PHYC10001 6 Specialrel Web

© David N. Jamieson 2011

By:

Prof. David N. Jamieson

School of PhysicsUniversity of Melbourne

PHYC10001 Physics 1: Advanced

The Special Theory of RelativityFaradayMaxwell

Lorentz

Einstein

Heaviside

Lecture 2

Lecture 3

Lecture 4

Lecture 5


At the end of this lecture you will be able to:

Explain how the theory of relativity arose from electromagnetism

Understand the concept of a reference frame

Explain how we know the speed of light is independent of the speed of the source and observer

Know the two postulates of special relativity

(Analyse the “light clock” and show that the postulates lead to time dilation)

Know when I say “c” I mean the speed of light– From celerity = swiftness of action or motion

Everything before this lecture was “Galilean Relativity”

1c + 1c = 2c


Two ways to make electricity!

NS

NS

Changing magnetic field makes an ELECTRIC field

Moving charge in wires feels MAGNETIC force

Achtung!: There does not need to be two laws of Physics

here!

Induction 1 & 2

BvF

q

. and

dt

ddq BsEEF


The Maxwell Equations (from semester 2)I Gauss’ Law for electrostatics

II Gauss’ Law for magnetism

III Faraday’s Law of induction

IV Ampere-Maxwell Law

Predict the speed of light as an electromagnetic wave

dt

di E

ooo

loop

B.ds

o surfaceclosed

q

E.dA

0 surfaceclosed

B.dA

dt

d B

loop

E.ds

oo

c

1


How fast are we going right now?

What about Galileo and Newton?

Galileo:– The laws of Physics do not depend on absolute

motion (does this include electromagnetism?)

Newton:– An object once set in motion remains in motion until

acted upon by an external force

– The universe is governed by a majestic clockwork where all clocks everywhere at all times tick in perfect synchronisation.


Do the laws of Physics depend on velocity?

Galileo: No! (but what about light?)

Newton: No! (but what about light?)

Maxwell: Magnetism is a velocity dependent force!

Michelson and Morley: Maybe but can’t find it

Einstein: No!

Very high speed

Galileo: 1634

There is no difference!


Mechanics Idea #1: Galilean Relativity

Galileo (and Newton) knew that an object, once set in motion, continues indefinitely at constant speed unless acted upon by an external force.

And this motion cannot be detected from “inside”

Twice the speed of sound and not a

drop spilled!...or parked at the

gate?

Lifestyle of a Concorde passenger


Electromagnetism - fast overview

Electromagnetism is DIFFERENT to ordinary mechanics (or is it?)

Electrostatic force:

Magnetic force:

...magnetic force depends on the speed, v !

q Qr

S NB

BvF

q

(into screen)F

rF ˆ2r

kQq

v

q


How fast are we going anyway?

Earth’s Orbit around Sun:30 km/s

Milky Way Galaxy trajectory towards Great Attractor:7000 km/s

Sun’s Orbit around galaxy:250 km/s

Rotation of Earth:1000 km/hr


How fast are we going right now?

Possible answers

We are not going right now, so zero

1000 km/hr as the Earth rotates

30 km/s as the Earth orbits

some other speed

??!!?!?!?

Do the laws of relative motion apply to light & electromagnetism?– Is Galilean Relativity correct for light?

Need to study how objects move– From this can develop laws of mechanics

– Need to study how light behaves

Is light consistent with the laws of mechanics?

Can we tell how fast we are going?

(No and No)


How does light behave?

Like tennis balls?

100 km/hr

Ball travels at speed hrkmuball /100

right) the to( sourceballball vuv

hrkm /1000100



Like tennis balls?

vsource=100 km/hr

0 km/hr

right) the to( sourceballball vuv

hrkm /0100100

Ball travels at speed hrkmuball /100

),,,( tzyx ),,,( tzyx

tvxx source

tt

x0

y

x

This is Galilean Relativity!



Like tennis balls?

double stars

v

v

? vc

? vc

right) the to( sourcelightlight vuv

Light travels at speed culight

Light travels at speed culight

Light signals would get out of sync!!

Galilean Stars



Like tennis balls?

double stars

v

v

c

c

right) the to( sourcelightlight vuv

True situation!

cvlight

No!

Independent of vsource



Like sound waves?

Speed of sound in air always Mach 1



Like sound waves?– Speed of sound in air is cs

– Speed relative to observer

Mach 2

Mach 1

right) the to ( observerssound vcv

scobserverv

ssssound cccv 2

observerv

sobserver cv 2

sc

Observer frame

Earth frame

Mach # = v/cs

cs=340 m/s



Like sound waves?

Propose a medium for light: Aether

Sound in Air– Air has pressure– Made of O, N, Ar– Has mass– You breath it– ...

Light in Aether?– Aether is insubstantial– Very elastic– Other properties???



Like sound waves?

Aether wind!

skmvobserver /30

skmcvcv observerlight /30

c

Michelson - Morley Experiment in 1887



Like sound waves?

NO!! Michelson

-Morleyexperimentfound nodifference!

Speed of light istherefore independent ofthe speed of theobserver! 30 km/s skmvobserver /30

c

c



Summary– Speed of light

independent of speed of source

– Speed of light independent of speed of observer

Object Us ThemUs 0Them 0Tennis Ball 100 km/hr + v 100 km/hrSound Wave

v

v

v

vcs sc

Light c c

smc /103 8 !

(+ to right)

Us

Them


Can you catch a beam of light?

Maxwell

Light is an electromagnetic

wave!


The Postulates of the Special Theory of Relativity

Laws of Physics the same for everybody

The speed of light the same for everybody

Sun’s light moves at speed c


Let us look at a startling consequence of these facts

2/c

2/c

2/3)2/( ccc

2/)2/( ccc

c

c

c

c

c



Know the two postulates of Special Relativity (SR)

Analyse the “light clock” and show that the postulates lead to time dilation

Analyse the “muon experiment” and show that time dilation leads to the Lorentz contraction

Explain why simultaneity is relative

Main conclusion from last lecture:The two postulates of Special Relativity (SR)

Laws of Physics same in all inertial frames

Speed of light same for all observers and independent of source


Things we have to give up

Universal time

Universal length

Relative velocity formula

Galilean Transform equations

vtxx

yy

zz

tt

Galilean transforms

)( vtxx

yy

zz

)/( 2cvxtt

22 /1/1 cvLorentz transforms

v

y

xt

y

x

t

A new term in the equation for the time transform!


The Light Clock

A photon bouncing between mirrors may be used as a clock


30 km/s

The light clock seen from space


The light clock seen from space


Speed c

Speed c

Lecture theatre at rest relative to Sun

Lecture theatre on Earth


Speed c

Speed c

Long distance

Short distance

Spaceship frame


Vector addition of velocities?

skmv /30

skmc /000,300

22 cvV

Wrong!

Speed of light is independent of speed of source!

Spaceship frame

We are going to need a new velocity addition formula!

photon


Vector addition of velocities?

cV

Speed of light is independent of speed of source!

skmv /30 See later for how to add velocity vectors consistent with relativity.

photon

Earth frame ,


Speed c

Speed c

Long distance

Short distance

Moving Clock is slow!d

cdt /2

Spaceship frame ,


22 /1/1)/2( cvcdt

Analysis by spaceship

Speed c

Speed c

d

v

dl l

22 dl 22 dl

cdt /2

The spaceship sees the time intervalon the Earth clock is longer than the timeinterval on the spaceship clock by a factor

22 /1

1

cv

Earth Clock

cdlt /2 22

cdvtt /)2/(2 22

2222 /)2/(4 cdvtt

222222 /4)/( cdtcvt

tt

22 /1 cv

tt

v

2//2 vtlvlt so but

t

Spaceship frame ,


Summary

The Special Theory of Relativity

First Result: Moving clocks run slow

tcv

tt

.

/1 22

22 /1

1

cv

Moving clock

Our (resting)

clock


Moving clocks run slow: Experimental Test

Many experimental tests available!

Will look at only one case study

Short lived radioactive particles are created by cosmic ray bombardment of the Earth’s atmosphere

These can be detected (see our third year physics lab)


View from Earth: Moving clocks run slow!

Cosmic Rays create fast muons in outer atmosphere

Identical muons created in the laboratory live for 2.2 millionths of a second

Even at speed of light, this is not long enough to reach sea level!

Fast muons live longer than slow muons

12 ttt events between interval Time

vLt /

Fast muons created here

Event 1

1t

See plenty here!

Event 22t

L Expect to decay here(660 m)

v

This is the proper height


View from muons: Moving objects contract!

How do the muons view the situation?

Muons always live for 2.2 millionths of a second by their own clocks

The fast mountain must shrink for it to get past in the short time available

Moving objects contract!

Stationary muons created

here

Fast mountain

I’m notgoing tolive longenoughto reach

sea level!

1t

Event 1







Stationary muons created

here

Fast mountain

1t

Event 1





The fast mountain must shrink for it to get past in the short time available!








This is the Lorentz contraction

Bottom of mountain reaches muon

2t

vLttt /12

vLttt /12 and

v

L

tLtLv // hence

22 /1 cv

tt

)/( ttLL

22 /1)/ cvLLtt :( eliminate

LL

Event 2

The measured length is always less than or equal to the proper length


Summary



Second Result: Moving objects contract

/LL

22 /1

1

cv


At the end of this lecture you will be able to: Know the two postulates of Special Relativity (SR)

– Laws of Physics same in all inertial frames– Speed of light same for all observers and independent of source

Lecture 21 Analyse the “muon experiment” and show that time dilation leads to

the Lorentz contraction Explain why simultaneity is relative Know the concept of a “reference frame” Describe the concept of the light cone and relate it to the past, future

and elsewhere Be perplexed at some of the phenomena of SR!

– The long car in the short garage problem– The short ruler in the even shorter inspection hatch– The Magnetic Force– The Twin Phenomenon

Lecture 22 Curved space and the General Theory of Relativity The Global positioning system

38

/LL

tt

22 /1/1 cv

21

mvm )(


Proton View

Galaxy crossing protons

Galaxy view

proton speed v < c (just) = 1010

Cross galaxy in t=105 years

Proton view

galaxy speed v < c (just)L’ = L/ = 105/1010 = 10-5 l.y.

= 95 million km = 0.7 A.U.Hence t’ = L’/c = 10-5 years

= 315 seconds

Galaxy View

L = 105 light years

v

-v


Earth View

Alpha Centauri

4.5 ly

Photon View

The universe seen by a photon

Only weightless objects can travel at the speed of light

All clocks freeze

All distances contract to zero

Where would you like to go today?

What if v > c ?

Violate causality

Cannot happen in our universe


The two postulates of the Special Theory of Relativity

Laws of Physics are the same in all inertial frames

The Speed of light is the same for all observers and independent of the speed source or observer


Relativity of Simultaneity

Lecture theatre experiment










I see it!I see it!

Event 2Event 1




Simultaneous!I see it!I see it!

Event 2Event 1



Space view of lecture theatre experiment

30 km/s




30 km/s




30 km/s

I see it!

Event 2




30 km/s




30 km/s




30 km/s




30 km/s

I see it!

Event 1




30 km/s

I see it!

NOT Simultaneous!

Spacetime simulation

Event 1 later than Event 2

Event 1


Summary



Second Result: Moving objects contract

Third Result: Simultaneity is Relative



Know that simultaneity is relative– Explain the relationship between simultaneity and measurement

Explain the concept of a “reference frame”– Plot maps of space and time and identify the past, the future and

…

– Describe a reference frame and the importance of synchronised clocks and rulers

Understand how to examine some perplexing problems of special relativity– … and how to resolve “paradoxes”

– The long car in the short garage problem

– The short ruler in the even shorter inspection hatch

– The Magnetic Force

(On to advanced topics...)


A reference frame

To specify an event,

an observer must assign – 3 space coordinates (x, y, z)

– and a time coordinate (t).

We need a calibrated reference frame.

Your clocks and rulers fill all space

You are here at origin

These clocks keep proper time and rulers are proper length


The problem of measurement Lecture theatre reference frame

Simultaneous!

Left end is at 0

Event 1 Event 2

Right end is at L’

22 /1

1

cv


The problem of measurement Spaceship reference frame

Left end is at 0

Event 1 Even

Right end is at L’

Not Simultaneous!L

L’=L/

22 /1

1

cv


time axis

space axis

home

uni

PAST

Future

Elsewhere Elsewhere

The Light Cone: Maps of space-time

Events that happen “elsewhere” can have no effect on

Supernova!

here and nowThis is a

“world line”


The Light Cone: History of Universe

Age of quasars

Here and now

The Big Bang

Earth forms

Milky Way forms

time


The car in the garage

Perplexing Problem #1


Will the car fit in the garage?22 /1

1

cv

2/L

Garage at rest

Car at rest

L


v

/L

v

/L

Will the car fit in the garage?

Garage Frame

EVENT “B”

Garage Frame

EVENT “F”

EVENT “B” occurs before EVENT “F”

2/L

2/L

22 /1

1

cv

4



Car Frame

EVENT “F”

v

Car Frame

EVENT “B”

v

EVENT “F” occurs before EVENT “B”

L

L

4


v

/L

v

/L


Garage Frame

EVENT “B”

Garage Frame

EVENT “F”

EVENT “B” occurs before EVENT “F”

2/L

2/L

22 /1

1

cv

4


Will the car get wet?

Car Frame

EVENT “F”

v

Car Frame

EVENT “B”

v

EVENT “F” occurs before EVENT “B”

L

L


The ruler in the drain



Will the ruler fall in the drain?

Street Frame

Ruler Frame

D r a i n

Drain

v

L

/L v

/L

L



Street Frame



Street Frame



Street Frame

YES!!



Ruler Frame



Ruler Frame

NO!!

WRONG! Unphysical simulation.



Ruler Frame



Ruler Frame



Ruler Frame



Ruler Frame

TRUE ANSWER: YES!!


The falling ruler

y'

x'

)0,0,( 1xSituation at t’ =0

)0,0,( 2x

Situation at t’ < 0

),,( 11 tyx ),,( 12 tyx

Speed u: y = -ut

Slides with this colour background are in , ,space

!

1y

1x 2x


The falling rulery'

x'

!

1x 2x


)0,0,( 2x

Speed u: y = -ut

)( vtxx yy

zz

)/( 2cvxtt 22 /1/1 cv

Lorentz transforms


The falling ruler

x'


)0,0,( 2x

!

1x 2x

Speed u: y = -ut

ct'

Event 1:The coordinates

(x’1, 0) transform to:

1111 )( xtvxx 2

12

111 / )/( cxcxvtt

)( tvxx yy

zz

)/( 2cxvtt 22 /1/1 cv

Lorentz transforms

2111 / cxuuty


(x’2, 0) transform to:

2222 )( xtvxx 2

22

222 / )/( cxcxvtt 2

222 / cxuuty

Note: Two velocities here

• u is velocity of falling ruler (in –y =y’ direction)

• v is speed of ruler along ground (in +x direction)


The falling ruler

)0,/ ,( 211 cxux

Situation at t = 0

)0,/ ,( 222 cxux

!

y

x


The magnetic force



Current in wire: Wire is electrically neutral

+ + + + + + +

– – – – – – –

+ + + + + + +

– – – – – – –

No current

With current dq /

Free electrons(1 per atom)

Metal ionsdq

q d

dq / v

C/m : of unit

! !


The origin of the magnetic force

Adjacent moving electron

+ + + + + + +

– – – – – – –

– vv

(can choose same speed without loss of generality)

d

d

–

+ + + + + + +

– – – – – –v

+

/d

d

Adjacent moving electron frame dqdq ///

)/( dq BvF

q

! !


Some numbers For a current of 10 Ampere in a wire of area 1 mm2

– Electron density in Copper Ne = 8.5x1022 atoms/cm3

– The atom density is the same as the electron density because each atom contributes 1 free electron to the metal

From semester 2 theory, the speed of electrons in this wire will be:

– e = electron charge = 1.6x10-19 Coulombs

– A = cross sectional area of wire = 1 mm2

– Gives

What is the gamma factor for this very slow speed?

eeAN

iv

mm/s 7.0105.8).101.(106.1

1022619

v

24

221031

/1

1

cv

This is incredibly small!

But we see these magnetic effects because of the enormous strength of the electric force!



Explain the magnetic force using the Special Theory of Relativity

Understand how to draw a Minkowski diagram – a map of space time– The time axis– The space axis– … using the magnetic force as an example

Understand (qualitatively) the origin of the Lorentz transformations

Understand Five advanced topics:

Derive the relativistic velocity addition formula

Show that, like time and space, mass depends on reference frame

Understand the subtleties of the twin paradox

Explain why brightness fluctuations constrain the size of an object

Be able to account for things that apparently go faster than the speed of light

(Link time, space and gravity)

(The Global Positioning System)

phenomenon


The magnetic force – revisited

This time with a spacetime diagram

Perplexing Problem #3 (again)


The inventor of the “Minkowski Diagram”

Hermann Minkowski

Young Einstein’smathematics lecturerat ETH* Zurich

Followed his youngprotégée's career with interest…

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

http

://e

n.w

ikip

ed

ia.o

rg/w

iki/H

erm

an

n_M

inko

wsk

i

*Swiss Federal Institute of Technology, Zurich


1: No current in wire

ct

x

Wor

ld L

ine

of o

rigin

All events at t=0

Positive ion Electron


2: With current in wire

ct

x

Wor

ld L

ine

of o

rigin

All events at t=0

x’

ct’


Exercise: Derive the Lorentz Transforms

ct

x

x’

ct’

1x

1ct

1x

1tc

)( vtxx yy

zz

)/( 2cvxtt



Example: Application of the Lorentz Transforms

ct

x

x’

ct’

Earth Frame

)( vtxx yy

zz

)/( 2cvxtt



Example: Application of the Lorentz Transforms

x’

ct’

Spaceship Frame

)( vtxx yy

zz

)/( 2cvxtt



)( vtxx yy

zz

)/( 2cvxtt


Example: What are the times on the spaceship clocks?

ct

x

x’

ct’


(x1, ct1) of this clock are: (0, 0)

Hence

0)00(1 vx 0)/00( 2

1 cvt


(x2, ct2) of this clock are: (L, 0)

HenceLvLx )0(2

222 /)/0( cvLcvLt

v

L0 Earth Frame


1. Relativistic addition of velocities

Advanced topics


The Postulates of the Special Theory of Relativity

Laws of Physics the same for everybody

The speed of light the same for everybody



How do we add velocities?

2/c

2/c

2/3)2/( ccc

2/)2/( ccc

c

c

c

c

c


Adding velocities in Special Relativity

cu 2/cv

Problem: If u=c in reference frame of sun, what is u’in frame of spaceship?

Note: Two velocity vectors

Light from Sun, u or u’

Relative motion, v

Reference frame S Reference frame S’


Adding velocities in Special Relativity

v

Problem: If object has velocity u in reference frame S, what is u’ in frame S’?

Note: Two velocity vectors

Object, u or u’

Relative motion, v

Reference frame S Reference frame S’

u=100 km/hr


The Lorentz Velocity Transformations

Consider an object moving at u in frame S, u´ in frame S´

Transform using Lorentz, and time derivatives

But dx/dt = u so:

and inverse (change sign of v):

222 //1

/

//'

''

cdtdxv

vdtdx

cvdxdt

vdtdx

cvxtd

vtxd

dt

dxu

21

'

c

uvvu

u

2

'1

'

c

vuvu

u

2'

'

c

vxtt

vtxx


The Lorentz Velocity Transformations

Example

A nucleus travelling at 0.50c in the laboratory frame decays and spits out an alpha particle at 0.80c with respect to the reference frame of the nucleus. What is the speed of the alpha particle, with respect to the lab?

Let nucleus frame be S´ and lab frame be S so we want:

0.5 + 0.8 = 0.93!

2

'1

'

c

vuvu

u

cccc

c

vuvu

u 93.04.1

3.1

8.05.01

80.050.0'

1

'

2

v = 0.5c u’ = 0.8c

Reference frame S’

Reference frame S


The spacetime interval

Clearly time dilation and length contraction are related

Consider light clock in two different reference frames, S´ and S´´

h is same in both reference frames

We define an invariant in all reference frames:The spacetime interval s is defined by

The spacetime interval has the same value in all inertial reference frames

2222 xtcs

h

Mirror in S’

''21

21 xtv

'21 tc h

Mirror in S’’

''''21

21 xtv

''21 tc Same in both!


2. Relativistic mass

Advanced topics


Length & Time change with speed: How about mass?

)(Vm

)(Vm



Earth Frame

L

v



Earth Frame

L

Earth’s

Throw to Catch analysis:

Time

Vertical momentum

u

L

u

Lt

2/2

v

Event 1: Throw

00 )()( uumuVm 0u

V

Event 2: Catch

v

)( 0um

)(Vm

uv



Spaceship Frame

L00

2/2

u

L

u

Lt

Event 1: Throw/ Event 2: Catch = same place

v

1/

u

Lt

tt

/0uu

u

Lt Earth’s

/tt

0u

v

Spaceship’s


Time



Earth Frame

u

Earth’s


Time

Vertical momentum

u

Lt

v

v

Event 1: Throw

0u

V

Event 2: Catch

v

)( 0um

)(Vm

00 )()( uumuVm

/0uu

000 )(/)( uumuVm

)()( 0umVm

22

0

/1

)()(

cv

umVm



Consider a symmetrical elastic collision between two balls

vv

)(vm

vVu then , Let 0022

0

/1

)()(

cv

umVm

)(vm

22 /1

)0()(

cv

mvm

mass rest mm )0(

Moving objects get heavier!

cvvm as )(


Constant force, F, implies constant acceleration, a from

Newton’s F = ma formula

Can any speed, v, be reached?

No! Cannot have v > c

Can show that E = mc2 where m is the “relativistic mass”

of an object and E is the total energy of the object

From this:

where m is the “rest mass” of the object.

Moving objects get heavier!

2

21

1)(

c

vmvm

mvm )(

22 /1/1 cv


3. E=mc2 & relativistic kinetic energy

Advanced topics


–v

What about E=mc2 ?

v

Mvc

E

c

v

vMc

E

c

v

2

)1(2

)1(

MMcE

2

2cE

MM

2mcE

MMm

M

M

EARTH

FRAM

ESP

ACES

HIP

FRAM

E

–v


Relativistic Kinetic Energy: .

Force is the time rate of change of the momentum:

where

where

and 1

What is the relativistically correct relation between force and acceleration?

…

.

We can now integrate . to get the kinetic energy


Relativistic Kinetic Energy continued

By definition: .

Using expression on previous page and a change of variable, get:

/ .

hence 1

(Exercise: Show this reduces to when ≪ )

Re-order:

≡ “Total Energy”

Total energy of a particle is Kinetic + Rest Mass energy


Summary: Total Energy & Momentum

Total energy of a particle is Kinetic + Rest Mass energy:

We already defined the total momentum of a particle:

And finally, it is possible to show:

which is the relativistically correct relation between momentum and total energy (replaces /2m )



Understand Six advanced topics:

1. Derive the relativistic velocity addition formula

2. Show that, like time and space, mass depends on reference frame

3. E=mc2

4. Understand the subtleties of the twin phenomenon

5. Explain why brightness fluctuations constrain the size of an object

6. Be able to account for things that apparently go faster than the speed of light

Light, time and gravity

Link time, space and gravity

The Global Positioning System


Example: Relativistic Kinetic Energy

The highest energy cosmic ray protons have 10

What is the kinetic energy of such a proton?

Answer:1

10 110

For a proton, rest mass ≅ 1so 10

Here the kinetic energy is ten orders of magnitude greater than the equivalent rest mass energy!

At CERN,maximum proton energy in LHC (lab frame) is

7 10


Some gamma factors

Object speed gamma

J/kg

Car on freeway 100 km/r 1.00 400

Fast proton c/2 1.15 1.4 x 1016

Electron in Melbourne

Synchrotron 3 GeV c 6.00 4.5 x 1016

Cosmic ray proton c 1010 9 x 1026

Australian electric power for 1 year 1.42 x 1018 J

(45 GW continuous)

22 /1/1 cvv

/ 1

63 billion years of Australian electric power

Kinetic Energy per kg

© A/Prof RE Scholten (2008)

Relativistic energy: pmc2

(pmc2) has units of energy

At low velocity, (pmc2) reduces to

Second term is classical kinetic energy, associated with the motion

First term suggests there is an inherent energy associated with mass itself

Define the total energy E of a particle as

Rest energy is E0=mc2 (also called mass energy)

Relativistic kinetic energy

Returning to eqn (*), we can substitute E0=mc2:

and

123

2222

2

22

22

p 2

1

2

11

/1mumcmc

c

u

cu

mcmc

energy kinetic energy rest 02

p KEmcE

0p2

p 11 EmcK

22222p mcpcmc

20

22 EpcE

Epc p


4. The Twin Phenomenon

Advanced topics


The twin phenomenonTi

me

axis

Arrive’

Depart’

Arrive/Depart High Earth clocks run FAST!

Start/Start’

Finish/Finish’

!


The twin phenomenon

Tim

e ax

is

Start/Start’

Finish/Finish’

!



me

axis

Start/Start’

Finish/Finish’

!


The twin phenomenon

Tim

e ax

is

Start/Start’

Finish/Finish’

Arrive’

Arrive/Depart

!



me

axis

Arrive’

Arrive/Depart

Start/Start’

Finish/Finish’

High Earth clocks run FAST!

!


The twin phenomenon

Tim

e ax

is

Arrive’

Arrive/Depart

Start/Start’

Finish/Finish’


!



me

axis

Arrive’

Arrive/Depart

Start/Start’

Finish/Finish’

Depart’


!


The twin phenomenon

Tim

e ax

is

Arrive’

Arrive/Depart

Start/Start’

Finish/Finish’

Depart’


!



me

axis

Arrive’

Arrive/Depart

Start/Start’

Finish/Finish’

Depart’


!


5. Brightness fluctuations and the size of objects

Advanced topics


Relativity and brightness fluctuations

Imagine all stars in a globular cluster simultaneously get brighter

A

BC

10 ly

Photons from A reach Earth 10 years before photons from C

Therefore takes 10 years for globular cluster to reach new brightness

Observations of brightness fluctuations provides information about the size of objects

e.g. Quasars fluctuate on timescales of months


6. Things that go faster than the speed of light (or at least look like it)

Advanced topics


6.1 Superluminal Interstellar Scissors


Superluminal scissors #1






6.2 Superluminal Jets


Superluminal jets from SS433 So what is going on???

18,000 light years from here

Eclipsing binary x-ray: black hole pulling matter from companion star

Emitting x-rays from accretion disc + jets of hot hydrogen


Superluminal jets from SS433

See: Mirabel, I.F, Rodriguez, L.F., 1994 Nature 371 page 46 “A superluminal source in the galaxy”

SS433

time

dist

ance


Day 1

Earth

SS433


Day 100

Earth

SS433


Day 200

Earth

SS433


Day 300

Earth

SS433


Day 365

Earth

SS433


Day 365 = 1Year

Earth

SS433

Distance=

2 light years


Day -6yr

SS433

Telescope viewfrom Earth

SS433

Earth

“Top view”


Day -5yr


SS433 SS433

Earth

“Top view”


Day -4yr

SS433


SS433

Earth

“Top view”


Day -3yr

SS433


SS433

Earth

“Top view”


Day -2yr

SS433


SS433

Earth

“Top view”


Day 0

SS433

Image of blob leaving SS433

Image of blob far from SS433


SS433

Earth

“Top view”


Day 100

SS433


SS433

Earth

“Top view”


Day 200

SS433


SS433

Earth

“Top view”


Day 300

SS433


SS433

Earth

“Top view”


Day 365

SS433

Image of blob far from SS433Telescope view

from Earth

SS433

Earth

“Top view”


Day 365

SS433


SS433

Earth

Distance=

2 light years

Distance=

1 light year

Image of blob far from SS433

“Top view”


Light, time and gravity

Introducing: The Theory of General Relativity

Advanced topics

Practitioners of General Relativity were once described as "magnificent cultural ornaments“ by Lord Martin Rees, President Royal Society, December 1999.


A reference frame

To specify an event,

an observer must assign – 3 space coordinates (x, y, z)

– and a time coordinate (t).

We need a calibrated reference frame.

Your clocks and rulers fill all space

You are here at origin

These clocks keep proper time and rulers are proper length


SR + Acceleration = Warped Space = GR

Mechanism for warping space

Use a rotating reference frame

Apply Special Relativity & Lorentz contraction

Apply Equivalence Principle

Get non-Euclidian geometry

Experiments in geometry:

Does C=2 r ?


WHY ARE YOU WEIGHTLESS WHEN YOU FALL?


Aside: Why are astronauts weightless?

Radius of Earth 6,350 km

Height of tower 400 km

Gravity at top of tower:– almost same as

on surface!

– “g”=8.71 m/s2

Why are astronauts weightless?

Answer: Because they are falling!

22

/71.8)400(

"" smkmr

Gmg

Earth

Earth

Data:mEarth = 61024 kgrEarth = 6350 kmG=6.6710–11 N.m2/kg2


Einstein’s Equivalence Principle of 1912

Inertial mass:

Gravitational mass:

Newton knew mI = mG

Eötvös experiment confirmed this to high accuracy

Why?

Arghhh!

Einstein: 1912

“The effects of a gravitational field are equivalent to those of an (upward) acceleration”

This is the Equivalence Principle & the birth of General Relativity.

2r

mmGF GEarth

gravity

amF Iinertia


The geometry of a rotating cylinder

Side view of rotating cylinder

Twelve 1 metre rulers laid down around rim by observer on ground

Circumference:

Velocity of rim

v

L

mLrC 12122

r


The geometry of a rotating cylinder

Observer on rim sees rulers going past at high speed

They are shrunken to less than 1 m by Lorentz contraction!

I am standing on the bottom of a

stationary cylinder watching lots of rulers go past

Fast moving shrunken rulerv

22 /1

1

cv

/LL

L


Down

Lorentz contraction makes observer on ground thin, observer on cylinder fat

I am standing on the bottom of a stationary cylinder watching lots

of rulers go past

L

/LL


Down



of rulers go past

L

/LL


Down



of rulers go past

L

/LL


Down



of rulers go past

L

/LL


Down



of rulers go past

L

/LL


Down



of rulers go past

L

/LL


Down



of rulers go past

L

/LL


Down



of rulers go past

L

/LL


Down



of rulers go past

L

/LL


Down



of rulers go past

L

/LL


Down



of rulers go past

L

/LL



A rotating cylinder provides “artificial gravity”

“Artificial gravity” in the control room of the “Discovery” (2001-A Space Odyssey)

Observer held onto wall by centripetal force

2001 centrifuge



“Gravity” points outwards on rim of rotating disk– C < 2R there

Gravity points inwards on Earth– C > 2R there

Earth’s gravity makes more space!

LLCLL 12/ so

CCL //12

mLrC 12122


Fast clock = slow by time

dilation


t

tt

v

Gravitational red shift for clocks at different heights

Resting clock



Centripetal acceleration

t

tt



Centripetal acceleration

t

tt

Gravitational red shift for clocks at different heights

Low slow clock

High fast clock


Kepler’s Laws

Data for the Solar System

Planet Eccentricity Orbit Radius R (AU)

Period T (years)

T2/R3

Mercury 0.206 0.387 0.241 1.002

Venus 0.007 0.723 0.615 1.001

Earth 0.017 1.000 1.000 1.000

Mars 0.093 1.524 1.881 1.000

Jupiter 0.048 5.203 11.862 0.999

Saturn 0.056 9.534 29.456 1.001

Largest Biggesterror

Closest

GMr

T

r

T

r

T 2

33

23

32

22

31

21 4

)(

)

)(

)

)(

)

( ( (


Curved Space: Advance in Perihelion of Mercury

Perihelion (closest approach to Sun) advances owing to gravitational influence from rest of solar system.

The observed advance:– 574” every 100 years.

Calculations using Newton’s universal gravity from the rest of the solar system:

– 531” every 100 years.

Could this be due to unknown planet Vulcan?

No! Residual 574 – 531 = 43” is due to General Relativity

Mercury


The Global Positioning System

Advanced topics


Array of 24 satellites that send precision time & position signals

In orbit 20,000 km above surface of Earth (12 hr orbit)

Ground station in US to maintain system

Satellites broadcast accurate time signals

Hand held receiver can locate your position to phenomenal accuracy

Orbits are inside Van Allen radiation belts

Need radiation hard electronics

What is the Global Positioning System?


Relativistic Problems for the GPS

1. GPS satellites clocks run slow by 6 millionths of a second per day (SR - time dilation)

2. Cannot synchronise by exchange of signals

3. GPS satellite clocks run fast by 45 millionths of a second per day (GR - gravitational blue shift)

Net effect: Run fast by 39 millionths of a second per day (=error of 12 kilometres)!

SOLUTION: Make the clocks run slow to compensate!


Relativistic formulae

Kinetic energy 2

2

1mvK 2)1( mcKrel

Momentum

Velocity addition

mvPrel mvP

m mMass

Pythagoras 2222 zyxd 222222 zyxtcs

)/1(

)(2cuv

vuu

vuu

22 /1

1

cv

v

uu ,

mm ,

Total energy UKE relKmcmcE 22


Conclusion

“Philosophy is written in this great book (by which I mean the universe) which stands always open to our view, but it cannot be understood unless one first learns how to comprehend the language and interpret the symbols in which it is written, and its symbols are triangles, circles and other geometric figures, without which it is not humanly possible to comprehend even one word of it; without these, one wanders in a dark labyrinth”– Galileo Galilei (Il Saggiatore,

1623)

The Hubble deep field south: 12 billion light years of galaxies at the south celestial polehttp://hubblesite.org/newscenter/newsdesk/archive/releases/1998/41/image/b

(October 1998)

Documents

PHYC10001 6 Specialrel Web