PHYS-3301

Sep. 12, 2017

Lecture 5

Announcement

Course webpage

http://www.phys.ttu.edu/~slee/3301/

Textbook

HW1 (due 9/14)

Chapter 220, 26, 36, 41, 45, 50, 51, 55, 58

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

What about y and z coordinates?

(x - direction of motion)

Parallel to the

Direction of

Relative motion

Orthogonal to the

Direction of

Relative motion

Doppler Effect

Relativistic Dynamics

Outline:

• Relativistic Momentum

• Relativistic Kinetic Energy

• Total Energy

• Momentum and Energy in Relativistic Mechanics

• General Theory of Relativity

• Next Week –Quantum Physics

Expressions for (total) Energy

and Momentum of a particle of mass m,

moving at velocity u

2mcEINTERNAL =

2222

)(mccpE +=

E = TOTAL

ENERGY

E = INTERNAL

ENERGY

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

Problems1. 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?

Problems1. 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

⎛ ⎞= − = =⎜ ⎟⎝ ⎠−

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

ProblemAn electron initially moving with momentum p=mc is passed through a retarding potential differenceof V volts which slows it down; it ends up with its final momentum being mc/2. (a) Calculate V involts. (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 = − ≈ × = ×

ProblemThe kinematic energy of a proton is half its internal energy. (a) What is the proton’s speed? (b)What is its total energy? (c) Determine the potential difference V through which the proton wouldhave to be accelerated to attain this speed.

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

before after

photon 1photon 2

Problem

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

( )1 2

21 /ph phE Emvc cv c

= −−before after

momentumconservation

( )

2

1 221 /ph ph

mc E Ev c

= +−

energyconservation

( )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 1photon 2

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

before afterenergy conservation2

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