Chapter 29 - PowerPoint Presentation for College Physics

Copyright © 2010 Pearson Education, Inc.

Lecture Outline

Chapter 29

Physics, 4th Edition

James S. Walker


Chapter 29

Relativity


Units of Chapter 29

• The Postulates of Special Relativity

• The Relativity of Time and Time Dilation

• The Relativity of Length and Length

Contraction

• The Relativistic Addition of Velocities

• Relativistic Momentum

• Relativistic Energy and E = mc2


Units of Chapter 29

• The Relativistic Universe

• General Relativity


29-1 The Postulates of Special Relativity

The postulates of relativity as stated by Einstein:

1. Equivalence of Physical Laws

The laws of physics are the same in all inertial frames of

reference.

2. Constancy of the Speed of Light

The speed of light in a vacuum, c = 3.00 x 108 m/s, is the

same in all inertial frames of reference, independent

of the motion of the source or the receiver.



The first postulate is certainly reasonable; it

would be hard to discover the laws of physics if

it were not true!

But why would the speed of light be constant? It

was thought that, like all other waves, light

propagated as a disturbance in some medium,

which was called the ether. The Earth’s motion

through the ether should be detectable by

experiment. Experiments showed, however, no

sign of the ether.



Other experiments and measurements have

been done, verifying that the speed of light is

indeed constant in all inertial frames of

reference.

With water waves,

our measurement

of the wave speed

depends on our

speed relative to

the water:



But with light, our measurements of its speed

always give the same result:



The fact that the speed of light is constant also

means that nothing can go faster than the

speed of light – it is the ultimate speed limit of

the universe.


29-2 The Relativity of Time and Time

Dilation

To begin to understand the implications of

relativity, consider a light clock:

The time it takes

for light to make a

round trip is:



Dilation

Now, look at the clock moving at a speed v:

The light has to travel farther. Now the round trip

time is:



Dilation

Therefore, a moving clock will appear to run

slowly.



Dilation

As the speed gets closer to the speed of light, the

clocks run slower and slower:



Dilation

This result applies to any kind of clock or

process that is time-dependent – if it did not, the

first postulate would be violated.

Definitions

Event: a physical occurrence that happens at a

specified location at a specified time.

Proper time: the amount of time separating two

events that occur at the same location.



Dilation

Time dilation has been measured with extremely

accurate atomic clocks in airplanes, and also is

frequently observed in subatomic particles.

Another consequence of time dilation is that

different observers will disagree about the

simultaneity of events occurring at different

places.


29-3 The Relativity of Length and Length

Contraction

The observer on Earth sees the astronaut’s

clock running slow; it takes him 25.6 years to

go from Earth to Vega, but only 3.61 years have

passed on the astronaut’s clock.



Contraction

But how does it appear to the astronaut, who

thinks his clock is fine? He sees the distance as

contracted instead – for him, Vega is only 3.57

light-years away.



Contraction

Proper length, L0: The proper length is the distance

between two points as measured by an observer who is

at rest with respect to them.

So in the above example, 25.3 light-years is the

proper length.

With some arithmetic, we find:



Contraction

Length contraction as a function of v:



Contraction

Important note:

Length contraction

occurs only in the

direction of motion.

Other directions are

unaffected.


29-4 The Relativistic Addition of Velocities

Suppose two space ships are heading towards

each other, each with a speed of 0.6 c with

respect to Earth. How fast do the astronauts in

one ship see the other ship approach? It can’t be

1.2 c, but what is it? Here we give the answer:


29-4 The Relativistic Addition of Velocities

So in the above example, the relative speed

would be 0.88 c.

Below is a plot of the speed a rocket would have

if it increased its speed by 0.1 c every time it

fired its rockets.


29-5 Relativistic Momentum

If adding more and more energy to a rocket only

brings its speed closer and closer to c, how can

energy and momentum be conserved?

The answer is that momentum is no longer

given by p = mv.


29-5 Relativistic Momentum

As the speed gets closer and closer to c, the

momentum increases without limit; note that the

speed must be close to the speed of light before

the difference

between

classical and

relativistic

momentum is

noticeable:


29-6 Relativistic Energy and E = mc2

If the momentum increases without limit, the

energy must increase without limit as well:



The rest energy of ordinary objects is immense!

In nuclear reactors, only a fraction of a percent

of the mass of fuel becomes kinetic energy, but

even that is enough to create enormous amounts

of power.



Every elementary particle, such as the electron,

has an antiparticle with the same mass but

opposite charge. The antiparticle of the electron

is called the positron.

Mass:

Charge:



When an electron

and a positron

collide, they

completely

annihilate each

other, emitting only

energy in the form

of electromagnetic

radiation.



We can find the relativistic kinetic energy by

subtracting the rest energy from the total

energy:



At ordinary speeds, the relativistic kinetic energy

and the classical kinetic energy are

indistinguishable.


29-7 The Relativistic Universe

It may seem as though relativity has nothing to

do with our daily lives. However, medicine

makes use of radioactive materials for imaging

and treatment; satellites must take relativistic

effects into account in order to function

properly; and space exploration would be a

disaster if relativistic effects were not handled

properly.


29-8 General Relativity

Einstein thought about the distinction between

gravitational force and acceleration, and

concluded that within a closed system one

could not tell the difference.



This leads to the principle of equivalence:

All physical experiments conducted in a uniform

gravitational field and in an accelerated frame of

reference give identical results.

Therefore, the people in the elevators on the

previous page cannot, unless they are able to

see outside the elevators, tell if they are in a

gravitational field or accelerating uniformly.



When the elevator is moving at a constant

speed, the light from the flashlight travels in a

straight line. When the elevator accelerates,

the light bends.



The principle of equivalence then tells us that

light should bend in a gravitational field as well.



This gravitational

bending of light can be

observed during a

solar eclipse, when

stars appearing very

close to the Sun can

be seen.



If the gravitational field is strong enough, light

may be bent so much that it cannot escape. An

object that is this dense is called a black hole.

Calculations show that the radius of a black

hole of a given mass will be:

Plugging in the numbers shows us that the

Earth would have to have a radius of about

0.9 cm in order to be a black hole.



One way to visualize the bending of light around

massive objects is to imagine that space itself is

bent (there is a deeper truth to this as well). The

region around a black hole then might look like

this:


Summary of Chapter 29

• The laws of physics are the same in all inertial

frames of reference.

• The speed of light in a vacuum is the same in all

inertial frames of reference, independent of the

motion of the source or the receiver.

• Clocks moving with respect to one another keep

time at different rates. An observer sees a

moving clock running slowly:



• Length in the direction of motion appears

contracted:

• Relativistic velocity addition:

• It is impossible to increase the speed of an

object from less than c to greater than c.



• Relativistic momentum:

• Total relativistic energy:



• Rest energy:

• Relativistic kinetic energy:

• Principle of equivalence: All physical

experiments conducted in a gravitational field

and in an accelerated frame of reference give

identical results.



• For an object of mass M and radius R to be a

black hole, its radius must be less than the

Schwarzschild radius:

Documents

Chapter 29 - PowerPoint Presentation for College Physics