PHY132H1/Unit 10

Unit 10 · Introduction to Physics II

Special Relativity

Two postulates, and everything else follows. The consequences — clocks disagreeing, lengths shrinking, simultaneity dissolving — sound like paradoxes and are simply what happens when the speed of light is the same for everyone.

By the end you can

• State the two postulates and see what forces them.
• Compute time dilation and length contraction.
• Explain the muon and twin cases without contradiction.
• Use $E = mc^2$ and relativistic momentum.

Why it matters & where it comes from

Why it exists

Maxwell's equations give one speed of light, with no reference to who is measuring. Either they held only in some special frame, or the speed really is the same for everyone. Experiment said the second, and Newtonian time had to give way.

In the real world

GPS satellites run clocks fast by about $38$ microseconds a day from the two relativistic effects combined. Uncorrected, positions would drift by kilometres within a day. Particle accelerators would not work at all without relativistic dynamics.

Story & open questions

Michelson and Morley tried in 1887 to detect Earth's motion through the ether and found nothing, to their own dissatisfaction. Einstein resolved it in 1905 by discarding absolute time rather than patching the ether.

The postulates

Definition

Relativity. The laws of physics are the same in every inertial frame. No experiment can tell you that you are "really" moving.

Light speed. Light travels at $c$ in vacuum for every observer, regardless of the motion of source or observer.

The first is unremarkable and dates to Galileo. The second is the one that breaks everything.

Example Why the second postulate is strange

Throw a ball at $10$ m/s from a train moving at $30$ m/s and someone on the platform measures $40$ m/s. Velocities add.

Shine a torch from the same train and the platform observer measures $c$ — not $c + 30$. Light does not add.

If both observers measure the same speed for the same beam while moving relative to each other, they cannot agree about distance and time. Something has to give, and it is simultaneity.

Rule The Lorentz factor

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

Always at least $1$, and rising steeply near $c$. Every relativistic formula is a Newtonian one with a $\gamma$ in it.

At $v = 0.1c$, $\gamma = 1.005$ — a half-percent effect.

At $v = 0.87c$, $\gamma = 2$.

At $v = 0.99c$, $\gamma = 7.1$.

This is why relativity is invisible in daily life: at everyday speeds $\gamma$ differs from $1$ by parts in a trillion.

Time and length

Rule Time dilation

$\Delta t = \gamma\,\Delta t_0$

where $\Delta t_0$ is the proper time, measured by a clock present at both events. Any other observer measures a longer interval: moving clocks run slow.

Rule Length contraction

$L = \dfrac{L_0}{\gamma}$

where $L_0$ is the proper length, measured in the object's own rest frame. Moving objects are shorter, along the direction of motion only.

Identify the proper quantity first

Half the errors in this unit come from mixing them up. Proper time is measured by the clock that is at both events. Proper length is measured in the frame where the object is at rest. Note they work oppositely: proper time is the shortest, proper length the longest.

Example The muon

Muons form about $10$ km up and live $2.2\,\mu$s. Even at $c$ they should travel only $660$ m. Yet they reach the ground.

Ground frame: at $v = 0.999c$, $\gamma = 22$, so the muon's clock runs slow and it survives $48\,\mu$s of our time — enough for $14$ km.

Muon frame: the muon lives its ordinary $2.2\,\mu$s, but the $10$ km of atmosphere is contracted to $10/22 = 450$ m, which it crosses easily.

Different explanations, same outcome. Both frames are correct and neither is preferred — which is exactly what the first postulate demands.

Example The twin case

One twin travels at high speed and returns younger. If motion is relative, why is it not symmetric?

Because it is not symmetric. The traveller turns around, and so changes inertial frame; the stay-at-home does not.

The postulates apply to inertial frames. The traveller accelerates and can feel it, which breaks the symmetry.

So there is no paradox, only an asymmetry that the informal statement of the puzzle hides.

Momentum, energy, mass

Rule The relativistic forms

$p = \gamma m v$

$E = \gamma m c^2$ — total energy

$E_0 = mc^2$ — rest energy, present even at rest

$KE = (\gamma - 1)mc^2$ — and this reduces to $\tfrac12 mv^2$ at low speed, as it must

Example Why nothing reaches $c$

As $v \to c$, $\gamma \to \infty$, so momentum and energy both diverge.

Accelerating a massive object to $c$ would take infinite energy. It is not a technological limit but an arithmetic one.

Massless particles are the exception: with $m = 0$ the formulas allow travel at exactly $c$, and photons must always move at $c$ — they are never at rest in any frame.

Example What $E = mc^2$ actually says

Mass is a form of energy. Because $c^2$ is enormous, a tiny mass carries a vast energy: one gram is about $9 \times 10^{13}$ J, roughly a day's electricity for a small city.

In the Sun, four hydrogen nuclei fuse into helium that is about $0.7$ per cent lighter, and that missing mass is the sunlight.

Chemical reactions do the same thing, but the mass change is around a billionth — far too small to notice, which is why chemistry could develop with mass treated as conserved.

Common pitfalls

• Confusing proper time with proper length. Identify which frame each is measured in.
• Applying length contraction perpendicular to the motion. Only the direction of travel contracts.
• Treating the twin case as symmetric. The traveller accelerates.
• Using $p = mv$ at relativistic speed.
• Forgetting that everything must reduce to the Newtonian result as $v \to 0$. That check catches most algebra errors.

Practice

Work each fully before opening the answer. Original exercises written for this guide in the style of the course's problem sets.

Practice 1 · Lorentz factor

Find $\gamma$ at $v = 0.6c$.

Answer

$\gamma = 1/\sqrt{1 - 0.36} = 1/0.8 = 1.25$.

Practice 2 · Dilation

A clock on that ship ticks $1$ s. How long on Earth?

Answer

$1.25$ s. The moving clock runs slow from our point of view.

Practice 3 · Contraction

The ship is $100$ m long at rest. How long do we measure it?

Answer

$100/1.25 = 80$ m.

Practice 4 · Which is proper

Who measures the proper lifetime of a particle?

Answer

An observer moving with it, for whom creation and decay happen at the same place.

Practice 5 · Rest energy

Find the rest energy of a $1$ kg mass.

Answer

$E_0 = (1)(3 \times 10^8)^2 = 9 \times 10^{16}$ J.

Practice 6 · The twin

Why is the twin situation not symmetric?

Answer

The traveller changes inertial frame when turning around and feels the acceleration. The stay-at-home never does, so the two are not interchangeable.

Practice 7 · Low-speed check

Show $KE = (\gamma - 1)mc^2$ becomes $\tfrac12 mv^2$ at small $v$.

Solution

For small $v/c$, $\gamma \approx 1 + \tfrac12 v^2/c^2$. Then $(\gamma - 1)mc^2 \approx \tfrac12 mv^2$. Every relativistic formula must pass this test, and it is the fastest way to check one.

Practice 8 · GPS

Why must GPS satellites correct for relativity?

Answer

Their clocks disagree with ground clocks by tens of microseconds a day. Since positions are computed from light travel times, an uncorrected microsecond is hundreds of metres of error, accumulating to kilometres within a day.

Expanded topic checklist

Course coverage. This unit is part of the complete PHY132H1 topic map. It should cover the following items before you move on:

• The postulates. Both statements, and why the second forces the rest.
• Dilation and contraction. The formulas and identifying the proper quantity.
• Resolving the puzzles. Muon and twin cases from both frames.
• Dynamics. Relativistic momentum and energy, rest energy, and the low-speed limit.

Textbook/source alignment

The calendar closes its topic list with "special relativity". Check the syllabus on Quercus for the assigned problems; this material usually occupies the last two weeks.

Core idea map

A constant light speed for all observers forces time and length to depend on the observer. Every formula is Newtonian with a factor of $\gamma$, and every one reduces to Newton at low speed. Nothing is discarded, only corrected.

Study routine. Name the two frames explicitly before computing, and say which measures the proper quantity. Then check the answer reduces sensibly as $v \to 0$.

Deep practice extensions

Use these after the main practice set. They push on definitions and edge cases rather than routine calculation.

Extension 1 · The radical postulate

Which postulate breaks Newtonian intuition?

Guide

The constancy of $c$. Light does not obey ordinary velocity addition, which forces time and space to adjust instead.

Extension 2 · Proper quantities

Which frame measures proper time, and which proper length?

Guide

Proper time: the frame where both events happen at the same place. Proper length: the frame where the object is at rest.

Extension 3 · Direction

Does an object contract in every direction?

Guide

No, only along the direction of motion. Perpendicular dimensions are unchanged.

Extension 4 · The speed limit

Why can a massive object not reach $c$?

Guide

Its energy $\gamma mc^2$ diverges as $v \to c$, so infinite energy would be required.

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