PHYS-2402

Feb. 3, 2015

Lecture 6

Announcement

Course webpage

http://highenergy.phys.ttu.edu/~slee/2402/

Textbook

Quiz 1: Thursday 30 min; Chap.2 HW1 (due 2/10) 20, 26, 36, 41, 46, 51, 55,

70, 76, 87, 92, 111

Special Relativity

1.  Basic Ideas 2.  Consequences of Einstein’s Postulates

3.  The Lorentz Transformation Equations

4.  The Twin Paradox 5.  The Doppler Effects

6.  Velocity Transformation 7.  Momentum & Energy

8.  General Relativity & a 1st Look at Cosmology

9.  The Light Barrier 10. The 4th Dimension

Chapter 2

Doppler Effect

Relativistic Dynamics

Outline:

•  Relativistic Momentum •  Relativistic Kinetic Energy •  Total Energy •  Momentum and Energy in Relativistic Mechanics •  General Theory of Relativity

2mcEINTERNAL =

2222

)(mccpE +=

E = TOTAL

ENERGY

E = INTERNAL

ENERGY

Expressions for (total) Energy

and Momentum of a particle of mass m,

moving at velocity u

Kinetic Energy = KE

!"u# !"

u# !"u#

!"u# !"u#

Is there Absolute Causality?

Might cause precede effect in one reference

frame but effect precede cause in different

reference frame(s)?

e.g. can someone see you

first die, and then

see you get born?

Is there Absolute Causality?

Might cause precede effect in one reference

frame but effect precede cause in different

reference frame(s)?

e.g. can someone see you

first die, and then

see you get born?

Let’s assume that the order of events is

changed in some reference frame S’

0'

0

<!

>!

t

t

Is that Possible?

!!!!t and !!!!t’ are the time intervals between

the same two events observed in

S and S’, respectively

Let’s assume that the order of events is

changed in some reference frame S’

0'

0

<!

>!

t

t

Is that Possible?

!!!!t and !!!!t’ are the time intervals between

the same two events observed in

S and S’, respectively

!"

#$%

&+

!

!"!=!

!"

#$%

&!+!"=!

1'

'

2

2

t

x

c

vtt

txc

vt

v

v

#

#

Using Lorentz transformations….

0'

0

<!

>!

t

tif then

!"

#$%

&+

!

!"!=!

!"

#$%

&!+!"=!

1'

'

2

2

t

x

c

vtt

txc

vt

v

v

#

#

Using Lorentz transformations….

0'

0

<!

>!

t

tif then

tv

cx

t

x

c

v

!>!

<!"

#$%

&+

!

!"

2

201

!"

#$%

&+

!

!"!=!

!"

#$%

&!+!"=!

1'

'

2

2

t

x

c

vtt

txc

vt

v

v

#

#

Using Lorentz transformations….

0'

0

<!

>!

t

tif then

tv

cx

t

x

c

v

!>!

<!"

#$%

&+

!

!"

2

201

Impossible

v2 > c2 ??

Is there Absolute Causality?

Might cause precede effect in one reference

frame but effect precede cause in different

reference frame(s)?

e.g. can someone see you

first die, and then

see you get born?

YES

NO

NO

Some Examples

Relativistic Dynamics

Problems 1. What is the momentum of an electron with K =mc2?

2. How fast is a proton traveling if its kinetic energy is 2/3 of its total energy?

Problems 1. What is the momentum of an electron with K =mc2?

2. How fast is a proton traveling if its kinetic energy is 2/3 of its total energy?

2 2 2 2 4E p c m c= +

22 22 2 2 2 2 2 2 24 3E mc Kp m c m c m c m c mc

c c! "+! "= − = − = − =$ %$ %

& ' & '

( )2 22 2 33 3

K E mc K E mc= = + =

( )

2

21 /

mcEV c

=−

( )

2

2

1 1 83 19 31 /

V V ccV c

! "= − = =$ %& '−

Problem An electron initially moving with momentum p=mc is passed through a retarding potential difference of V volts which slows it down; it ends up with its final momentum being mc/2. (a) Calculate V in volts. (b) What would V have to be in order to bring the electron to rest?

Problem An electron initially moving with momentum p=mc is passed through a retarding potential difference of V volts which slows it down; it ends up with its final momentum being mc/2. (a) Calculate V in volts. (b) What would V have to be in order to bring the electron to rest?

p=mc: ( ) ( ) ( )2 2 22 2 2 2 2 21 2E p c mc mc mc mc= + = + =

Thus, the retarding potential difference

p=mc/2: ( )2

22 2 22

1 52 2

E mc mc mc! "= + =# $% &

( )2 2 6 51 2

52 0.3 0.3 0.5 10 1.5 102

E E E mc mc eV eV! "

Δ = − = − ≈ ≈ × = ×' (' () *

51.5 10V V= ×

(a)

(b) ( )2 2 2 5 51 22 2 1 2.1 10 2.1 10E mc E mc E mc eV V V= = Δ = − ≈ × = ×

An unstable particle of mass m moving with velocity v relative to an inertial lab RF disintegrates into two gamma-ray photons. The first photon has energy 8 MeV in the lab RF and travels in the same direction as the initial particle; the second photon has energy 4 MeV and travels in the direction opposite to that of the first. Write the relativistic equations for conservation of momentum and energy and use the data given to find the velocity v and rest energy, in MeV, of the unstable particle.

before after

photon 1 photon 2

Problem An unstable particle of mass m moving with velocity v relative to an inertial lab RF disintegrates into two gamma-ray photons. The first photon has energy 8 MeV in the lab RF and travels in the same direction as the initial particle; the second photon has energy 4 MeV and travels in the direction opposite to that of the first. Write the relativistic equations for conservation of momentum and energy and use the data given to find the velocity v and rest energy, in MeV, of the unstable particle.

( )1 2

21 /ph phE Emvc cv c

= −−before after

momentum conservation

( )

2

1 221 /ph ph

mc E Ev c

= +−

energy conservation

( )1 22

8 4 41 /

ph phmvc E E MeV MeV MeVv c

= − = − =−

( )

2

1 228 4 12

1 /ph ph

mc E E MeV MeV MeVv c

= + = + =−

( ) 4 1( ) 12 3

v ac b= = =

(a)

(b)

( ) ( ) ( )221 2 1 / 12 1 1/ 9 11.3ph phmc E E v c MeV MeV= + − = − ≈

photon 1 photon 2

Problem

Problem A moving electron collides with a stationary electron and an electron-positron pair comes into being as a result. When all four particles have the same velocity after the collision, the kinetic energy required for this process is a minimum. Use a relativistic calculation to show that Kmin=6mc2, where m is the electron mass.

before after energy conservation 2

1 24E mc E+ =

1 24p p= momentum conservation

( ) ( )2 22 2

1 1E mc p c= +

( ) ( )2 22 2

2 2E mc p c= +

21 24E mc E+ =

1 24p p=

( ) ( ) ( ) ( ) ( )2 22 2 22 2 2

1 1 2 112 16 1616

E E mc mc E mc p c! "+ + = = +# $% &

( ) ( ) ( ) ( )2 22 2 2 2 21 1 12 16E p c E mc mc mc− + + =

( )22mc

( )22 2 21 14 / 2 7E mc mc mc= =

1p 24p

2 21 1 6K E mc mc= − =

In the center-of-mass RF:

before after

1 'p 1 'p2

12 ' 4E mc= ( )2 21 ' ' 2 ' 2E mc mcγ γ= = → =

( )22

' 1 31 '4 2

vv c

c− = → = relative speed

'2vV =

( )2' 3 4 3 48

1 3 / 4 7 491 '/v Vvvv c+

= = = =++ ( )

22 2

1 61 48 / 49mcK mc mc= − =

−

•  General relativity is the geometric theory of gravitation published by Albert Einstein in 1916.

•  It is the current description of gravitation in modern physics.

•  It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a geometric property of space and time.

•  In particular, the curvature of space-time is directly related to the four-momentum (mass-energy and momentum).

•  The relation is specified by the Einstein’s field equations, a system of partial differential equations. (graduate level course)

General Relativity

•  Many predictions of general relativity differ significantly from those of classical physics.

•  Examples of such differences include gravitational time dilation, the gravitational red-shift of light, and the gravitational time delay.

•  General relativity's predictions have been confirmed in all observations and experiments to date.

•  However, unanswered questions remain, solution is the quantum gravity!! ----- sounds quite complicate..

General Relativity Special Theory of RelativityThe two postulates:

earthBUT:Accellerating

frames

?

Special Theory of Relativity:

General Theory of Relativity:

locally

Special Theory of Relativity:

General Theory of Relativity:

locally

Deals exclusively with

globally

INERTIAL FRAMES -

v = constant

Deals also with

Accelerating -

LOCALLY INERTIAL FRAMES

Acceleration

Gravitational Force

Profound

Link

Equivalence Principle

Accelerating reference frames are indistinguishable from a gravitational force ???

See what this means!!

Try some experiments

Constant velocity

Cannot do any experiment to distinguish accelerating frame from gravitational field

t=0 t=to t=2to

Floor accelerates upward to meet ball

Constant accel.

t=0 t=to t=2to

? ?

Light follows the same path Path of light beam in our frame

Velocity = v

t =0

Velocity = v +ato

t =to

Path of light beam in accelerating frame

Velocity = v+2ato

t =2to

Light

37

Is light bent by gravity?

•  If we can’t distinguish an accelerating reference frame from gravity…

•  and light bends in an accelerating reference frame…

•  then light must bend in a gravitational field But light doesn’t have any mass. How can gravity affect light?

Maybe we are confused about what a straight line is ?

38

Which of these is a straight line?

A

B

C

A.  A B.  B C.  C D.  All of them

39

Which of these is a straight line?

A.  A B.  B C.  C D.  All of them

A

B

C

40

Straight is shortest distance!! •  They are the shortest distances determined by

wrapping string around a globe. On a globe, they are called ‘great circles’.

•  This can be a general definition of straight, and is in fact an intuitive one on curved surfaces

•  It is the one Einstein used for the path of all objects in curved space-time

•  The confusion comes in when you don’t know you are on a curved surface.

41

Mass and Curvature •  General relativity says

that any mass will give space-time a curvature

•  Motion of objects in space-time is determined by that curvature

43!

Idea behind geometric theory •  Matter bends space and time. •  Bending on a two-dimensional

surface is characterized by the radius of curvature: r

•  Einstein deduced that 1/r2 is proportional to the the local energy and momentum density

•  The proportionality constant is

•  where G is Newton's constant

€

8πGc 2

44

A test of General Relativity

•  Can test to see if the path of light appears curved to us

•  Local massive object is the sun

•  Can observe apparent position of stars with and without the sun

45!

Eddington and the Total Eclipse of 1919

Apparent position of star

Actual position of star

Measure this angle to be about 1.75 arcseconds

Q:: Can we test to see if the path of light appears curved to us?

Curved Space-time:

(Intercontinental flights)

47!

Space is Curved? •  Einstein said to picture gravity as a warp in space

•  Kepler’s Laws can all be explained by movement around these “puckers”

•  Everything moving is affected, regardless of mass

48

Other Consequences of GR •  Time dilation from gravity effects •  Gravitational Radiation!

–  Created when big gravity sources are moved around quickly

–  Similar to the electromagnetic waves that were caused by moving electron charges quickly

•  Black Holes •  Expanding Universe (although Einstein

missed the chance to predict it! – He didn’t believed)

49!

Gravitational time dilation •  Gravity warps both space and time!

•  At 10,000 km above the Earth’s surface, a clock should run 4.5 parts in 1010 faster than one on the Earth

•  Comparing timing pulses from atomic oscillator clocks confirms the gravitational time dilation in 1976 to within 0.01%.

•  Corrections are now standard in the synchronizing satellites

•  This correction needed in addition to the special relativity correction for GPS

50

Gravitational Radiation •  When a mass is moved, the curvature of

space-time changes •  Gravitational radiation carries energy and

momentum and wiggles mass in its path

51

Evidence for Gravity Waves •  In 1974, Joseph Taylor and his student Russell

Hulse discovered a binary neutron star system losing energy as expected from gravitational radiation

Phy107 Fall 2006 52

Direct Detection of Gravity Waves LIGO is a collection of large laser interferometers searching for gravity waves generated by exploding stars or colliding black holes

Phy107 Fall 2006 53!

The Big Bang •  In 1929 Observation of nearby and far away

galaxies indicate that everything is receding from us. –  Key physics needed to understand this is the

simple Doppler shift of light waves. Waves from sources moving away from us are stretched out or lower frequency.

•  Extrapolating backwards indicates that all the galaxies originated from the same source 14 billion years ago.

•  In 1964 radiation from the early stages of that explosion was detected. –  Again the Doppler shift was the key since the

waves were shifted to low frequency - microwave

Nobel Prize in 2006 •  For the universe to start small and expand space and

time must be thing that can expand(or contract) –  General relativity was key physics needed to understand

that process

•  However, a simple model of that would predict such a universe would not have clumps of matter(stars, galaxies)

•  Unless those clumping were present very early on

•  2006 Nobel prize was given to the people who designed the COBE experiment which was sensitive enough to see those clumping in the CMB