Special Relativity — Inertial Frames and Postulates
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.
Practise this lesson
Four printable worksheets that build from the foundations up to exam-style questions — start at whatever level suits you.
You are on a train moving at 100 km/h. You shine a flashlight forward, and the light travels at $c$ relative to you.
- According to classical (Galilean) physics, how fast does the light travel relative to a person standing on the platform?
- If the speed of light is truly constant for all observers, what would the platform observer measure?
- 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:
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 $\gamma$
Core Content
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.
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)
Einstein's solution was radical: discard the aether entirely. In his 1905 paper "On the Electrodynamics of Moving Bodies," he proposed two postulates:
- The principle of relativity: The laws of physics are identical in all inertial frames of reference. No experiment can detect absolute motion.
- 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.
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 relativity: velocities add, but Maxwell's equations give a fixed $c$ for all observers. Michelson-Morley (1887) null result disproved the aether. Einstein's two postulates (1905): (1) laws of physics are identical in all inertial frames; (2) $c$ is constant for all observers. These lead directly to time dilation, length contraction and $E = mc^2$.
Write both postulates in full and note the Michelson-Morley significance.
The Michelson-Morley experiment was designed to detect:
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.
$$\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \dfrac{1}{\sqrt{1 - \beta^2}}$$
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)
- $v = 0.99c$: $\gamma = 7.09$
- $v = 0.999c$: $\gamma = 22.4$
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
The Lorentz factor appears in time dilation, length contraction, relativistic mass, and energy 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)
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!
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:
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 $v = 0.98c$.
An electron is accelerated to 0.98$c$ in a particle accelerator.
- Calculate the Lorentz factor $\gamma$.
- If the electron's proper lifetime is $\tau_0 = 2.0\,\mu\text{s}$, calculate its dilated lifetime in the lab frame.
- Calculate the electron's total energy and kinetic energy. ($m_e = 9.11\times10^{-31}$ kg)
- Through what potential difference must the electron be accelerated to reach this speed?
$\gamma = 1/\sqrt{1 - 0.98^2} = 1/\sqrt{1 - 0.9604} = 1/\sqrt{0.0396} = 1/0.199 = $ 5.03
$\Delta t = \gamma \tau_0 = 5.03 \times 2.0 = $ 10.1 $\mu$s
$E = \gamma m c^2 = 5.03 \times (9.11\times10^{-31})(3.00\times10^8)^2 = 5.03 \times 8.20\times10^{-14} = 4.12\times10^{-13}$ 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
$E_k = qV$, so $V = E_k/e = 2.06\times10^6$ eV $/ e$ = 2.06 MV
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:
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
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
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.
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.
Build fluency with $\gamma$ across the range of relativistic speeds
- Calculate $\gamma$ for $v = 0.5c$, $0.9c$, and $0.99c$. Record your results and describe the trend.
- 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?
- 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.
- Explain why $\gamma \to \infty$ as $v \to c$, and why this means massive objects cannot reach the speed of light.
Evaluate the experiment's role in building special relativity
- Describe the apparatus and what Michelson and Morley expected to observe if the aether existed.
- What did they actually observe? Why was this result surprising?
- How did Einstein's postulates resolve the contradiction between classical mechanics and the Michelson-Morley result?
- Is a null result in physics ever "unimportant"? Justify your answer using this experiment as an example.
A fresh five-question set drawn from this lesson's bank — feedback shown immediately. +5 XP per correct · +25 XP all correct
Pick your answer, then rate your confidence — that tells the system what to drill next.
AnalyseBand 5(3 marks) 1. (a) Distinguish between an inertial and a non-inertial frame of reference, giving one example of each. (b) State Einstein's two postulates of special relativity. (c) Explain how the Michelson-Morley experiment contributed to the development of special relativity. (3 marks)
1 mark: correct distinction with examples · 1 mark: both postulates clearly stated · 1 mark: null result + how it motivated discarding the aether
EvaluateBand 6(4 marks) 2. An electron ($m_e = 9.11\times10^{-31}$ kg) is accelerated to $v = 0.95c$. (a) Calculate $\gamma$. (b) Calculate its total relativistic energy in joules. (c) Calculate its relativistic kinetic energy. (d) Compare this with the classical kinetic energy $\frac{1}{2}mv^2$ and explain why classical mechanics fails at relativistic speeds. (4 marks)
1 mark: correct $\gamma$ · 1 mark: correct total energy · 1 mark: correct relativistic KE · 1 mark: classical comparison and explanation
ApplyBand 4(2 marks) 3. A particle has $\gamma = 3.0$. (a) Calculate its speed as a fraction of $c$. (b) If this particle's proper lifetime is $1.0\,\mu\text{s}$, what is its lifetime as measured in the laboratory frame? (2 marks)
1 mark: correct speed (v/c ≈ 0.943) · 1 mark: correct dilated lifetime (3.0 µs)
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Multiple choice
MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set drawn from the lesson bank.
Short Answer — Model Answers
Q1 (a): An inertial frame moves at constant velocity — Newton's first law holds with no fictitious forces (e.g. a train coasting at 200 km/h). A non-inertial frame accelerates or rotates — fictitious forces appear (e.g. a car braking). (1 mark)
Q1 (b): Postulate 1: The laws of physics are identical in all inertial frames — no experiment can detect absolute uniform motion. Postulate 2: The speed of light in vacuum is $c$ for all observers, regardless of the motion of the source or observer. (1 mark)
Q1 (c): Michelson and Morley found no path difference between perpendicular light beams (null result), showing no aether wind existed. This meant the speed of light was the same in all directions regardless of Earth's motion — directly contradicting classical aether theory and supporting Einstein's second postulate. (1 mark)
Q2 (a): $\gamma = 1/\sqrt{1-0.95^2} = 1/\sqrt{1-0.9025} = 1/\sqrt{0.0975} = 1/0.3122 = $ 3.20 (1 mark)
Q2 (b): $E = \gamma mc^2 = 3.20 \times (9.11\times10^{-31}) \times (3.00\times10^8)^2 = 3.20 \times 8.20\times10^{-14} = $ $2.62\times10^{-13}$ J (1 mark)
Q2 (c): $E_k = (\gamma-1)mc^2 = 2.20 \times 8.20\times10^{-14} = $ $1.80\times10^{-13}$ J (1 mark)
Q2 (d): Classical: $\frac{1}{2}mv^2 = \frac{1}{2}(9.11\times10^{-31})(0.95\times3\times10^8)^2 = 3.70\times10^{-14}$ J — about 5 times smaller than the relativistic value. Classical mechanics fails because it assumes mass is constant and speeds can be added simply, which breaks down near $c$ where $\gamma$ diverges and energy grows without bound. (1 mark)
Q3 (a): $\gamma = 3.0$: $1/\sqrt{1-\beta^2} = 3 \Rightarrow \beta^2 = 8/9 \Rightarrow v/c = \sqrt{8/9} \approx$ 0.943 (1 mark)
Q3 (b): $\Delta t = \gamma\Delta t_0 = 3.0 \times 1.0 = $ 3.0 µs (1 mark)
Five timed questions on special relativity foundations. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
Enter the arenaLook back at your Think First answers:
- Did you predict that classical physics gives $c + 100\,\text{km/h}$ for light speed relative to the platform? That is Galilean velocity addition: $u' = u + v$.
- Did you predict that both observers measure exactly $c$? This is Einstein's second postulate — and it defies classical intuition entirely.
- Did you predict that time must slow down and lengths must contract to keep $c$ constant? Time dilation and length contraction are the direct consequences of the constancy of the speed of light.
The hook answer: the platform observer also measures the laser at $c$ — not $c + 100\,\text{km/h}$. This forces time and space to be relative to the observer, not absolute.
The historical anchor: in 1887, Albert Michelson and Edward Morley at Case Western Reserve University, Cleveland, used an 11 m optical path interferometer and expected aether-drift fringe shifts of 0.4. They measured less than 0.01 — a profound null result. In June 1905, Albert Einstein at the Swiss Patent Office, Bern, published "On the Electrodynamics of Moving Bodies," using this null result to argue that the speed of light is constant for all inertial observers. Einstein's two postulates — the principle of relativity and the constancy of $c$ — are the foundation of everything in L12–L15.