Chapter 02 Special Relativity General Bibliography 1) Various wikipedia, as specified 2) Thornton-Rex, Modern Physics for Scientists & Eng, as indicated

Chapter 02Special Relativity

General Bibliography1) Various wikipedia, as specified

2) Thornton-Rex, Modern Physics for Scientists & Eng, as indicated

Version 110906, 110907, 110908, 110913

Outline

• Galilean Transformations• Names & Reference Frames• The Ether River• Michelson-Morley Experiments• Einstein Postulates• Lorentz Transformations

– Position– Velocity

• Space-Time Diagrams• Relativistic Forces & Momentum• Relativistic Mass• Relativistic Energy

CLASSICAL / GALILEAN / NEWTONIANTRANSFORMATIONS

Galilean Transformations

t

z

y

x

z

y

x

v

v

v

v

K’ frame moving with speed v

K frame fixed

K & K’ coincided at t=0. Sketch shown at time t later.

How do the position, velocity, acceleration, & time between the 2 frames compare?

'

'

'

'

t

z

y

x

'

'

'

z

y

x

v

v

v

z

y

x

a

a

a

'

'

'

z

y

x

a

a

a

Fam Fam

'

Galilean Transformations

v

K’ frame moving with speed v

K frame fixed

K & K’ coincided at t=0. Sketch shown at time t later.

How do the position, velocity, acceleration, & time between the 2 frames compare?

K K’

x = x’ + vt

K’ K

x = x’ - vt

Newtonian Principle of Relativity

• If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system.

• This is referred to as the Newtonian principle of relativity or Galilean invariance.

• K is at rest and K’ is moving with velocity

• Axes are parallel

• K and K’ are said to be INERTIAL COORDINATE SYSTEMS

Inertial Frames K and K’

The Galilean Transformation

For a point P In system K: P = (x, y, z, t) In system K’: P = (x’, y’, z’, t’)

x

K

P

K’ x’-axis

x-axis

Conditions of the Galilean Transformation

• Parallel axes (for convenience)

• K’ has a constant relative velocity in the x-direction with respect to K

• Time (t) for all observers is a Fundamental invariant, i.e., the same for all inertial observers

speed of frameNOT speed of object

x’ = x – v t

y’ = y

z’ = z

t’ = t

Galilean Transformation Inverse Relations

Step 1. Replace with .

Step 2. Replace “primed” quantities with

“unprimed” and “unprimed” with “primed.”

speed of frameNOT speed of object

x = x’ + v t

y = y’

z = z’

t = t’

General Galilean Transformations

'

'

'

tt

yy

vtxx

11'

''

''

'

dt

dt

dt

dt

vvdt

dy

dt

dy

vvvvdt

dx

dt

dx

samethearetandt

yy

xx

yyyy

xxxx

aadt

dv

dt

dv

aadt

dv

dt

dv

samethearetandt

''

'0'

'

inertial reference frame

FamFam '

11'

''

''

'

dt

dt

dt

dt

ttdt

dy

dt

dy

vuuvdt

dx

dt

dx

samethearetandt

yy

xx

frame K frame K’

Newton’s Eqn of Motion the same at face-value in both reference frames

Pos

ition

Vel

ocity

Acc

eler

atio

n

Classical Reference Frames

• Inertial Reference Frame– Non-accelerating– Newton’s Laws apply at face-value

• Non-Inertial Reference Frame– Examples:

• Rocket during acceleration phase• Rotating merry-go-round• Rotating Earth

Youtube clips (part 1)

• Galilean/Classical Relativity Part 1 – The Cassiopeia Project http://www.youtube.com/watch?v=6rl3Z9yCTn8

The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind.

http://www.cassiopeiaproject.com/

To read more about the Theory of Special Relativity, you can start here:

http://www.phys.unsw.edu.au/einsteinlight/

http://www.einstein-online.info/en/elementary/index.html

http://en.wikipedia.org/wiki/Special_relativity

http://www.youtube.com/watch?v=6rl3Z9yCTn8





THE ETHER RIVERHISTORY OF ETHER

MICHELSON-MORLEY EXPTS

The Ether River

A

C

Dv

Maximum speed of the boat is ‘c’ meters/sec

The Ether River

Time t1 from A to C and back:

Time t2 from A to D and back:

So that the difference in trip times is:

2

2

2

22

22

1

122

cvcvc

t

2

21

2

2

212

11

2

cv

cvc

ttt

down river

Timeline of luminiferous aether(http://en.wikipedia.org/wiki/Timeline_of_luminiferous_aether)

• 4th cent BC – Light propagates in air – Aristole• 1704 – Aetheral medium for light & heat – Newton• 1818 – aether – Fresnel wave theory• 1830 – problems emerge, explained by “aether drag”, Fresnel &

Stokes

• 1830 – ~1955 – mixed experimental conclusions

Cronholm144, http://en.wikipedia.org/wiki/File:AetherWind.svg

Timeline of luminiferous aether(http://en.wikipedia.org/wiki/Timeline_of_luminiferous_aether)

• 1830 – ~1955 – mixed experimental conclusions

• 1887 – 1st Michelson-Morley expt doesn’t find aether• 1889(1895) – Fitzgerald hypothesis (Lorentz)• 1902-1904 – Refined Michelson-Morley measurements• 1905 – Trouton-Rankine expt doesn’t support Fitz-Loentz hypothesis• 1958- nearly all measurements do not find evidence for aether


Michelson-Morley Expt “the most famous failed experiment”


Michelson-Morley: Ether River - Revisited

A

C

Dv

Measure two orientations because don’t know direction of aether river

A C

D

v

Ether River - Revisited

2

21

2

2

2121

11

2

cv

cvc

tttorient

Orientation 1

Orientation 2

2

2

1

2

22

122

11

2

cv

cvc

tttorient

Difference in Orientations

2

2

21

2

2

21

2

221

21

1

11

2

c

v

cc

vc

vctt orientorient

down river

down river

Michelson-Morley Measurements

l1+l2

(m)

to1-to2

(sec)

c[to1-to2]

1887 2*1.5 1E-16 30 nm

~1903 2*30 2E-15 600 nm

Earth-Moon 3.8E8 1.3E-8 3.8 m

v=30 km/s c=3E8 m/s

http://en.wikipedia.org/wiki/Lunar_Laser_Ranging_Experiment

Apollo 11 Apollo 15

~2002 accuracy ~1 mm

Crises with Reference Frame Xformations

• Can’t find the Ether• Maxwell’s Equations not Galilean Invariant

Speed of Light fixed by EM constants

1

c

Fitzgerald-Lorentz Hypothesis1889 (1895)

{only a partial explanation}

2

21

1

cv

LL

POSTULATE: the null results are due to changes in length in the direction of travel.

2

21

2

2

2

22

2121

11

22

cv

cvcvc

tttorient

2

2

1

2

22

22

2122

11

22

cv

cvcvc

tttorient

021 orientorient tt

EINSTEIN’s 1905 POSTULATES

• All laws for physics have the same functional form in any inertial reference frame

• Speed of Light (in vacuum) is same in any inertial reference frame.

LORENTZ POSITION-TIME

TRANSFORMATIONS

Lorentz Transformations

c

xtt

zz

yy

vtxx

'

'

'

'

21

1

x

K

P

K’

x’-axis

x-axis

x’

v

c

v

K’ K

Lorentz Transformations

c

xtt

zz

yy

vtxx

''

'

'

''

21

1

x

K

P

K’

x’-axis

x-axis

x’

v

c

v

K K’

ExampleAs observed from a large asteroid, an explosion occurs at x=3000, y=500, z=-500 and t=5 us.

P

v

A spaceship approaches at a high speed v=0.6c .The reference frames coincided at t=0, t’=0

At what position does the spaceship observe the explosion to occur?

K: 3km, 5usK’: 2.6km, -1.25us

ExampleThe reference frames coincide at x=0, x’=0 & t=0, t’=0

A spaceship has indicator lights which are flashed at the same time. At t’=0 the lights flash. The locations of the lights are x’rear=4km & x’front=+4km.

The spaceship is observed from the spacestation.The spaceship is observed to move at v=0.6c .

At what position does the spacestation observe the lights to flash?

K rear -5km, -10 us front +5km, +10 us

KK’

x’-axis

x-axis

v

ExampleThe reference frames coincide at x=0, x’=0 & t=0, t’=0

As viewed from the Earth, a meteorite impacts the lunar surface at 3E8m and 2.5s .

The impact is observed from 2 passing spaceships, one traveling to the right at 40% c and the other to the left at 40% c.

Where do the 2 spaceships observe the impact to occur ?

0m, 2.3 s6.5E8, 3.2 s

Length Contraction(Lorentz-Fitzgerald)

A meter stick, lying parallel to the x-axis, is moving with speed v

v

How long does the stick appear to be to a stationary observer who makes the observation of the length at t=0?

xleft & tleft=0 xright & tright=0

Moving objects appear shorter

Time Dihilation(distinct from the L-F)

A clock, located at x’=0, makes ticks at t’1, t’2, …

v

What is the interval between ticks to a stationary observer, who observes the clock to move at speed v?

x’1=0 & t’1 x’2=0 & t’2

Moving clocks

run slow

Distorted Picturesstationary moving to the right

Our brain records photographs (frames in a movie) – light rays arriving at the same time.

“Jump to Light Speed”

Distorted Pictures

Lorentz Transformation - DerivationLight propagates with speed c in all inertial reference frames

Spherical wavefronts in K:

Spherical wavefronts in K’:

K K’

ct ct’

1) Let x’ = (x – vt) so that x = (x’ + vt’)

2) By Einstein’s first postulate:

3) The wavefront along the x,x’- axis must satisfy:x = ct and x’ = ct’

4) Thus ct’ = (ct – vt) and ct = (ct’ + vt’)

5) Solving the first one above for t’ and substituting into the second...

Derivation – see pages 30-31


• Galilean/Classical Relativity Part 2 – The Cassiopeia Project http://www.youtube.com/watch?v=WgsKlSnUO0k







http://www.youtube.com/watch?v=WgsKlSnUO0k

LORENTZ VELOCITY

TRANSFORMATIONS

Lorentz Velocity Transformationsee page 40

c

xtt

zz

yy

vtxx

''

'

'

''

c

dxdtdt

dzdz

dydy

vdtdxdx

''

'

'

''


c

dxdtdt

dzdz

dydy

vdtdxdx

''

'

'

''

cdx

dt

dy

dt

dyu

cdx

dt

vdtdx

dt

dxu

y

x

''

'

''

''


x

yy

x

xx

uc

u

dtdx

cdt

dtdy

dt

cdx

dt

dy

dt

dyu

uc

vu

dtdx

cdt

vdtdx

dt

cdx

dt

vdtdx

dt

dxu

'1

'

''

1'

''

'

''

'

'1

'

''

1'

''

'

''

''

Note that because of the time transformation, the y- and z-components get messed up.

A spaceship traveling at 60%c shoots a proton with a muzzle speed of 99%c at an asteroid.

What is the velocity of the proton as viewed from a ‘stationary’ space station?

MISC. LORENTZ TRANSFORMATION

EXAMPLES

Cosmic Ray Muon Lifetime electron mo=9.1e-31 kg halflife = inf

muon mo=207 * (mass e) halflife = 1.5e-6 sec

http://landshape.org/enm/cosmic-ray-basics/http://www.windows2universe.org/physical_science/physics/ atom_particle/cosmic_rays.html

Cosmic Rays

Susan BaileyNuclear NewsJan 2000, pg 32

Cosmic Ray references

http://www.ans.org/pubs/magazines/nn/docs/2000-1-3.pdf

http://pdg.lbl.gov/2011/reviews/rpp2011-rev-cosmic-rays.pdf

http://hyperphysics.phy-astr.gsu.edu cosmic rays

ashsd.afacwa.org/ radation

Cosmic Ray Muon Measurementshttp://www.youtube.com/watch?v=yjE5LHfqEQI

Cosmic Ray Muon Lifetime muon mo=207 (9.1e-31 kg) halflife = 1.5e-6 sec

Q1. Classically, how far could the muon travel during a time 1.5e-6 sec ?

Suppose muon traveling at 0.98c

Q2. What do we observe the lifetime to be ?

Q3. How far do we observe the muon to travel during that time ?

2000

met

ers

Q4. How high does the muon think the mountain is?

Simultaneity

• http://www.youtube.com/watch?v=wteiuxyqtoM

• http://www.youtube.com/watch?v=KYWM2oZgi4E

Atomic Clock Measurements

• http://www.youtube.com/watch?v=cDvmN_Pw96A

• d

Twin Paradox

http://www.youtube.com/watch?v=A0jiY-CZ6YA

http://www.phys.unsw.edu.au/einsteinlight/jw/module4_twin_paradox.htm

Video Clip

What’s the correct explanation of the paradox?

Reliable Discussion at

Spacetime DiagramsMinkowski Diagrams

x

t x

t

In SP211 course:

ux

ux

Forbidden region

Allowed region

Two events plotted on a space time diagramP=(x,y,z,t)

Simultaneity in a Stationary System

#1 Measuring location #2 Measuring location

Watching a moving system

Animated Minkowski Diagrams

• http://www.youtube.com/watch?v=C2VMO7pcWhg – (uses Minkowski space diagrams, but with time axis pointing

down, opposite from figures in textbook.)

Analysis of theTwin Paradox

usingMinkowski Diagrams

Frank sends a signal once a year.

Mary sends a signal once a year.

Invariant Quantities

Define s2 = x2 – (ct)2

Then can show s’2 = x’2 – (ct’)2 x2 – (ct)2 = s2

s2 is independent of choice of reference frame.

We can use s2 to discuss the ability for ‘events’ to impact one another.

s2 = x2 – (ct)2

If s2 < 0 then x2 < (ct)2 = c2 t2

The distance between ‘events’ is less than the time required for light to propagate between the spatial locations.

RELATIVISTIC MOMENTUM

Style 1. Sandin’s Development

Style 2. Rex & Thorton’s Development

The following 4 slides present Sandin’s treatment of momentum

in Special Relativity

Forces and Momentum (Sandin version)

- a first look

y

x

yy

x

xx

uuc

uu

vuc

vuu

'1

'1

'

'1

'

yoy

yoy

umum

ummu

''1

'

v u’x , u’y , u’z

ux , uy , uz

p’x , p’y , p’z

px , py , pz

mo

If want p to ‘look like’ “mv” ;

Then are forced to let omm BUT


amFtot

dt

pdFtot

u

dt

dm

dt

dum

dt

mudFtot

Suppose F ┴ u

maamdt

dum

dt

dumF

dt

dm

changetdoesnmm

constconstspeed

oo

o

0

'

So why are we complaining about the phrase ‘relativistic mass’?


amFtot

dt

pdFtot

u

dt

dm

dt

dum

dt

mudFtot

Suppose F || u

mau

dt

dm

dt

dumF

mdt

dm

dt

d

dt

dm

constconstspeed

3

2/12

...

...1

BECAUSE Newton’s Eqn Motion is different depending on the direction of the force.

Forces & Momentum (Sandin version)

PRO-

Relativistic Mass People

ANTI-

Relativistic Mass People

Definitions m = mo

p=mu

No such thing

p=mu

Newton’s Eqn of Motion

if F ┴ u

F=ma

with m mo

F=dp/dt

Newton’s Eqn of Motion

if F || u

F=ma

with m mo

but throw in an extra 2F=dp/dt

The following 9 slides present Rex & Thorton’s

treatment of momentum in Special Relativity

2.11: Relativistic Momentum

Because physicists believe that the conservation of momentum is fundamental, we begin by considering collisions where there do not exist external forces and

dP/dt = Fext = 0

Frank (fixed or stationary system) is at rest in system K holding a ball of mass m. Mary (moving system) holds a similar ball in system K that is moving in the x direction with velocity v with respect to system K.

Relativistic Momentum

• If we use the definition of momentum, the momentum of the ball thrown by Frank is entirely in the y direction:

pFy = mu0

The change of momentum as observed by Frank is

ΔpF = ΔpFy = −2mu0


According to Mary

• Mary measures the initial velocity of her own ball to be u’Mx = 0 and u’My = −u0.

In order to determine the velocity of Mary’s ball as measured by Frank we use the velocity transformation equations:

K’

K

Relativistic MomentumBefore the collision, the momentum of Mary’s ball as measured by Frank becomes

Before

Before

For a perfectly elastic collision, the momentum after the collision is

After

After

The change in momentum of Mary’s ball according to Frank is

(2.42)

(2.43)

(2.44)

K

K

The conservation of linear momentum requires the total change in momentum of the collision, ΔpF + ΔpM, to be zero. The addition of Equations (2.40) and (2.44) clearly does not give zero.

Linear momentum is not conserved if we use the conventions for momentum from classical physics even if we use the velocity transformation equations from the special theory of relativity.

There is no problem with the x direction, but there is a problem with the y direction along the direction the ball is thrown in each system.


• Rather than abandon the conservation of linear momentum, let us look for a modification of the definition of linear momentum that preserves both it and Newton’s second law.

• To do so requires reexamining mass to conclude that:


Relativistic momentum (2.48)

Some physicists like to refer to the mass in Equation (2.48) as the rest mass m0 and call the term m = γm0 the relativistic mass. In this manner the classical form of momentum, m, is retained. The mass is then imagined to increase at high speeds.

Most physicists prefer to keep the concept of mass as an invariant, intrinsic property of an object. We adopt this latter approach and will use the term mass exclusively to mean rest mass. Although we may use the terms mass and rest mass synonymously, we will not use the term relativistic mass. The use of relativistic mass to often leads the student into mistakenly inserting the term into classical expressions where it does not apply.


RELATIVISTIC KINETIC ENERGY

The following 5 slides present Rex & Thorton’s

treatment of kinetic energy

in Special Relativity

2.12: Relativistic Kinetic Energy

Newtonian KE=1/2 m u2 which came from

KE = Work = ∫ F•ds with F = dp/dt = m dv/dt = ma

Relativistic Kinetic EnergyStart from rest and accelerate until u

duvuvdvu

partsbynIntegratio

Relativistic Kinetic Energy

duvuvdvu

partsbynIntegratio

u

duumuumKE

0

Start from rest and accelerate until u

22

2/12

22/1

2

21

11

cd

c

cu

duu

22

22 1 mcc

umcimitsevaluate

222

2 1mcmcmuKE


222

2 1mcmcmuKE

22222

2222

222 111

ucucc

uucucu

2

22

1 mcKE

mcmcKE

Which reduces to the Newtonian expression for u small

ComparisonRelativistic and Classical Kinetic Energy

Formula

Equation (2.58) does not seem to resemble the classical result for kinetic energy, K = ½mu2. However, if it is correct, we expect it to reduce to the classical result for low speeds. Let’s see if it does. For speeds u << c, we expand in a binomial series as follows:

where we have neglected all terms of power (u/c)4 and greater, because u << c. This gives the following equation for the relativistic kinetic energy at low speeds:

which is the expected classical result. We show both the relativistic and classical kinetic energies in Figure 2.31. They diverge considerably above a velocity of 0.6c.

(2.59)


Total Energy

21 mcKE

22 mcmcKE

22 mcKEmc

2mcEEnergyTot tot

Relationship between Total Energy & Momentummup

4242222 cmcmcp

Square, mult c2, convert u, use =(1-2)1/2 to subst 4

42222 cmEcp tot

42222 cmcpEtot

an invariant


• Galilean/Classical Relativity Part 3 – The Cassiopeia Project http://www.youtube.com/watch?v=W6o_-yTa168







Examples

Example 2.11Electrons in a television set are accelerated by a potential difference of 25000 Volts before striking the screen.

a). Calculate the speed of the electrons and b). Determine the error in using the classical kinetic energy result.

http://express.howstuffworks.com/exp-tv1.htm

http://www.o-digital.com/wholesale-products/2227/2285-4/LCD-TV-LDT32-225837.html

mc2 = 0.511 MeV m = 9.1e-31 kg |q| = 1.6e-19 Coul

Example 2.13A 2-GeV proton hits another 2-GeV proton in a head-on collision in order to create top quarks.

• For each of the initial protons, calculate– Speed v– – Momentum p– Rest-mass Energy– Kinetic Energy KE

– Total Energy Etot

http://www.fnal.gov

mc2=938 MeV

Example 2.16The helium nucleus is built from 2 protons and 2 neutrons.The binding energy is the difference in rest mass-energy of the nucleus from the total rest mass-energy of it’s component parts.

Calculate the nuclear binding energy of helium.

mHe = 4.002603 amu mp = 1.007825 amu mn = 1.008665 amu

http://www.dbxsoftware.com/helium/

Hints: 1 amu = 1.67e-27 kg or c2 = 931.5 MeV/amu

Example 2.17The molecular binding energy is called dissociation energy.It is the energy required to separate the atoms in a molecule. The dissociation energy of the NaCl molecule is 4.24 eV.

Determine the fractional mass increase of the Na and Cl atoms when they are not bound together in NaCl.

http://www.ionizers.org/water.html

Hints: 1 amu = 1.67e-27 kg or c2 = 931.5 MeV/amu

mNa = 22.98976928 amu

Average mCl = 35.453 amu

Sandin 5.30

A spaceship has a length of 100 m and a mass of 4e+9 kg as measured by the crew. When it passes us, we measure the spaceship to be 75 m long.

What do we measure its momentum to be?

RHICThe diameter of an gold nucleus is 14 fm.

If a Au nucleus has a kinetic energy of 4000 GeV, what is the apparent ‘thickness’ of the nucleus in the laboratory?

http://www.bnl.gov/rhic/

Length contraction

mc2=197*931.5 MeV

Sandin 5.22At the Stanford Linear Accelerator, 50 GeV electrons are produced

• For one of these electrons, calculate– Speed v– – Momentum p– Rest-mass Energy– Kinetic Energy KE


http://www.flickr.com/photos/kqedquest/3268446670/

http://www.daviddarling.info/encyclopedia/L/linear_accelerator.html mc2 = 0.511 MeV

Sandin 5.25A cosmic ray pion (rest mass 140 MeV/c2) has a momentum of 100 MeV/c.

• Calculate– Speed v– – Momentum p– Rest-mass Energy– Kinetic Energy KE


http://www.mpi-hd.mpg.de/hfm/CosmicRay/Showers.html

http://www2.slac.stanford.edu/vvc/cosmicrays/cratmos.html

Sandin 4.26

Spaceship A moves past us at 0.6c followed by Spaceship B in the same direction at 0.8c

What do they measure as their relative speed of approach?

What do we measure as their relative speed of approach?

B A

Sandin 4.28

Spaceship A approaches us from the right at at 0.8c

Spaceship B approaches us from the left at 0.6c

What do they measure as their relative speed of approach?

What do we measure as their relative speed of approach?

B A

Documents

Chapter 02 Special Relativity General Bibliography 1) Various wikipedia, as specified 2) Thornton-Rex, Modern Physics for Scientists & Eng, as indicated