Relativity

Chapter 26

Introduction

• Major Physics accomplishments by the end of the 19th century– Newton’s laws– Universal gravitation– Kinetic-molecular theory– Laws of thermodynamics– Maxwell’s theories which unified electricity

and magnetism

• A major revolution shook the world of Physics at the start of the 20th century.– What was discovered?

• Thomson discovered the electron• The quantum theory was introduced• Einstein’s proposed his special theory of

relativity

The Thomson Atom

The Rutherford Atom

The Speed of Light

• Everyday speeds are much slower than the speed of light.– Newton’s laws describe the motion of

objects at such speeds.• They don’t work with things traveling near

the speed of light.– Newton’s laws don’t place a speed limit on

particles.

Special Relativity

• Einstein published his special theory of relativity in 1905, at age 26.– It was one of the greatest intellectual

achievements of the 20th century.

– He felt that it was simple and consistent in explaining deep contradictions in the old theories.

Relativity Postulates

• Two postulates form the basis of the special theory of relativity.– The laws of physics are the same in all

inertial reference systems.

– The speed of light is constant and does not vary with the motion of the source or the observer. (c = 2.99 792 458 x 108 m/s)

Consequences of the Special Theory of Relativity

• Time slows down when speed increases.

• Object lengths are shortened in the direction of travel.

Understanding Special Relativity

Mr. Tompkins in Wonderland

Concepts Of Special Relativity

An event is a physical happening that occurs at a particular place and time.

The Principle Of Galilean Relativity

• What is a frame of reference?– It is a set of objects that are assumed to be

at rest with respect to the observed event.

– The laws of mechanics are the same in all inertial (non-accelerated) reference frames.• Newton’s law of inertia is valid.

• Common observations– There is no preferred frame of reference

for describing the laws of mechanics.• Examples:

– A ball thrown straight up in a plane

– A ball rolled across the aisle in a train

• Galileo showed this in a boat.

The Speed Of Light Paradox

• If relativity applies to Newtonian mechanics, does it also apply to electricity, magnetism, optics, and other areas?– The speed of light paradox

• A pulse of light sent forward by an observer in a moving boxcar.

26.2

• Resolving the paradox– Either:

• The addition law for velocities is incorrect.

– Or:• The laws of electricity and magnetism are

not the same in all inertial frames.

• What was the conclusion?– The addition law for velocities is

incorrect!

The Speed Of Light

• Does light require a medium to travel through?– The luminiferous ether theory of the 19th

century said “Yes!”.• The ether was supposed to be a massless

fluid in space.• Scientists tried to prove or disprove its

existence.

The Michelson-Morley Experiment

• It was designed to determine the existence of the ether.– It proved that there was no ether.

• The Michelson interferometer did not detect any change in the speed of light when it was rotated 90o.

Einstein’s Principle Of Relativity

The laws of physics are the same in all inertial reference systems.

The speed of light is always constant and does not vary with the motion of the source or the observer.– Anyone who measures the speed of light

will get the same value.• Michelson-Morley experiment

Consequences Of Special Relativity

• Our basic notions of space and time must change. (Newton was wrong!!!)– There is no absolute length.– There is no absolute time.

• Events at different locations that occur simultaneously in one frame are not necessarily simultaneous in another frame.

• Time interval measurements depend on the reference frame in which they are made.– This contradicted Newton’s concept of

time being unchanging.– Example: A boxcar struck by lightning

at both ends “simultaneously”.26.7

Frames of Reference

• There is no preferred frame of reference.– Both observers are correct in their own

reference frames.

Time Dilation

• Example: A vehicle moving to the right at a speed v– The time measured by an observer in a

stationary frame is longer than that measured by the moving observer in his own reference frame.

• A moving clock runs more slowly than an identical stationary clock.

271, 26.8 272

Proper Time

• Proper time (tp)

– The time interval between two events as measured by an observer who sees the events occur at the same place.• Measured with a single clock at rest in the

frame in which the events take place at the same position

– All physical processes slow down .

Equation for Time Dilation

€

t =Δt p

1−v

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

• Time dilation has been verified.– Unstable muons entering our atmosphere

• They travel farther than they should.

– Muons accelerated in the laboratory• They “live” 30 times longer.

– Clocks in jets– Clocks with the lunar astronauts

269

The Twin Paradox

• Speedo and Goslo

Length Contraction

• Proper length (Lp)

– The length of the object measured in the reference frame in which the object is at rest

• Relativistic length contraction (L)– The length contraction only takes place in the

direction of motion.

• Applications for space travel to the stars

26.11, 273

Equation for Length Contraction

€

L = L p 1−v

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

Conservation of Momentum

• We must now generalize Newton’s laws of motion, and the definitions of momentum and energy.– Momentum is not always conserved if we

always use p = mv.

267

Relativistic Momentum

• Relativistic momentum must be conserved in all collisions.

• As v approaches zero, relativistic momentum must approach classical momentum.

Relativistic Momentum Equation

2

c

v1

mvp

⎟⎠⎞

⎜⎝⎛−

=

Relativistic Addition Of Velocities

• Velocities can no longer be added together as we did in Newtonian mechanics.– Objects cannot travel faster than the

speed of light.

268

Relativistic Mass and Energy

• Einstein said that mass and energy are equivalent.

€

E = mc2

Rest Energy Equation

€

ER = mc2

Einstein’s Total Mass-Energy Equation

2

2

T

cv

1

mcE

⎟⎠⎞

⎜⎝⎛−

=

Relativistic Energy and the Equivalence of Mass and Energy• The definition of kinetic energy must also

be modified.

€

KE = ET - ER

Mass-Energy

• Mass is one possible manifestation of energy.– A small amount of mass corresponds to

an enormous amount of energy.• Nuclear reactions

Energy of Subatomic Particles

• Energy involving subatomic particles

€

1 eV = 1.6 x 10−19 J

Electron Rest Energy

• The rest energy of an electron is:

or

J 10 x 8.20 14−

MeV 0.511

Pair Production

• What is Pair Production?– It is the process in which a photon splits

into a particle and an antiparticle.

Pair Annihilation

• What is Pair Annihilation?– It is the process in which a particle and

an antiparticle collide to form two photons.

• Pair annihilation is the opposite of pair production.– In Pair Annihilation, an electron and a

positron can combine to form two photons.• This is necessary for momentum to be

conserved.

Electron-Positron Pairs

• The rest energy of an electron has already been found to be 0.511 MeV.– Therefore, the electron-positron pair

would require a photon whose energy was at least 1.02 MeV.• Gamma rays• Short wavelength x-rays

• Pair production cannot take place in vacuum.– A more massive particle must be

involved for energy and momentum to be conserved.

Properties of Mass

• Mass has two seemingly different properties.– Gravitational attraction

• Causes acceleration

– Inertia• Resists acceleration

Einstein’s Concept of Gravity

• According to Einstein, there is no such thing as gravitational force.– Masses cause a curvature of space-time.

• Gravitational wells

– Mass one tells space-time how to curve and space-time tells mass two how to move.

Verifying General Relativity

• General relativity predicts that light passing a large mass in space should bend toward the mass.– This was verified after World War II.

A Gravitational Lens

Black holes

• The largest concentrations of known mass in the universe– Formed when stars collapse

• Gravity is so great that no matter or light could ever escape.

A Black Hole

Light Years

• One light year is the distance that light travels in one year. – Think of one ly as 1 (c)y

– For a person traveling a distance of 20 ly at 0.5c, the time measured on earth would be:

€

t =d

v=

(20c)y

0.5c= 40 y

General Relativity

Summary Of Important Equations

270

Questions

1, 2, 5 - 8, 10

Pg. 866