Lesson 1 of 7Revolutions in Modern Physics

Unit 5 · Lesson 1 of 7

Special Relativity

Einstein's two postulates, and the surprising results that follow from taking them seriously: simultaneity that depends on the observer, moving clocks that run slow, moving objects that shorten, and mass that is a form of energy.

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Key things to remember

1Postulates: the laws of physics are the same in all inertial frames, and light in a vacuum travels at $c$ for every inertial observer.
2Lorentz factor: $γ = \frac{1}{1 - v ^{2} / c ^{2}} \geq 1$ .
3Time dilation: $Δ t = γ Δ t_{0}$ , where the proper time $Δ t_{0}$ is measured by a clock present at both events.
4Length contraction: $L = L_{0} / γ$ , only along the direction of motion; $L_{0}$ is the proper length.
5Relativistic momentum $p = γm v$ ; rest energy $E = m c^{2}$ ; kinetic energy $(γ - 1) m c^{2}$ .
6Events simultaneous in one frame need not be simultaneous in another.

Everything you have done in mechanics so far assumes that time ticks at the same rate for everyone and that a metre stick is a metre long no matter who measures it. Those assumptions work perfectly for cars, planets and baseballs. They fail — measurably — for anything moving at a large fraction of the speed of light. Special relativity is the theory that replaces them, and it rests on just two statements that sound harmless until you follow them to their conclusions.

A frame of reference is the point of view from which you measure positions and times. An inertial frame is one that is not accelerating — a train gliding at constant velocity, or the ground (to a good approximation). Newton's laws work the same way in every inertial frame: drop a ball inside a smoothly moving train and it falls straight down, exactly as it would on the platform. In the 1860s, though, Maxwell's equations predicted that electromagnetic waves travel at one definite speed, $c = 3.00 \times 1 0^{8} m/s$ . Speed relative to what? Physicists assumed an invisible medium, the ether, filled space. In 1887 the Michelson–Morley experiment looked for the change in light's speed as Earth moved through that ether and found none.

Einstein's two postulates (1905)

1. The relativity postulate: the laws of physics are the same in every inertial frame of reference. No experiment done inside a closed, non-accelerating lab can tell you whether the lab is 'really' moving. 2. The constancy of the speed of light: light in a vacuum travels at

c

for every inertial observer, regardless of the motion of the source or of the observer.

The second postulate is the strange one. If you run toward a ball thrown at you, you measure it coming faster. If you race toward a beam of light at half the speed of light, you still measure the light arriving at exactly $c$ . Speed is distance divided by time, so if the speed cannot change, distance and time themselves must be what differs between observers. Every result in this lesson comes from that idea.

Simultaneity depends on the observer

Einstein's thought experiment: a train moves past a platform, and a flash of light goes off at the exact centre of one carriage. A passenger at the centre sees the light reach the front and back walls at the same time — the walls are equal distances away and light travels at $c$ in both directions. A person on the platform also measures both pulses moving at $c$ , but in their frame the back wall is moving toward the light and the front wall is moving away from it. So for the platform observer, the light hits the back wall first. Two events that are simultaneous in one frame are not simultaneous in another. Neither observer is wrong; 'at the same time' is not a universal fact.

Time dilation

Fig 1The same light clock seen at rest and moving. The moving path is longer, but light still travels at c, so each tick takes longer.

A light clock is a pulse of light bouncing between two mirrors a distance $d$ apart. In the clock's own frame, one tick takes $Δ t_{0} = 2 d / c$ . Now watch the clock fly past at speed $v$ . From your frame the mirrors move sideways while the light is travelling, so the pulse follows a longer, diagonal zigzag. Light must still move at $c$ along that longer path, so each tick takes you longer to see complete. Applying the Pythagorean theorem to the diagonal path gives the time dilation equation. The clock is not broken — every process in the moving frame, from atoms vibrating to hearts beating, runs slow by the same factor as seen from your frame.

Δ t = \frac{Δ t _{0}}{1 - \frac{v ^{2}}{c ^{2}}} = γ Δ t_{0} γ = \frac{1}{1 - \frac{v ^{2}}{c ^{2}}}

Delta t0 is the proper time, measured by a clock at rest relative to the events. The Lorentz factor gamma is always 1 or more.

Which time is the proper time?

The proper time

Δ t_{0}

is measured by the one observer for whom both events happen at the same place — the person carrying the clock. It is always the shortest interval. Everyone moving relative to that clock measures the longer, dilated time

Δ t

. Some resources write

Δ t_{s}

and

Δ t_{m}

for the same two quantities.

Length contraction, momentum and mass-energy

L = L_{0} 1 - \frac{v ^{2}}{c ^{2}} = \frac{L _{0}}{γ} p = γm v E = m c^{2}

L0 is the proper length, measured at rest relative to the object. Contraction happens only along the direction of motion.

Length contraction: an object moving past you is shorter along its direction of motion than its proper length $L_{0}$ , measured by someone riding with it. Its width and height are unchanged. Relativistic momentum: as speed approaches $c$ , momentum grows without limit because $γ$ does, so pushing a particle faster takes ever more impulse and it can never reach $c$ . Mass-energy equivalence: an object at rest has rest energy $E = m c^{2}$ . Because $c^{2}$ is enormous, a tiny mass corresponds to a huge energy — the total energy of a moving object is $γm c^{2}$ , and its kinetic energy is $(γ - 1) m c^{2}$ .

Evidence	What was observed	What it supports
Michelson–Morley experiment (1887)	No change in the speed of light as Earth moves in different directions	The speed of light is the same for all observers; no ether is needed
Cosmic-ray muons	Muons made high in the atmosphere reach the ground, even though their mean lifetime is only about $2.2 μ s$	Time dilation (in the ground frame) or length contraction (in the muon's frame)
Atomic clocks flown on aircraft and in GPS satellites	Moving clocks drift from identical clocks on the ground; GPS must correct for this (along with a separate gravitational effect from general relativity)	Time dilation is real, not an illusion of measurement
Particle accelerators	Particles near $c$ need far more energy for tiny speed gains and never exceed $c$	Relativistic momentum $p = γm v$
Nuclear reactors, such as Ontario's CANDU stations	The products of fission have slightly less mass than the fuel	Mass converts to energy according to $E = m c^{2}$

Why you never notice any of this

At highway speed,

v / c

is about

1 0^{- 7}

, so

γ

differs from 1 only in the fifteenth decimal place. Even at

0.10 c

γ = 1.005

. Relativity does not overturn Newton for everyday motion — it contains Newtonian mechanics as the low-speed limit. The effects only become large above roughly

0.5 c

, where

γ

starts climbing steeply toward infinity.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Muons have a mean lifetime of

2.2 μ s

in their own rest frame. A muon created high in the atmosphere travels toward the ground at

0.995 c

. (a) What lifetime does an observer on the ground measure? (b) How far does it travel in that time, and how far would it get without time dilation?

Example 2

A spaceship is

120 m

long when measured at rest. How long does an observer on a space station measure it to be as it flies past at

0.80 c

Example 3

An astronaut travels to a distant station and back at a steady

0.60 c

. Mission control on Earth measures the round trip as

10.0

years. How much does the astronaut age? (Ignore the short periods of acceleration.)

Example 4

A proton (

m = 1.67 \times 1 0^{- 27} kg

) in an accelerator moves at

0.90 c

. Find its relativistic momentum and compare it with the classical value

m v

Where marks get lost

Mixing up which time is proper. The proper time $Δ t_{0}$ belongs to the observer who sees both events at the same place, and it is always the shorter one. If your 'dilated' time comes out smaller than the proper time, you have them reversed.
Multiplying the length by $γ$ instead of dividing. Moving objects get shorter, so $L = L_{0} / γ$ must be less than $L_{0}$ .
Contracting the dimensions perpendicular to the motion. Only the length along the direction of travel changes.
Forgetting to square the ratio: using $1 - v / c$ instead of $1 - v^{2} / c^{2}$ . At $0.80 c$ that gives $γ = 2.24$ instead of the correct $1.67$ .
Thinking time dilation is only an illusion caused by light taking time to arrive. It remains after that delay is accounted for, and clocks brought back together really do disagree.

Flashcards 9

Inertial frame of reference

Einstein's two postulates

Lorentz factor

Proper time

+ 5 more

Practice

Which statement is one of Einstein's two postulates of special relativity?

Calculate the Lorentz factor

γ

for an object moving at

0.80 c

. Give two decimal places.

A clock on a spacecraft measures

5.00 s

between two ticks. The spacecraft passes Earth at

0.60 c

. What time interval does an observer on Earth measure between those ticks, in seconds?

A rod has a proper length of

50.0 m

. How long is it, in metres, as measured by an observer it passes lengthwise at

0.60 c

How much energy, in joules, is equivalent to a mass of

2.0 g

? (You may type an answer like 1.2e14.)

Muons created about 10 km up in the atmosphere are detected in large numbers at sea level, even though a muon's mean lifetime at rest is only about

2.2 μ s

. Why?

What is the name for the time interval between two events measured by a clock for which both events occur at the same place?

A light flashes at the centre of a moving train carriage. A passenger at the centre sees the light reach both ends at once. What does an observer on the platform conclude?

You are an astronaut in a spaceship moving at

0.50 c

relative to Earth. You time your own heart rate using the ship's clock. What do you find?

Order these speeds from the smallest Lorentz factor to the largest.

1 $0.50 c$
2 $0.99 c$
3 $0.80 c$
4 $0.10 c$

Match each piece of evidence to the part of relativity it supports.

Michelson–Morley experiment

Muons reaching the ground

Particles in accelerators never exceeding c

Energy released by nuclear fission

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