Announcements 1/22 (Fri) at 1:30 pm (Sci 12)

Special Seminar on HEP by Dr. Damgov (TTU)

1/22 (Fri) at 3:00 pm (Langford Lab –EE)

Joint Physics-EE Colloquium “Reaching for Green” by Prof. Wetzel (Rensselaer Polytechnic Institute)

*** Course Web Page **

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

Lecture Notes, HW Assignments,

Schedule for thePhysics Colloquium, etc..

Lecture 3 – Chapter. 2 Special Relativity

Outline:

•! Basic Ideas

•! Consequences of Einstein’s Postulates

•! The Lorenz Transformation Equations

•! The Twin Paradox

•! The Doppler Effect

•! Velocity Transformation

•! Momentum & Energy

•! General Relativity & a 1st Look at Cosmology

3

Galilean-Newtonian Relativity

•! Galilean-Newtonian relativity is known as a

“classical” theory.

•! Einstein’s special theory of relativity is known

as a “modern” theory.

4

Galilean Transformation

x

x´

y´

z

y

v

x´ = x – vt

y´ = y

z´ = z

t´ = t Time is absolute

z´

K

K´

O´

O

5

Galilean Transformation

x x´

y´ y

v

x´ = x – vt

y´ = y

z´ = z

t´ = t Time is absolute

K K´

O´ O

vt x´

x

EVENT

6

Historical Perspective

•! Light is a wave.

•! Waves require a medium through which to

propagate.

•! Medium as called the “ether.” (from the

Greek , meaning upper air)

•! Maxwell’s equations assume that light obeys

the Newtonian-Galilean transformation.

7

The Michelson-Morley Experiment

•! Experiment designed to measure small changes in the speed of light was performed by Albert A. Michelson (1852 – 1931, Nobel ) and Edward W. Morley (1838 – 1923).

•! Used an optical instrument called an interferometer that Michelson invented.

•! Device was to detect the presence of the ether.

•! Outcome of the experiment was negative, thus contradicting the ether hypothesis.

•! A.A. Michelson and E.W. Morley, American Journal of Science, 134 – 333, 1887)

8

Einstein’s Postulates

Big problem at the turn of the century:

1.! Michelson-Morley showed that the Galilean

transformation did not hold for Maxwell’s

equation.

2.! Maxwell’s equations could not be wrong.

3.! Galilean transformation did hold for the laws

of mechanics.

4.! Einstein proposed a solution.

9

Inertial Reference Frame

An inertial reference frame is one

–!in which no accelerations are observed in

the absence of external forces.

–!that is not accelerating.

–!Newton’s laws hold in all inertial reference

frames.

10

Principle of Simultaneity

Two events that are simultaneous in one

reference frame (K) are not necessarily

simultaneous in another reference frame (K

´) moving with respect to the first frame.

Recall that in the Galilean transformation time is

considered absolute regardless of the relative

motion of the reference inertial reference frames.

11

Einstein’s Postulates

1.! The Principle of Relativity

2.! The constancy of the speed of light.

12

The Principle of Relativity

All the laws of physics are the same in all

inertial systems.* There is no way to detect

absolute motion, and no preferred inertial

system exists.

*Particular quantities (velocity, momentum, kinetic

energy, …) have different values in different inertial

reference frames, but the laws of physics (conservation

of energy and momentum, …) are the same.

13

The Constancy of the Speed of

Light

Observers in all inertial systems measure

the same value for the speed of light in a

vacuum. (c = 2.9979 x 108 m/s)

14

The Ultimate Speed

•! The speed of light has been defined to be exactly:

c = 299 792 458 m/s •! Light travels at this ultimate speed, as do any

massless particles.

•! No entity that carries energy or information can exceed this speed limit.

•! No particle that does have a mass, can actually reach c.

•! Electrons have been accelerated to at least 0.999 999 999 95 times the speed of light—still less than c.

15

2.4 The Lorentz Transformation

•! Einstein needed to find a new transformation (the

old one being the Galilean transformation).

•! It must fit both the laws of mechanics and

Maxwell’s electrodynamic equations.

•! It must allow time to be relative.

16

Galilean Transformation

x x´

y´ y

v

x´ = x – vt

y´ = y

z´ = z

t´ = t Time is absolute

K K´

O´ O

vt x´

x

EVENT

17

Lorentz Transformation Equations

x´ =

y´ = y

z´ = z

t´ =

x - vt

1- ! 2

1- ! 2

t - vx c2

where

! = v/c!

18

Lorentz Transformation Equations

x´ = " ( x – vt )

y´ = y

z´ = z

t´ = " ( )

1- ! 2

t - vx

c2 where

! = v/c!

where " =

19

Time Dilation and Length

Contraction

20

Time Dilation

#t´ =!#t0!

1 – v2/c2

Clocks moving relative to an observer

are measured by that observer to run

more slowly, as compared to the clock at

rest.

Time where clock is at rest relative to

the observer. Proper time. Time where clock is moving relative to

the observer.

21

Time Dilation

# = " T0´$

For clock in the K´ frame

22

Length Contraction

L0 = " L

L0 > L

23

Length Contraction

L = L0 1- v2/c2

The length of an object is measured to be

shorter when it is moving relative to the

observer than when it is at rest.

Length were observer is at rest

relative to the length being

measured.

Length were observer is moving relative to the

length being measured.

24

Einstein’s Postulates of Relativity:

• Michelson- Morley Experiment – NO AETHER !

• Consequences of Einstein’s Postulates:

1. Relative Simultaneity

2. Time Dilation

3. Length Contraction

SUMMARY

IMPORTANT: space and time coordinates mix

together!

(Not in Classical Physics)

In the “Classical Limit”

?

Nonclassical Physics

Nonclassical Physics

CLASSICAL LIMIT

“Classical Limit”:

v << c

“Classical Limit”:

v << c

<< 1

“Classical Limit”:

v << c

<< 1

~ 1

~1

~1~1

~1~0

~0

~0

~0

x’ ~ x t’ ~ t

In the “Classical Limit”

First case: v = 0, ! = 1

~1

~1~1

~1 ~0

~0

x’ ~ x – vt t’ ~ t

In the “Classical Limit”

General case: v << c, ! ~ 1

OK

Lorentz Transformation of

Distances and Time Intervals

IMPORTANT: space and time distances mix together

This time using Lorentz Transformations

Lorentz Transformation of

Distances and Time Intervals

0 =

0

0

Lorentz Transformation of

Distances and Time Intervals

0 =

-2 m =

-2 m

-2 m

0

0

Einstein’s Postulates of Relativity:

• Michelson- Morley Experiment – NO AETHER !

• Consequences of Einstein’s Postulates:

1. Relative Simultaneity

2. Time Dilation

3. Length Contraction

SUMMARY

EXPERIMENT

Muon Lifetime

Muon (µ) is an elementary

particle

similar to electron, but heavier

(will learn more in Chapter 11)

Muon Lifetime

Muons are created abundantly in

elementary particle showers in the

atmosphere, initiated by energetic cosmic

rays (photons, particles and nuclei).

Muons originate from decays of particles

called “pions” (!) that are the primary

products in these showers

Muon Lifetime

Earth

Cosmic Ray

Neutrinos

Atmosphere

Particle shower

CONCLUSION: According to Classical

Physics,muons should not reach the ground!

BUT THEY DO

The distance between the

creation and detection points

There are many-many

experiments that prove the

consequences of the two

postulates of relativity

Twin Paradox

As an object approaches the speed of light, time slows down.

(Moving clocks are slow) & (Moving rulers are short)

The Theory of Relativity

52

Twin Paradox

•! One twin stays at home.

•! One twin travels on a spaceship at very high speeds.

•! Relativity says traveling twin will age more slowly.

•! But one can say the twin on Earth is traveling w.r.t. the twin in the spaceship and should be the younger.

•! This is the paradox. Who is really younger.

•! Answer: Traveling twin because of accelerations for the traveling twin—non inertial frame..

Earth

Homer

Loner

Planet

Hollywood

Effect of Time on Spaceship

•! Velocity (v) = 0.8c therefore…

•! ! =v/c= 0.8

"= 1

1 - !# 1 - !# = 1 - 0.64 = 0.36 and the $ of 0.36 = 0.6 !!

(!# = 0.8# = 0.64)

" = 1 ÷ 0.6 = 1#/% = 5/3

Physical Results of Trip

!! Homer on Earth ages

25 years!!

!! Loner, traveling at

80% the speed of light

ages 15 years!!

The traveling twin is younger!

As Viewed From Spaceship

•! Loner sees distance of planets contracted by "=

5/3

•! In Loner’s frame distance is 10 light years ÷ 5/3

10 ÷ 5/3 = 6 light years.

•! Therefore t = x/v= 6/0.8 = 7.5 years each way.

•! There and back is 7.5 x 2 = 15 year trip for

Loner!!

Anna travels away

and back at

v = 0.8 c

(! = 5/3)

Bob stays home