Special RelativitySpecial Relativity1.1. Quiz 9.4 and a few comments on quiz Quiz 9.4 and a few comments on quiz

8.24.8.24.2.2. Topics in Special Relativity in this course:Topics in Special Relativity in this course:

Inertial frame of reference and the definition of Inertial frame of reference and the definition of an “event”.an “event”.

The Lorentz Transformation equations of spatial The Lorentz Transformation equations of spatial coordinates or time.coordinates or time.

The Doppler effect: transformation of spatial The Doppler effect: transformation of spatial coordinates and time.coordinates and time.

Velocity transformation: the derivative of Velocity transformation: the derivative of coordinates with respect to time.coordinates with respect to time.

Momentum and Energy, a step into dynamics. Momentum and Energy, a step into dynamics.

3.3. A glimpse into General Relativity if we A glimpse into General Relativity if we have time.have time.

√√

√√

The Doppler effect: transformation The Doppler effect: transformation of spatial coordinates and timeof spatial coordinates and time

Doppler effect (review):Doppler effect (review):

When the light source is moving away at a velocity When the light source is moving away at a velocity vv with with respect to the observer, the frequency the observer respect to the observer, the frequency the observer measures relates to the frequency the light source emits measures relates to the frequency the light source emits through this formula:through this formula:

Here Here θ is the angle between the velocity is the angle between the velocity vv and the line and the line defined by the observer and the source. When defined by the observer and the source. When θ =0, the =0, the course is moving away from the observer. course is moving away from the observer.

A discussion about redshift and the measurement of stars A discussion about redshift and the measurement of stars motion relative to the Earth.motion relative to the Earth.

Example 2.6: direct application of the above formula. Example 2.6: direct application of the above formula.

1 cossource

obsv

ff

2

1

1v

v,

c

Reminder:

SOwave

obs sourcewave source,radial component

vf f

v v

Velocity transformation: the Velocity transformation: the derivative of coordinates with derivative of coordinates with

respect to timerespect to time When a particle moves in frame S with a velocity When a particle moves in frame S with a velocity uu, and in frame S’ with a velocity , and in frame S’ with a velocity u’u’, and S’ moves , and S’ moves in frame S with a velocity in frame S with a velocity vv: : Classical mechanics:Classical mechanics: Special relativity:Special relativity:

The 3 dimensional space: still assume S’ moves The 3 dimensional space: still assume S’ moves in S, along its x-axis with velocity in S, along its x-axis with velocity vv::

Example 2.7Example 2.7

u' u v 1

21 uvu' u vc

Derive on the blackboard

vx' x vt

y' y

z' z

2v

vt' x t

c

1

21 xx x

u vu ' u vc

1

21 xy y v

u vu ' uc

1

21 xz z v

u vu ' uc

Momentum and Energy, a step Momentum and Energy, a step into dynamicsinto dynamics

Momentum of a particle of mass Momentum of a particle of mass mm, , velocity in frame S:velocity in frame S:

The total energy of a particle mass The total energy of a particle mass mm, , velocity in frame S:velocity in frame S:

When , that is ,the When , that is ,the particle has an energy that is its particle has an energy that is its mass:mass:

So the kinetic energy of the particle So the kinetic energy of the particle is:is:

p uum

u

1

2 21u u c

u

2uE mc

2E mcu = 0

12 21 1u u c

21uKE mc 21

2KE muHow do you get to

remember When x is small.

1 1121x

x

Momentum and Energy, a step Momentum and Energy, a step into dynamicsinto dynamics

Example 2.9: Example 2.9: Example 2.10: momentum Example 2.10: momentum

conservation and conservation and

Example 2.11:Example 2.11:

The reference independent energy The reference independent energy and momentum formula: and momentum formula:

21uKE mc EU q V

p uum

2uE mc

2 2 2 2 4E p c m c

22 2 2 4 2 2 4 2 2 2 2 2 42 2

2 2

1 1

1 1up c m c mu c m c m c u c m c

u uc c

Derive it:

Review questionsReview questions

You accelerate two protons with mass You accelerate two protons with mass mm to a to a speed of 0.98speed of 0.98cc and then make them collide and then make them collide head-on. What is the approaching speed one head-on. What is the approaching speed one proton sees the other? What are the total proton sees the other? What are the total momentum and energy of this two particle momentum and energy of this two particle system? (a real example: system? (a real example: http://lhc2008.web.cern.ch/lhc2008/http://lhc2008.web.cern.ch/lhc2008/) )

By what factor would a star’s characteristic By what factor would a star’s characteristic frequencies of light be shifted if it were frequencies of light be shifted if it were moving away from the Earth at 0.01moving away from the Earth at 0.01cc? ?

Preview for the next Preview for the next classclass

Text to be read:Text to be read: In chapter 3:In chapter 3:

Section 3.1Section 3.1 Section 3.2Section 3.2 Section 3.3Section 3.3

Questions:Questions: What is wave-particle duality?What is wave-particle duality? How Planck propose to modify the classical spectral energy How Planck propose to modify the classical spectral energy

density formula to make it match experimental data? density formula to make it match experimental data? What is the formula that brings Einstein the Nobel Prize in What is the formula that brings Einstein the Nobel Prize in

physics in 1921?physics in 1921? Check those that are correct:Check those that are correct:

Roentgen discovered X-ray and obtained a patent for it to make Roentgen discovered X-ray and obtained a patent for it to make him rich.him rich.

Roentgen won the Nobel Prize in physics in 1901 for his discovery Roentgen won the Nobel Prize in physics in 1901 for his discovery of the radiation he named the X-rays. of the radiation he named the X-rays.

The X-rays are produced by smashing a laser beam on a target.The X-rays are produced by smashing a laser beam on a target.

Homework 3, due by 9/11Homework 3, due by 9/11

1.1. Derive this formula Derive this formula with the condition in slide 3.with the condition in slide 3.

2.2. Problem 54 on page 66.Problem 54 on page 66.

3.3. Problem 59 on page 66.Problem 59 on page 66.

4.4. Problem 81 on page 67.Problem 81 on page 67.

1

21 xy y v

u vu ' uc