General theory of relativity: Principle of equivalence

12( 1 10 )

I

G G

G

I

G I

F m a

F m a

ma g

m

m m

proved to in

Principle of equivalence: There is no experiment that can be done in a small confined space that can detect the difference between a uniform gravitational field and an equivalent uniform acceleration.

Bending of light: acceleration, gravitation

According to the astronaut inside the spacecraft, the light curved downward and must have been affected by the acceleration.

Due to the equivalence principle, the same thing must happen on the Earch because of gravity.

Distortions of space-time by a mass

Mass-energy tells spacetime how to curve .

Spacetime curvature tells matter how to move.

Changing of light frequency in gravitational field, gravitational redshift

2 1

2 1

2 2/ /

E E W

E E mgH

for light E hf

and effective mass

m E c hf c

For large changes of altitude

Bending of light and its propagation retardation

Black holes2

2

2

2

mv GMm

r

GMr

v

Escape velocity: initial kinetic energy =potential barrier of the gravitational field.

v c

Gravitational wavesInterferometer

An accelerated mass produces gravitational waves.

Best chance to observe them from collapse into ablack hole, for first fraction of a second of Big Bang, collapse of two neutron stars rotating around each other.

Expected effects on Earth are less than21(1 10 )part in

Review for exam 11. Work independently.

2. Use only a calculator and provided formula sheet,

but no textbook or anything else.

3. The problems will be quite similar to those discussed

on the lectures or given as a homework.

4. Make up exams only with medical excuse. As a rule

make up exams are more difficult, so don’t miss

exam on Monday.

Exam problems1. Standard multiple choice problems for which you don’t need to present your

work.2. Questions or problems where you have to decide whether the statement is True

or False.Example:

Verify whether the statement below are false or true, circle the correct answer:(Max. 2 pts.) The laser light from Earth received for an by an Earth satellite is redshifted.In case of correct answer you will get 2* [1-1/2]=1, in case of wrong answer 2* [0-1/2]= -1; for random responses and several questions number of points close to 0.

3. Problems, where you will have to present you work to get credit. You will also have to select the answer that is closest to the one you obtained, so please avoid arithmetic errors.

False True

Response# 1 2 3 4 5 6 7 8 9 10

Answer 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5

Formula sheet

It will be posted on the web! Also see solutions to HW #4,5 in the end of these lecture notes.

CHAPTER 1

1.1 Classical Physics of the 1890sIn particular mechanics: Newton’s laws, Neutonian principle of relativity, Galilean invariance, also in Chapter 2.1.2 The Kinetic Theory of Gases1.3 Waves and Particles1.4 Conservation Laws and Fundamental Forces1.5 The Atomic Theory of Matter1.6 Unresolved Questions of 1895 and New Horizons

General knowledge, some questions, but no problems!

Three laws describing the relationship between

mass and acceleration.

Newton’s first law (law of inertia): An object in motion with a constant velocity will continue in motion unless acted upon by some net external force.

Newton’s second law: Introduces force (F) as responsible for the change in linear momentum (p):

Newton’s third law (law of action and reaction): The force exerted by body 1 on body 2 is equal in magnitude and opposite in direction to the force that body 2 exerts on body 1.

Isaac Newton (1642-1727)

Cavendish balance

Henry Cavendish(1798) announced that he has

weighted the earth.

Light source

( )

Gravitation

1 2

2

-11 2 2

G=gravitational constant = 6.673(10) 10 /

g

Gm mF

r

Nm kg

E

The weight of a body of mass m on the earth's surface with

radius

Note:

R is

2 2 or E E

E E

Gm m Gmmg g

R R

Newton’s Law of

Gravitation

Primary Results

• Average molecular kinetic energy directly related to absolute temperature

• Internal energy U directly related to the average molecular kinetic energy

• Internal energy equally distributed among the number of degrees of freedom (f ) of the system

(NA = 6.022×1023 mol−1

Avogadro’s Number, number of molecules in 1 mole)

1.4: Conservation Laws and Fundamental Forces

• Recall the fundamental conservation laws:• Conservation of energy

• Conservation of linear momentum

• Conservation of angular momentum

• Conservation of electric charge

• Later we will establish the conservation of mass as part of the conservation of energy

Also in the Modern Context…

• The three fundamental forces are introduced

• Gravitational:

• Electroweak• Weak: Responsible for nuclear beta decay and effective only

over distances of ~10−15 m

• Electromagnetic: (Coulomb force)

• Strong: Responsible for “holding” the nucleus together (interaction between protons and neutrons) and effective at distances less than ~10−15 m.

• 2.1 The Apparent Need for Ether• 2.2 The Michelson-Morley Experiment• 2.3 Einstein’s Postulates• 2.4 The Lorentz Transformation• 2.5 Time Dilation and Length Contraction• 2.6 Addition of Velocities• 2.7 Experimental Verification• 2.8 Twin Paradox• 2.9 Space-time• 2.10 Doppler Effect• 2.11 Relativistic Momentum• 2.12 Relativistic Energy• 2.13 Computations in Modern Physics• 2.14 Electromagnetism and Relativity

CHAPTER 2Special Theory of Relativity

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 

 

v t

Conditions of the Galilean Transformation

• Parallel axes

• 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

1. AC is parallel to the motion of the Earth inducing an “ether wind”

2. Light from source S is split by mirror A and travels to mirrors C and D in mutually perpendicular directions

3. After reflection the beams recombine at A slightly out of phase due to the “ether wind” as viewed by telescope E.

The Michelson Interferometer

The system was set on a rotatable platform

The Analysis (continued)

and upon a binomial expansion, assuming

v/c << 1, this reduces to

Thus a time difference between rotations is given by:

Einstein’s Two Postulates

With the belief that Maxwell’s equations must bevalid in all inertial frames, Einstein proposes thefollowing postulates:

1) The principle of relativity: 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.

2) The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.

The Problem of Simultaneity: “Gedanken” (German) (i.e. thought) experiment

Frank at rest is equidistant from events A and B:

A B

−1 m +1 m

0

Frank “sees” both flashbulbs go off simultaneously.

The Problem of Simultaneity

Mary, moving to the right with speed

is at the same 0 position when flashbulbs go off, but she sees event B and then event A.

−1 m 0 +1 m

A B

Thus, the order of events in K’ can be different!

 

v

 

v

 

v

 

v

 

v

 

v

 

v

 

v

 

v

Lorentz Transformation Equations

Lorentz Transformation Equations

A more symmetric form:

According to Mary and Melinda…

• Mary and Melinda measure the two times for the

sparkler to be lit and to go out in system K’ as times t’1

and t’2 so that by the Lorentz transformation:

• Note here that Frank records x2– x1 = 0 in K with a

proper time: T0 = t2 – t1 or

with T ’ = t’2 - t’1

1) T ’ > T0 or the time measured between two events in moving system K’ is greater than the time between the same events in the system K, where they are at rest: time dilation.

2) The events do not occur at the same space and time coordinates in the two systems

3) System K requires 1 clock and K’ requires 2 clocks.

Time Dilation:Moving Clocks Run Slow

2.7: Experimental Verification

Time Dilation and Muon Decay

Figure 2.18: The number of muons detected with speeds near 0.98c is much different (a) on top of a mountain than (b) at sea level, because of the muon’s decay. The experimental result agrees with our time dilation equation.

Frank’s measurement

So Frank measures the moving length as L given by

but since both Mary and Frank in their respective frames measure L’0 = L0

i.e. the measured length for the moving stick shrinks

and L0 > L.

2.6: Addition of VelocitiesTaking differentials of the Lorentz transformation,

relative velocities may be calculated:

So that…

defining velocities as: ux = dx/dt, uy = dy/dt, u’x = dx’/dt’, etc. it is easily shown that:

With similar relations for uy and uz:

The Lorentz Velocity Transformations

In addition to the previous relations, the Lorentz velocitytransformations for u’x, u’y , and u’z can be obtained by switching primed and unprimed and changing v to –v:

Particular case when

, 0x y

v c u u

Then 'x

u c

The Lorentz Velocity Transformations: an object moves with the speed of light

u’x = c (light or, if neutrinos are massless, they must travel at the speed of light)

22

(1 )'

1 11

xx

x

vcu v c v cu cvu vc v

ccc

Doppler effectsource

v

light, x=ct=n

source, x=vt

x=(c-v)t=nλ’

λ=(c-v)t/n=λ0(1-v/c)

f=1/T=1/(λ/c)=f0/(1-v/c)

Light wave proper period seen in the

source frame T0 (f0=1/ T0)

observer

0

(1 / )

(1 / )

v cf f

v c

period in the observer’s frame T=T0

Astronomy: redshift 0

(1 / )

(1 / )

v cf f

v c

0

f f

/cT c f

Spacetime diagrams

Courtesy: Wikimedia Commons and John Walker

2 2 2 2 2( )s x y z ct Invariant:

2 2 2( )s x c t

For two events

2 0s

2 0s

2 0s timelike

separation

spacelike

separation

lightlike

separation

1 1 2 2 2 1 2 1( , ),( , ); ;x t x t x x x t t t

lightlines

Spacetime interval

Causality: cause-and-effect relations

• 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

Relativistic momentum (2.48)

Total Energy and Rest Energy, Mass-energy Equivalence

We rewrite the energy equation in the form

The term mc2 is called the rest energy and is denoted by E0.

This leaves the sum of the kinetic energy and rest energy to be interpreted as the total energy of the particle. The total energy is denoted by E and is given by

(2.63)

(2.64)

(2.65)

Relativistic Kinetic Energy

The Equivalence of Mass and Energy• By virtue of the relation for the rest mass of a

particle:

• we see that there is an equivalence of mass and energy in the sense that “mass and energy are interchangeable”

• Thus the terms mass-energy and energy are sometimes used interchangeably.

Energy and MomentumThe first term on the right-hand side is just E2, and the second term is E0

2. The last equation becomes

We rearrange this last equation to find the result we are seeking, a relation between energy and momentum.

or

Equation (2.70) is a useful result to relate the total energy of a particle with its momentum. The quantities (E2 – p2c2) and m are invariant quantities. Note that when a particle’s velocity is zero and it has no momentum, Equation (2.70) correctly gives E0 as the particle’s total energy.

(2.71)

(2.70)

Useful formulas

/pc E

1/22

2 2 1/2 0

0

1( ) 1

E Ep E E

c c E

from p mu and2E mc

1/2 1/22

0

2

11 1

E

E

Other Units

1) Rest energy of a particle:Example: E0 (proton)

2) Atomic mass unit (amu):

Example: carbon-12

Mass (12C atom)

Mass (12C atom)

The binding energy is the difference between the rest energy of the individual particles and the rest energy of the combined bound system.

Binding Energy

In the fission of 235U, the masses of

the final products are less than the

mass of 235U. Does this make sense?

What happens to the mass?