping a ball, bouncing a laser off a mirror, measuring the weight of an object, can tell you whether the train is moving or stationary. Open the blinds and you can measure your speed relative to a tree or a platform, but there is no absolute "at rest." This is Galilean relativity: the laws of mechanics are the same in all inertial frames. Velocities simply add: throw a ball at 10 m/s forward on a train moving at 20 m/s and a platform observer measures the ball at 30 m/s.

But when James Clerk Maxwell derived his electromagnetic wave equations, they predicted a single speed for light: $c = 1/\sqrt{\mu_0 \varepsilon_0} \approx 3.00\times10^8$ m/s. This speed appeared without reference to any medium or observer, a profound departure from Galilean relativity.

Physicists assumed light travelled through an invisible medium called the luminiferous aether. If so, Earth's motion through the aether should create a detectable "aether wind." In 1887, Albert Michelson and Edward Morley performed an exquisitely precise interferometer experiment to measure this. Their result: no aether wind was detected. The speed of light was the same in all directions, regardless of Earth's motion.

The Michelson-Morley interferometer: light is split into two perpendicular arms, reflected by mirrors and recombined at a detector; Earth’s motion should have made a time difference, but none was found.SourcesplitterMirrorMirrorDetectorEarth’s orbital motionNo detectable time differencebetween the two paths (null result)
Figure 1, Michelson-Morley interferometer: light split into two perpendicular paths and recombined. Earth's motion should have created a detectable path difference, but none was found (null result)
Text description
  • Light from a source reaches a half-silvered splitter, which sends part of the beam along a horizontal arm to one mirror and part up a perpendicular arm to another mirror.
  • Both beams reflect back to the splitter and recombine, travelling down to a detector opposite the upper mirror.
  • Earth’s orbital motion was expected to make the two paths take different times. No detectable time difference was found: the null result.

Einstein's solution was radical: discard the aether entirely. In his 1905 paper "On the Electrodynamics of Moving Bodies," he proposed two postulates:

  1. The principle of relativity: The laws of physics are identical in all inertial frames of reference. No experiment can detect absolute motion.
  2. The constancy of the speed of light: The speed of light in vacuum, $c$, is the same for all observers, regardless of their motion or the motion of the light source.

These postulates, seemingly simple, have extraordinary consequences: time dilation, length contraction, relativity of simultaneity, and the equivalence of mass and energy.

Stop & Check

A spaceship travels past Earth at 0.8$c$. The captain shines a laser forward. How fast does the laser light travel (a) relative to the captain, and (b) relative to an observer on Earth? Explain how this differs from Galilean velocity addition.

Galilean velocity addition conflicts with the invariant wave speed suggested by Maxwell's equations. Michelson and Morley's null result was inconsistent with the expected stationary-aether wind, within experimental sensitivity; it did not alone prove Einstein's later theory. Einstein's two postulates (1905) state that (1) the laws of physics are identical in all inertial frames and (2) every inertial observer measures the same vacuum light speed $c$. Relativistic effects follow from these postulates.

Write both postulates in full and note the Michelson-Morley significance.

The Michelson-Morley experiment was designed to detect:

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Build the Lorentz factor and its domain

See how gamma behaves from everyday speeds up to a large fraction of $c$, and where it is defined.

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The Lorentz Factor
+5 XP

The key to all relativistic effects

We just saw that Einstein's two postulates demand $c$ is constant for all observers. That raises a question: if $c$ never changes, what mathematical quantity tells us by how much time, length and energy change at high speeds? This card answers it → the Lorentz factor $\gamma$, which appears in every relativistic equation.

All relativistic effects are governed by the Lorentz factor $\gamma$ (gamma). It quantifies how much time dilation, length contraction, and energy increase occur at a given velocity.

Lorentz factor

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

where $\beta = v/c$ is the velocity as a fraction of the speed of light.

At everyday speeds ($v \ll c$), $\gamma \approx 1$ and relativity reduces to classical physics. As $v$ approaches $c$, $\gamma$ grows without bound:

  • $v = 0.1c$: $\gamma = 1.005$ (0.5% effect)
  • $v = 0.5c$: $\gamma = 1.155$ (15.5% effect)
  • $v = 0.9c$: $\gamma = 2.29$ (129% effect)
  • $v = 0.99c$: $\gamma = 7.09$
  • $v = 0.999c$: $\gamma = 22.4$
The Lorentz factor γ against v/c: close to 1 at low speed, 2 at v = 0.866c, and rising without limit as v approaches c.2468100.51.0v = 0.866c, γ = 2v/cγcγ → ∞ as v → c:massive objects cannot reach c
Figure 2, The Lorentz factor $\gamma$ as a function of $v/c$. At low speeds $\gamma \approx 1$; it diverges as $v \to c$, making it impossible for a massive object to reach the speed of light
Text description
  • A graph of the Lorentz factor γ = 1/√(1 − v²/c²) against v/c, from 0 to 1, with γ from 1 to 8.
  • At low speeds γ ≈ 1. At v = 0.866c, γ = 2 (marked).
  • As v approaches c (the dashed line at v/c = 1), γ rises without limit: γ → ∞, so massive objects cannot reach c.

The Lorentz factor appears in time dilation, length contraction, relativistic momentum and energy equations; invariant mass does not increase with speed:

Key relativistic equations

$\Delta t = \gamma \,\Delta t_0$   (time dilation; $\Delta t_0$ = proper time)

$L = L_0/\gamma$   (length contraction; $L_0$ = proper length)

$E = \gamma mc^2$   (total relativistic energy)

$E_k = (\gamma-1)mc^2$   (relativistic kinetic energy)

HSC Tip, Calculator Discipline

Always calculate $\gamma$ first, then use it in subsequent equations. Work with $\beta = v/c$ to avoid large powers of 10. Never use classical KE $\frac{1}{2}mv^2$ for relativistic speeds, at $v = 0.98c$ the classical answer is off by nearly a factor of 10!

Stop & Check

Calculate $\gamma$ for a proton travelling at $0.95c$. If the proton's rest mass is $1.67\times10^{-27}$ kg, calculate its total energy and kinetic energy. ($c = 3.00\times10^8$ m/s)

Lorentz factor: $\gamma = 1/\sqrt{1-v^2/c^2}$; always $\geq 1$; $\to \infty$ as $v\to c$. Key values: $v=0.5c\Rightarrow\gamma=1.155$; $v=0.9c\Rightarrow\gamma=2.29$; $v=0.99c\Rightarrow\gamma=7.09$. It governs: $\Delta t = \gamma\Delta t_0$, $L = L_0/\gamma$, $E = \gamma mc^2$, $E_k = (\gamma-1)mc^2$, never use $\tfrac{1}{2}mv^2$ at relativistic speeds.

Record the formula and the four key relativistic equations that use $\gamma$.

A particle moves at $v = 0.6c$. Its Lorentz factor $\gamma$ is:

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Solve Lorentz factor worked examples

Work gamma calculations step by step, including reading a speed back out of a given gamma.

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Worked Example, Lorentz Factor Calculations
+5 XP

Getting comfortable with gamma

We just saw the Lorentz factor formula and the four equations it governs. That raises a question: how do you chain these equations together correctly in a multi-step exam problem involving time dilation, energy and potential difference? This card answers it → a four-step worked example at $v = 0.98c$.

Problem

An unstable muon, and separately an electron, are each accelerated to 0.98$c$ in a particle accelerator.

  1. Calculate the Lorentz factor $\gamma$.
  2. If the muon's proper lifetime is $\tau_0 = 2.0\,\mu\text{s}$ for this worked example, calculate its dilated lifetime in the lab frame.
  3. Calculate the electron's total energy and kinetic energy. ($m_e = 9.11\times10^{-31}$ kg)
  4. Through what potential difference must the electron be accelerated to reach this speed?
Step 1, Lorentz factor

$\gamma = 1/\sqrt{1 - 0.98^2} = 1/\sqrt{1 - 0.9604} = 1/\sqrt{0.0396} = 1/0.199 = $ 5.03

Step 2, Time dilation

$\Delta t = \gamma \tau_0 = 5.03 \times 2.0 = $ 10.1 $\mu$s

Step 3, Energy

$E = \gamma m c^2 = 5.03 \times (9.11\times10^{-31})(3.00\times10^8)^2 = 5.03 \times 8.20\times10^{-14} = 4.12\times10^{-13}$ J

In eV: $E = 5.03 \times 511$ keV = 2.57 MeV

$E_k = (\gamma - 1)mc^2 = 4.03 \times 511$ keV = 2.06 MeV

Step 4, Potential difference

$E_k = qV$, so $V = E_k/e = 2.06\times10^6$ eV $/ e$ = 2.06 MV

Stop & Check

A proton travels at $0.87c$. Calculate $\gamma$, its total energy, and its kinetic energy in MeV. ($m_p c^2 = 938$ MeV)

Worked strategy at $v = 0.98c$: step 1, $\gamma = 5.03$. Step 2, time dilation: $\Delta t = 5.03\times 2.0\,\mu\text{s} = 10.1\,\mu\text{s}$. Step 3, total energy: $E = \gamma mc^2 = 2.57$ MeV; $E_k = (\gamma-1)mc^2 = 2.06$ MeV. Step 4, potential difference: $V = E_k/e = 2.06$ MV.

Write the four-step checklist: $\gamma$ first, then time dilation, energy, potential difference.

A muon travels at $0.99c$ ($\gamma \approx 7.09$). If its proper lifetime is 2.2 µs, its lifetime in Earth's frame is approximately:

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Tell inertial from non-inertial frames

Decide which frames special relativity applies to, and what an accelerating observer detects.

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Inertial vs Non-Inertial Frames
+5 XP

Recognising the difference matters for the HSC

We just saw how to calculate $\gamma$ and apply it in multi-step problems. That raises a question: special relativity only applies to inertial frames, so how do you identify whether a given frame qualifies? This card answers it → the definition of inertial vs non-inertial frames with HSC examples.

An inertial reference frame is one in which an object with no net force remains at rest or in uniform motion, Newton's first law holds. Any frame moving at constant velocity relative to an inertial frame is itself inertial.

A non-inertial frame is one that accelerates or rotates. Newton's first law does not hold without introducing fictitious forces (such as the centrifugal or Coriolis forces). Examples:

  • Inertial: a train moving at constant 200 km/h on a straight track; the surface of the Earth (approximately); a spacecraft coasting in deep space
  • Non-inertial: a car accelerating away from traffic lights; a spinning roundabout; a rocket thrusting through space
Inertial versus non-inertial frames: in a train at constant velocity a ball stays put; in an accelerating car a ball appears to slide backward, as if pushed by a fictitious force.INERTIAL FRAMEconstant velocity (or at rest)Train, 200 km/hball stays putNewton’s 1st law holdsNON-INERTIAL FRAMEaccelerating or rotatingCar, acceleratingaball slides backa fictitious force appears
Figure 3, In an inertial frame, a ball remains stationary with no net force; in an accelerating (non-inertial) frame, a fictitious backward force appears to act on the ball
Text description
  • Inertial frame: a train moving at a constant 200 km/h (or at rest). A ball on the floor stays put, because no net force acts on it: Newton’s first law holds.
  • Non-inertial frame: a car accelerating forward (or a rotating frame). A ball on the floor appears to slide backward, as if a fictitious force pushed it; in the ground frame the ball simply keeps its velocity while the car speeds up.

Special relativity applies only to inertial frames. General relativity (Einstein's later theory, 1915) extends this to include gravity and accelerating frames, but that is beyond the HSC scope.

Stop & Check

Classify each as inertial or non-inertial, with a reason: (a) ISS in circular orbit, (b) a car at constant speed on a curved road, (c) a spacecraft drifting at 0.5$c$ with engines off.

Inertial frame: constant velocity, Newton's 1st law holds, no fictitious forces. Non-inertial frame: accelerating or rotating, fictitious forces (centrifugal, Coriolis) appear. Special relativity applies only to inertial frames. A car cornering at constant speed is non-inertial (centripetal acceleration). A coasting spacecraft is inertial.

Copy the definition of inertial frame and three inertial/non-inertial examples.

An inertial frame must be at rest relative to Earth's surface.

Einstein's second postulate means that a platform observer and a train passenger both measure the same speed for a laser beam fired from the train.

The Michelson-Morley experiment provided evidence against the existence of the luminiferous aether.

6

Apply and consolidate

Build gamma fluency across a range of speeds, then evaluate what Michelson-Morley did and did not show.

Activity 1, Lorentz Factor Calculations
ApplyBand 4

Build fluency with $\gamma$ across the range of relativistic speeds

  1. Calculate $\gamma$ for $v = 0.5c$, $0.9c$, and $0.99c$. Record your results and describe the trend.
  2. At $v = 0.866c$, $\gamma = 2$. How much does a 1 m rod contract? How slow does a moving clock tick compared with a stationary one?
  3. A muon is created at $0.99c$ with a proper lifetime of 2.2 µs. Calculate its lifetime in Earth's frame and the distance it travels before decaying.
  4. Explain why $\gamma \to \infty$ as $v \to c$, and why this means massive objects cannot reach the speed of light.
Activity 2, Analyse the Michelson-Morley Experiment
UnderstandBand 5

Evaluate the experiment's role in building special relativity

  1. Describe the apparatus and what Michelson and Morley expected to observe if the aether existed.
  2. What did they actually observe? Why was this result surprising?
  3. How did Einstein's postulates resolve the contradiction between classical mechanics and the Michelson-Morley result?
  4. Is a null result in physics ever "unimportant"? Justify your answer using this experiment as an example.
Evaluate7 marks

Evaluate the evidence provided by the Michelson–Morley experiment for and against the existence of the luminiferous aether.

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Unpack the verb

Evaluate requires a judgement supported by criteria, and the question explicitly asks for both sides. A null result is easy to misreport, so the marks turn on stating precisely what was and was not established.

7 marks. Plan: what the aether hypothesis required (1), the method and why it was sensitive enough (2), the result (1), what it counts against (1), what it does not prove (1), judgement (1).

The single most common error is writing that the experiment “proved the aether does not exist”. A null result cannot prove non-existence, and saying so is where the top band starts.
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Analyse6 marks

Analyse why Einstein’s principle of relativity applies only to inertial frames of reference.

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Unpack the verb

Analyse means take the restriction apart: what an inertial frame is, what the principle claims, and what goes wrong when the claim is extended to an accelerating frame. Defining the terms is necessary but is only the first mark.

6 marks. Plan: define the frame (1), state the principle (1), show the failure in a non-inertial frame with a worked instance (3), close (1).

The marks are in the failure case. Plan a specific example — an accelerating train or a rotating platform — before writing a word, because a general claim about “fictitious forces” is not worth much on its own.
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