Before 1905 Albert Einstein was an unknown 25-year-old clerk in the Swiss patent office. Newton’s laws ruled supreme in the past 200 years. Many people.

Chapter 9Special Relativity(Time and length)

Before 1905Albert Einstein was an unknown 25-year-old clerk in the Swiss patent office.

Newton’s laws ruled supreme in the past 200 years.Many people were of the opinions that anything worth discovering in Physics had been discovered.

Annus Mirabilis[The Extraordinary

Year]Published several papers:1. Brownian motion (settled the debate on the

existence of atoms)2. Photoelectric effect (kick-started quantum

mechanics, eventually won the Nobel Prize for the work)

3. Special relativity (profound change in the foundation of physics)

The only other annus mirabilis in history:1666 when Newton wrote Principia

Ended the year with a paper with the equation E=mc2, arguably the most famous equation in physics.

The Special Theory of Relativity

Special relativity is the study of motion. It led to astonishing discovery about the nature of space and time.

It laid the foundation for an even more profound work of Einstein, the General Theory of Relativity.

It will take him another 10 years to complete General Relativity. Einstein himself said that compared with General Relativity, Special Relativity was a child’s play.

Some Implications1. Events that are simultaneous for one observer

may not be simultaneous for another.2. When two observers measure a time interval or

a length, they may not get the same results.3. Newton’s Laws and equations for kinetic energy

needs to be revised.4. The equivalence of energy and mass (E=mc2)5. Space and time are no longer considered

separate entities, and is combined into four-dimensional space-time.

6. Expansion or contraction of the universe.7. The existence of exotic objects like black holes,

worm holes.

What is time?

You will learn why Einstein is regarded as a great genius. He discovered the most profound things about the universe by asking some of the most deceptively simple questions.

Some questions

• What does a beam of light look like if you are traveling at the speed of light? In other words, what does a “stationary” beam of light look like.

• If you are holding a mirror as you fly, what will you see in the mirror? Will you see your own face?

We will discuss the answer to these questions later.

Some facts about light

Light is an electromagnetic wave, described by Maxwell’s equation. According to Maxwell’s equation the speed of light is given by:

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c =1

ε 0μ 0

≈ 3×108 m /s

This has been confirmed by experiment, but some puzzles remains.If light is a wave, what exactly is waving? Water waves travels on water, sound waves travels in the air, what is the medium of light? Ether??? If so, how can we detect ether?

Can light goes faster?

We just mentioned c=3×108m/s, can we somehow make light go faster?

Can light go faster?

MotionWhen you say you are driving at 60mph, what exactly does that mean?It means your car is moving at 60mph relative to the surface of the earth.

If you are piloting a spaceship, suppose you are moving at 1000km/s, what exactly does that mean? Relative to earth? What if you are far from earth? What if the earth is destroyed? Perhaps relative to space??? But then what is space?

Motion is relative

Einstein believes there is no absolute space, and hence no absolute motion. It makes no sense to say one is moving at 100m/s without saying “relative to what”.

If that is true, then what does it mean to say the speed of light is c=3×108m/s? What is the speed relative to???

Einstein’s answerWhat is the speed c=3×108m/s relative to?

Einstein says the speed is 3×108m/s relative to everything![actually it is relative to any inertia frame, but more about that later]If that doesn’t sound strange to you, suppose someone tells you Einstein is driving a car on the freeway at 50mph. The odd thing is, whether you are standing on the ground, driving behind him or driving toward him, you will still find his speed to be exactly 50mph. How is that possible?

The constancy of cEinstein believed Maxwell’s equations dictated the constancy of the speed of light, and he was also influenced by philosophy that there is no absolute space.

In fact, around the time Einstein was coming to this conclusion, Albert Michelson performed the famous Michelson-Morley experiment, experimentally proving that c=3×108m/s independent of your state of motion.

Einstein, however, claimed that he had not heard of the experiment at the time.

Inertial frameWhen you are accelerating in a car (when you are stepping on the gas or making a sharp turn), you will feel a force acting on you.

Inertial frames are observers who do not experience such force. Example:• An spaceship moving freely in outer space with

all the engines turned off.• An object under free fall in a gravitational field.

Two Postulates1. The principle of relativity: All the laws of

physics are the same in all inertial reference frames.

2. The constancy of the speed of light: The speed of light in vacuum has the same value in all inertial frames, regardless of the velocity of the observer or the velocity of the source emitting the light.

From these two simple postulates Einstein constructed special relativity and started a revolution in physics.

So what is wrong here?

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What exactly is velocity anyway?

v =Δx

ΔtSo what exactly is space (x) and time (t)?

The problem with time

Two events E1, E2:E1: Sam jumps on the right side of the roomE2: Sally jumps on the left side of the room

Somebody says:E1 and E2 happened at the same time.

What is wrong with this statement?

The relativity of simultaneity

The Relativity of Simultaneity

Simultaneous for Sam but not for Sally. Einstein decided there was something wrong with our understanding of time.

The relativity of simultaneity

The relativity of simultaneity

Einstein’s light clockSince the speed of light is fixed, we can bounce a photon up and down to measure time.

Δt0 =2D

c

Sally put such a light clock in her spaceship. From her point of view:

What happens if the clock is moving?

Δt0 =2D

c

Sam is on the ground watching Sally and her clock flies by.

The situation

Sally

Sam

Sam’s viewSam:

L2 =D2 + v2 (Δt2)2

L =cΔt2

⇒14

c2Δt2 =D2 +14

v2Δt2

⇒14(c2 −v2 )Δt2 =D2

⇒ Δt=2Dc

1

1−(v/ c)2=

Δt01−(v/ c)2

Sally:

Δt0 =2Dc

Time dilationSally:

Δt0 =2Dc

Sam:

Δt=Δt0

1−(v/ c)2

Δt0 < Δt

Two points of viewAccording to Sally, the time passed is:

Δt0 =2Dc

According to Sam, a longer time has passed:

Δt=Δt0

1−(v/ c)2=γΔt0

where γ ≡1

1−(v/ c)2≥1

Define β =v/ c⇒ γ ≡1

1−β 2

Δt0 < Δt

The γ factorγ ≡

1

1− β 2 where β = v / c

γ ≥ 1 always, γ = 1 when β = 0

Examples (lab frame vs moving frame) :

Δt = γΔt0

ΔL =1

γΔL0 (more on this later)

What if I use a different clock?

• The first postulate ensures that there cannot be any difference between a light clock and a “normal” clock, including your biological clock.

• If your wrist watch and the light clock do not show the same time, by comparing a light clock with a wrist watch you could tell you are in absolute motion or not (when there really is no such thing).

• Conclusion: Time really does slow down!

The laws of physics are the same for observers in all inertial reference frames. No one frame is preferred over any other.

Time Dilation Verification – Muon

DecaysMuon decay in 2.2μs when it is at rest (relative to the lab).

But when it travel fast, it decays much slower (and hence travel much longer than expected).

This is time dilation for fast moving muons.

Time Dilation Verification – Muon Decays

Muons are unstable particles that have the same charge as an electron, but a mass 207 times more than an electron

Muons have a half-life of ∆tp = 2.2 µs when measured in a reference frame at rest with respect to them (a)

Relative to an observer on the Earth, muons should have a lifetime of

γ ∆tp (b)

A CERN experiment measured lifetimes in agreement with the predictions of relativity

ExampleCalculate the γ factor for a muon traveling at 0.9994c.

How much time has passed in the lab for a muon of lifetime 2.2μs to decay?

γ =1

1 − (v / c)2= 28.87

⇒ Δt = γΔt0 = 28.87(2.2μ s) = 63.51μ s

It travels 28.87 times longer than expected.

Twins ParadoxΔt = γΔt0

where γ ≡1

1 − β 2

Δt : Time on Earth

Δt0 : Time on spaceship

GPS and Relativity• GPS (Global Positioning System) satellites are moving

at 14000km/h. Each satellite carries atomic clock within 30ns precision. Fix your position to within 10m.

• Special relativity tells us time on the satellites slows down by 7μs per day.

• General relativity (gravitational effect) tells us time on the satellite goes faster by 45μs per day.

• Overall clocks on satellites gain 38μs per day, which is 38000ns.

• If not taken into account, GPS fails within minutes, generating an error of 10km per day.

Calculations with LightyearIn this chapter, you will often encounter the unit

of lightyear. A lightyear is defined as the distance light travels in a year. It is a unit of distance, not a unit of time.

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1ly = 1c ⋅1y = (3×108 m /s)(365 × 24 × 60 × 60s) = 9.46 ×1015 m

In a calculation in relativity, however, you will almost never convert lightyear into meters. You are almost always better off writing 1ly=(1c)(1y).For example, how long does it take for a spaceship travelling at 0.6c to get to a galaxy 10ly away (from earth’s view)?

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t =10ly

0.6c=

(10c)(1y)

0.6c=

10

0.6y = 16.67y

You should never do the following :

t =10(9.46 ×1015 m)

0.6(3 ×108 m /s)= 5.26 ×108 s = 16.67y

Relativity of lengthFrom the platform:

Length of the platform = L0

Time for man to run past=ΔtL0 =vΔt

From the man's view:

Length of the platform = LTime for man to run past=Δt0L =vΔt0

v

L0

L=

ΔtΔt0

=γ ⇒ L =1γ

L0

Therefore, the platform is shorter by a factor of γ from the man’s point of view.

Length Contraction The measured distance between two

points depends on the frame of reference of the observer

The proper length, L0, of an object is the length of the object measured by someone at rest relative to the object

The length of an object measured in a reference frame that is moving with respect to the object is always less than the proper length This effect is known as length contraction

Length Contraction – Equation

Length contraction takes place only along the direction of motion

Things shrink as they move

Trip to a StarPlanet X is at a distance 10ly away from Earth (as observed in the Earth’s frame). How long does it take for an astronaut traveling at 0.95c (relative to Earth) to get to X?

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Method 1 (by time dilation) :

tE =lE

v=

10c ⋅yr

0.95c= 10.52yr ⇒ tA =

tE

γ=

lE

vγ= 3.3yr

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Method 2 (by length contraction) :

lA =lE

γ= 3.125ly ⇒ tA =

lA

v=

lE

vγ= 3.3yr

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γ =1

1− 0.952= 3.2

Trip to a StarPlanet X is at a distance 10ly away from Earth (as observed in the Earth’s frame). How fast does an astronaut need to travel (relative to Earth) to get to X in 1 year (from his view)?

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tA =lE

vγ

⇒ tA2 =

lE2

v 2 (1− β 2) =lE

2

β 2c 2 (1− β 2)

⇒tA

2c 2

lE2 =

1− β 2

β 2 =1

β 2 −1

⇒1

β 2 = 1+tA

2c 2

lE2

⇒ β =1

1+tA

2c 2

lE2

=1

1+1yr2c 2

(10c ⋅yr)2

=1

1+1

102

= 0.995

Ladder & Garage Paradox

Can the garage trap the ladder?

Ladder’s view

Garage with swing doors

Ladder’s viewFrom the ladder’s point of view, the doors do not close simultaneously. The door on the right closes first.

Will the ladder fall into the hole?