What is the difference between proper time and dilated time?

Proper time t0 is measured by a clock present at both events, so the events happen at the same place in that clock's frame. Any observer who sees that clock moving measures a longer, dilated time t = t0/√(1 − v^2/c^2).

Which relativity equations are on the HSC data sheet?

The 2019 data sheet gives time dilation t = t0/√(1 − v^2/c^2), length contraction l = l0√(1 − v^2/c^2), relativistic momentum pv = m0v/√(1 − v^2/c^2) and E = mc^2, plus c = 3.00 × 10^8 m s^−1. The Lorentz factor γ is not printed.

Is the twin paradox evidence for special relativity?

No. It is a thought experiment, and the travelling twin accelerates, so that twin is not in an inertial frame. For evidence, use real experiments: muons reaching the ground, the Hafele–Keating clocks and particle lifetimes in accelerators.

Does mass increase at high speed?

The HSC course uses relativistic momentum with a fixed rest mass m0: the momentum grows without limit as v approaches c. Say that, rather than 'the mass increases'.

Why can't anything travel faster than light?

For anything with mass, the momentum and energy needed grow without limit as the speed approaches c, so reaching c would take infinite energy. Light itself has no rest mass and always travels at c.

A spaceship moving at 0.60c towards Earth shines a laser beam forwards. What speed do observers on Earth measure for the laser light?

It depends on the frequency of the light.

A spaceship flies past a space station at 0.5c. Who measures the proper length of the spaceship?

The crew on board the spaceship

Both measure the same length, so both measure proper length

Nobody, because the ship is moving

Why can't a proton in an accelerator be pushed to the speed of light?

Its relativistic momentum grows without limit as v approaches c, so infinite energy would be needed.

The accelerator's magnets cannot work faster than light.

Its charge falls to zero as it approaches c.

Time stops for the proton, so it stops speeding up.

An electron and a positron annihilate, and all of their mass becomes gamma-ray energy. How much energy is released? (me = 9.109 × 10^−31 kg)

0 J, because their charges cancel

HSC Physics · Module 7 · The Nature of Light · 15 min read

Special relativity

Special relativity is what happens when you take one fact seriously: every observer measures the same speed of light. From that, moving clocks run slow, moving objects are shorter, nothing with mass can reach the speed of light, and mass is a store of energy. HSC questions test all four, and they nearly always turn on one decision: which observer measures which quantity.

This guide covers the two postulates and the evidence for them, time dilation and length contraction (with proper time and proper length made clear), the muon evidence worked in both frames, relativistic momentum and E = mc². Then the mistakes that cost marks, a quick quiz and a full worked example.

What you need to know

For each of Einstein's two postulates, analyse and evaluate the evidence for or against it: the speed of light in a vacuum is the same for every observer, and no inertial frame of reference is preferred over another.
Investigate how length contraction, l = l₀√(1 − v²/c²), and time dilation, t = t₀/√(1 − v²/c²), are supported by Einstein's thought experiments and later experiments, and analyse cases quantitatively: cosmic-ray muons reaching the ground, the Hafele–Keating atomic clocks, particle accelerator results and cosmological evidence.
Describe the consequences and uses of relativistic momentum, p_v = m₀v/√(1 − v²/c²), including the upper limit special relativity puts on a particle's speed.
Calculate the energy released when mass becomes energy with E = mc²: for example the Sun's output, particle–antiparticle annihilation and burning ordinary fuels.

The equations (from the 2019 HSC data sheet unless marked):

t = t₀ / √(1 − v²/c²)Time dilation. t₀ is the proper time; t, measured by an observer the clock moves past, is always longer.

l = l₀ √(1 − v²/c²)Length contraction along the direction of motion. l₀ is the proper length; l is always shorter.

p_v = m₀v / √(1 − v²/c²)Relativistic momentum. m₀ is the rest mass. It grows without limit as v approaches c.

E = mc²Mass–energy equivalence. c = 3.00 × 10⁸ m s⁻¹ (data sheet). Mass in kg gives energy in J.

γ = 1 / √(1 − v²/c²) not on the data sheetThe Lorentz factor, a shorthand: t = γt₀, l = l₀/γ, p_v = γm₀v. It is 1 at low speed and grows quickly near c.

Dot points: PH-M7-15, PH-M7-16, PH-M7-17, PH-M7-18 (2017 Physics syllabus, HSC 2026 and 2027), in our own words.

Mini lesson

Einstein's two postulates and the evidence

An inertial frame of reference is one that is not accelerating: a smooth train at constant velocity, not one braking or going round a bend. Special relativity rests on two postulates:

1. The laws of physics are the same in every inertial frame. No experiment inside a closed, steadily moving lab can tell you how fast it is moving: no inertial frame is preferred.
2. The speed of light in a vacuum, c, is the same for every inertial observer, whatever the motion of the source or the observer.

Evidence: the Michelson–Morley experiment (1887) looked for a change in the speed of light as Earth moved through a supposed 'aether' and found none (a null result), which fits the second postulate and removes the need for a preferred frame. In 1964 at CERN, gamma rays given off by neutral pions travelling at more than 0.999c were measured moving at c, not nearly 2c. No experiment since has found a preferred frame or a light speed that depends on the source.

Postulate 2 clashes with everyday (Galilean) velocity addition: a torch on a fast train should give light at c + v. Einstein kept the postulates and gave up absolute time and space instead. Time dilation and length contraction follow.

Time dilation and proper time

Einstein's light clock is a pulse of light bouncing between two mirrors. On board, the light goes straight up and down. To someone the clock moves past, the light follows a longer zigzag path, but at the same speed c, so each tick takes longer. Moving clocks run slow: t = t₀/√(1 − v²/c²).

Proper time t₀ is the time between two events measured by a clock that is present at both of them, so the events happen at the same place in that clock's frame. It is always the shortest time any observer measures. Every other inertial observer, who sees the clock move, measures the longer, dilated time t. There is more on choosing it in our page on proper time in relativity.

The effect is tiny at everyday speeds: at 300 m s⁻¹ (a fast jet) √(1 − v²/c²) differs from 1 by about 5 parts in 10¹³. Even at 0.14c it is only a 1% effect; it becomes large only close to c.

Same light speed, longer path: the observer who sees the clock moving measures a longer time between ticks (t > t₀).

Length contraction and proper length

Moving objects are shorter along their direction of motion: l = l₀√(1 − v²/c²). Lengths at right angles to the motion do not change.

Proper length l₀ is the length measured in the frame where the object is at rest. A spaceship's proper length is measured by its crew; a stretch of road's proper length is measured by people standing beside it. Anyone the object moves past measures it shorter.

Time dilation and length contraction are two views of the same thing. If a ship's crew time a trip with their own clock (proper time), the distance they cover, as they see it, is the contracted length of the route.

The evidence: muons, clocks, accelerators and supernovae

Cosmic-ray muons. Muons are made when cosmic rays hit the upper atmosphere, about 15 km up. Their mean lifetime at rest is about 2.2 μs, and even at 0.995c that is enough to travel only about 660 m. Yet large numbers reach the ground. Both frames explain this, and a full answer gives both:

Earth's frame (time dilation): at 0.995c the Lorentz factor is about 10, so the muons' mean lifetime as measured on Earth is about 22 μs, long enough for many to cover 15 km.
The muon's frame (length contraction): the muon lives its normal 2.2 μs, but the atmosphere rushing past is contracted to about 1.5 km, short enough for many to get through.

Same result, two explanations: Earth sees a longer muon lifetime; the muon sees a thinner atmosphere (not to scale).

Hafele–Keating (1971). Caesium atomic clocks were flown around the world on airliners, eastward and westward, and compared with clocks left on the ground. The eastward clocks lost about 59 ns and the westward clocks gained about 273 ns, close to the predicted values (the predictions include a gravitational effect from general relativity as well as time dilation).

Particle accelerators. Fast unstable particles live longer by exactly the predicted factor. In a 1977 CERN experiment, muons circling a storage ring with a Lorentz factor of about 29 lived about 29 times longer than muons at rest.

Cosmological evidence. The brightening and fading of very distant supernovae is stretched out in time by the amount predicted for their recession, a time-dilation effect seen across the universe.

The twin paradox is a thought experiment, not evidence, and the travelling twin accelerates (a non-inertial frame), so don't use it as an example of experimental support.

Relativistic momentum and the speed limit

At low speeds momentum is mv. Near the speed of light it is p_v = m₀v/√(1 − v²/c²), where m₀ is the rest mass. As v gets close to c the denominator heads to zero, so the momentum grows without limit.

That sets the speed limit. Pushing a particle with a force keeps increasing its momentum and its energy, but the speed only creeps closer to c. Reaching c would need infinite momentum and so infinite energy, so nothing with mass can reach or pass the speed of light. Particle accelerators show this: adding more energy to electrons that are already very close to c barely changes their speed.

Say 'its momentum (and energy) increase without limit', rather than 'its mass becomes infinite'. HSC questions use relativistic momentum, not a changing mass.

At low speed the two agree. Near c the relativistic momentum rises without limit, so v = c can't be reached.

E = mc²: mass is a store of energy

Einstein showed that mass and energy are equivalent: E = mc². When a system gives out energy, its mass drops by Δm = E/c². Because c² is huge (9.0 × 10¹⁶ m² s⁻²), a tiny mass is a vast amount of energy.

The Sun gives out about 3.8 × 10²⁶ W, so its mass falls by 3.8 × 10²⁶ ÷ (3.00 × 10⁸)² ≈ 4.2 × 10⁹ kg every second, through fusion in its core.
Annihilation: an electron and a positron (each 9.109 × 10⁻³¹ kg) meet and their whole mass becomes gamma-ray energy: 2 × 9.109 × 10⁻³¹ × (3.00 × 10⁸)² = 1.64 × 10⁻¹³ J (about 1.02 MeV).
Burning fuel: burning 1 kg of petrol releases about 46 MJ, so the products are lighter by only about 5 × 10⁻¹⁰ kg. True, but far too small to measure, which is why chemistry seems to conserve mass.

Common mistakes

Putting the time measured on Earth in as t₀ when the clock is on the spaceship (swapping t and t₀).t₀ is measured by the clock present at both events. The dilated time t is always longer: check your answer is bigger than t₀.
Saying 'the stationary observer measures proper time'.Every inertial observer thinks they are stationary. Proper time belongs to the frame where both events happen at the same place, so name that frame.
Contracting lengths at right angles to the motion, or using l₀ for the length seen by a passing observer.Only lengths along the motion contract. l₀ is measured in the object's own rest frame and is the longest length anyone measures.
Explaining the muon evidence with time dilation only.Give both frames: Earth sees the muon's lifetime dilated; the muon sees the atmosphere contracted. Both predict that many muons reach the ground.
Saying a particle can't reach c because 'its mass becomes infinite'.Use relativistic momentum: p_v = m₀v/√(1 − v²/c²) grows without limit as v approaches c, so infinite energy would be needed.
Using the twin paradox as experimental evidence, or applying special relativity to an accelerating rocket.Cite real experiments (muons, Hafele–Keating, accelerators). Special relativity's equations are for inertial frames, so the speed must be constant.

Worked example

In a particle accelerator, a beam of charged pions travels along a straight beam pipe at a constant 0.980c. In the pions' own rest frame their mean lifetime is 2.60 × 10⁻⁸ s.

(a) Calculate the mean lifetime of the pions as measured in the laboratory. (2 marks)

(b) Calculate the mean distance a pion travels along the pipe, as measured in the laboratory, before it decays. (2 marks)

(c) In its own frame a pion lives only 2.60 × 10⁻⁸ s. Use length contraction to show that an observer moving with the pion agrees it reaches the point where the laboratory sees it decay. (2 marks)

(6 marks)

I'll think out loud, the way you should in the exam, using KNOW-UD: Knowns, Not known, Organise, Work it out, Units, Direction.

Prediction first. At 0.980c, 1 − v²/c² ≈ 1 − 0.96 = 0.04, and √0.04 = 0.2, so the Lorentz factor is about 5. I predict a lab lifetime about 5 × 26 ns ≈ 130 ns. The pion covers about 0.3 m every nanosecond, so in 130 ns it goes about 38 m. Without relativity it would manage only about 7.6 m.

K · Knowns

I sketch the beam pipe, which is at rest in the lab, and the pion moving along it. Then the key decision: which time is proper? The pion's lifetime in its own rest frame is measured by a 'clock' that travels with it, present at both its creation and its decay. So 2.60 × 10⁻⁸ s is t₀.

v = 0.980c, c = 3.00 × 10⁸ m s⁻¹
t₀ = 2.60 × 10⁻⁸ s (pion's rest frame: proper time)
the pipe's length is a proper length in the lab frame

Knowns on the sketch: v = 0.980c, t₀ = 2.60 × 10⁻⁸ s in the pion's frame. Unknowns: the lab lifetime t and the lab distance d.

N · Not known

Part (a) wants a time in the lab frame, part (b) a distance in the lab frame, part (c) a length in the pion's frame to compare.

(a) t = ? s
(b) d = ? m
(c) l (lab distance as seen by the pion) = ? m

O · Organise

I write each step in symbols. In (b) I use the lab's own time and speed together, so I stay in one frame. In (c) the lab distance d is a proper length (it's measured in the frame where the pipe is at rest), so the pion sees it contracted.

t = t₀ / √(1 − v²/c²)
d = vt (all lab-frame quantities)
l = d √(1 − v²/c²), compare with v × t₀ (all pion-frame quantities)

W · Work it out

The denominator first, kept to full calculator precision, then each part.

√(1 − 0.980²) = √(1 − 0.9604) = √0.0396 = 0.19900
(a) t = 2.60 × 10⁻⁸ ÷ 0.19900 = 1.3065 × 10⁻⁷ s
(b) d = 0.980 × 3.00 × 10⁸ × 1.3065 × 10⁻⁷ = 38.41 m
(c) l = 38.41 × 0.19900 = 7.644 m
distance the lab moves past the pion in its lifetime: v × t₀ = 0.980 × 3.00 × 10⁸ × 2.60 × 10⁻⁸ = 7.644 m
The two agree: in the pion's frame the 38.4 m stretch of pipe is contracted to 7.64 m, and that is exactly how far the pipe moves past the pion in its 26.0 ns life. Both frames agree on where it decays.

U · Units

Seconds for the time, metres for the distances. The data are to three significant figures, so I give three.

(a) t = 1.31 × 10⁻⁷ s
(b) d = 38.4 m
(c) l = 7.64 m = v × t₀

D · Direction

These are a time and lengths, so instead of a direction I do the relativity sense checks: the dilated time must be longer than t₀ (1.31 × 10⁻⁷ s > 2.60 × 10⁻⁸ s, yes), and the contracted length must be shorter than the proper length (7.64 m < 38.4 m, yes). The speed used was below c throughout.

t > t₀ and l < l₀, as they must be.

(a) The lab measures a mean lifetime of 1.31 × 10⁻⁷ s. (b) A pion travels a mean distance of 38.4 m along the pipe. (c) In the pion's frame that 38.4 m is contracted to 7.64 m, exactly the distance the pipe moves past the pion in 2.60 × 10⁻⁸ s, so both frames agree.

Check against the prediction. I predicted about 130 ns and about 38 m, with a Lorentz factor near 5. The answers are 131 ns and 38.4 m (the factor is 5.03), so they match. The 7.6 m 'no relativity' distance turns up again in (c) as the contracted length, which is the point of the question.

HSC advice

What markers keep saying

Name the frames of reference and say which observer measures the proper time or the proper length, with a reason (the events happen at the same place in that frame; the object is at rest in that frame).NESA marking feedback, 2022, summarised
For muons, explain both frames: time dilation of the muon's lifetime in Earth's frame and length contraction of the atmosphere in the muon's frame. Use the full names of the effects, not abbreviations.NESA marking feedback, 2024, summarised
For a thought experiment on time dilation, use the light clock: the twin paradox was not accepted. For experimental evidence, name a specific experiment and say what its results showed.NESA marking feedback, 2019, summarised
Frame a long relativity answer with Einstein's two postulates, stated precisely (the speed of light in a vacuum is the same for all inertial observers; the laws of physics are the same in all inertial frames), then back each consequence with specific experimental data.NESA marking feedback, 2025, summarised
Explain the speed limit with relativistic momentum (it grows without limit as v approaches c), rather than leaning on the word 'infinite'.NESA marking feedback, 2024, summarised
Decide which time is dilated (t) and which is proper (t₀) before you substitute, and work out the square-root denominator carefully. Check that the dilated time comes out longer than the proper time.NESA marking feedback, 2023, summarised

Summaries in our own words. Read NESA's own feedback with the exam packs: 2019 · 2022 · 2023 · 2024 · 2025.

How often it is examined

Special relativity questions were worth 49 marks across the seven HSC Physics exams from 2019 to 2025 (about 7.0 marks a paper), and appeared in 7 of 7 papers.

Marks per paper (Module 7, Light and Special Relativity). Source: CortexHSC's own tagging of the NESA HSC Physics exams 2019–2025, each question counted under its main topic, so treat the numbers as close, not exact.

FAQ

What is the difference between proper time and dilated time?: Proper time t₀ is measured by a clock present at both events, so the events happen at the same place in that clock's frame. Any observer who sees that clock moving measures a longer, dilated time t = t₀/√(1 − v²/c²).
Which relativity equations are on the HSC data sheet?: The 2019 data sheet gives time dilation t = t₀/√(1 − v²/c²), length contraction l = l₀√(1 − v²/c²), relativistic momentum p_v = m₀v/√(1 − v²/c²) and E = mc², plus c = 3.00 × 10⁸ m s⁻¹. The Lorentz factor γ is not printed.
Is the twin paradox evidence for special relativity?: No. It is a thought experiment, and the travelling twin accelerates, so that twin is not in an inertial frame. For evidence, use real experiments: muons reaching the ground, the Hafele–Keating clocks and particle lifetimes in accelerators.
Does mass increase at high speed?: The HSC course uses relativistic momentum with a fixed rest mass m₀: the momentum grows without limit as v approaches c. Say that, rather than 'the mass increases'.
Why can't anything travel faster than light?: For anything with mass, the momentum and energy needed grow without limit as the speed approaches c, so reaching c would take infinite energy. Light itself has no rest mass and always travels at c.

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