UNIT SIXUNIT SIXRelativity and Relativity and BeyondBeyond

Chapter 20Chapter 20RelativityRelativity

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“Everything is relative”

Or, motion depends

upon your point of

view: your frame of reference

Relative Motion in Classical Physics

A floating twig in the water of a moving stream is carried along by the current.

What is its velocity with respect to the stream bank?

What is its velocity with respect to the water?

How are these two velocities related?

All motion is measured with respect to some frame of reference.

From the point of view of someone on the bank, the twig is moving past them with the same velocity as the current.

From the point of view of someone on a boat drifting with the current, the twig is not moving.

From the point of view of someone on a motorboat moving relative to the water, the twig may even be moving backward.

While the boat moves a distance dbe with respect to the earth, the wood moves a distance dwe with respect to the earth.

If the boat and the stream are moving in the same direction, the boat’s velocity relative to the water adds to the water’s velocity relative to the earth to yield the overall velocity of the boat relative to the earth.

vbe vbw vwe

If the boat travels at 4 m/s upstream relative to the water, and the water is

moving at 6 m/s downstream relative to the earth, what is the velocity of the boat

relative to the earth?a) 2 m/s upstreamb) 2 m/s downstreamc) 10 m/s upstreamd) 10 m/s

downstream

b) Since (with respect to the earth) the water is traveling downstream faster than the boat is traveling upstream, the velocity of the boat relative to the earth is 4 m/s - 6 m/s = -2 m/s (downstream).

A motorboat pointed straight across the stream is carried downstream relative to the earth as it moves across the stream.

It ends up at a point on the opposite bank that is somewhat downstream.

The two velocities add as vectors.

If you wanted to hit a point on the bank directly across the stream from your starting point, you would have to point the boat somewhat upstream.

Principle of Relativity

Velocity addition can also be applied to events happening in moving vehicles. For example, if you are walking down the aisle of an

airplane traveling at constant velocity to the earth, your velocity relative to the earth would be the vector sum of your velocity relative to the airplane and the airplane’s velocity relative to the earth.

In practice, you are usually much more aware of your velocity relative to the airplane than your velocity relative to the earth.

The airplane is your frame of reference.

The Speed of Light and Einstein’s Postulates

Recall that light is an electromagnetic wave, consisting of oscillating electric and magnetic fields.

Does light require a medium in which to oscillate, even when traveling through seemingly empty space? An invisible, elastic, and apparently massless medium

hypothesized to exist in the vacuum was called the luminiferous ether.

It was the medium that light waves supposedly traveled through.

It could conceivably provide an absolute reference frame against which all other motion could be measured.

Measurements of the speed of light should reflect this motion through the ether.

If the earth is moving through the ether, the ether is also flowing past the earth.

As the earth travels around the sun, there should be times when the earth is moving in different directions relative to the ether.

The velocity of light should be affected by this motion, much as the velocity of a wave relative to a stream adds to the velocity of the stream to yield the velocity of the wave relative to the earth.

Accurate measurement of the speed of light relative to the earth at different times of the year would let us determine whether or not the earth is moving in a certain direction relative to the ether.

The Michelson-Morley Experiment

Michelson and Morley used an interferometer to detect small differences in the velocity of light or in the distance that the light traveled.

Light waves traveling along the two perpendicular arms interfere to form a pattern of light and dark fringes.

The Michelson-Morley Experiment

At some time during the year the earth should be moving relative to the ether.

No fringe shift was observed; the experiment failed to detect any motion of the earth relative to the ether.

This “failure” was a very important result!

Einstein’s Postulates of Special Relativity

Einstein’s solution to the dilemma of the ether and the speed of light was both simple and radical. Postulate 1: The laws of physics are the same in any

inertial frame of reference. Postulate 2: The speed of light in a vacuum is the

same in any inertial frame of reference, regardless of the relative motion of the source and observer.

The first is just a reaffirmation of the principle of relativity stated earlier.

The second is much more radical: light does not behave like most waves or moving objects.

On a moving vehicle such as an airplane, the velocity of a thrown ball relative to the earth would be the vector sum of the ball’s velocity relative to the plane and the plane’s velocity relative to the earth.

Likewise, a sound wave’s velocity relative to the earth would be the vector sum of the plane’s velocity relative to the earth and the sound wave’s velocity relative to the plane.

However, the plane’s velocity does not add to the light’s velocity to yield the speed of light relative to the earth.

The speed of light measured on the plane must be the same as the speed of that same light measured by an observer at rest on the earth.

This has some very surprising implications.

Time Dilation and Length Contraction

In a light clock, the time taken for light to travel the distance 2d to the mirror and back becomes the basic measure of time t0.

An observer on the earth can also see the time taken for the light beam to travel to the mirror and back.The earthbound observer interprets things differently.

t0 2d

c

If the spaceship is moving with a velocity v relative to the earth, the mirror also moves at that velocity.

According to the earthbound observer, the light must travel along a diagonal path as it moves with the spaceship.

She sees the light beam traveling a longer distance than the spaceship’s observer sees.

If she agrees that the light travels at the speed c, then she will measure a longer time for the light beam’s trip than the spaceship’s observer will measure.

What is the difference in the time intervals measured by the two observers?

t2

2

c 2 v 2 d2

t2

2

d2

c 2 v 2

t 2d2

c 2 v 2

2d

c

1

1 v2

c 2

t0

1 v2

c 2

t0 2d

c

c 2 t2

2

d2 v 2 t2

2

This is the time-dilation formula. The observer on the earth measures a

longer, dilated time than the spaceship’s observer measures.

“Moving clocks run slow.” The time, t0, that the spaceship’s observer

measures is called the proper time.

t t0

1 v2

c 2

t0

where 1

1 v2

c 2

The two observers will also disagree on the distance the spaceship and mirror traveled during this time. The observer on earth marks the position of the spaceship when the

light pulse is emitted and when it returns. The distance L0 between these positions is easily measured. The spaceship’s observer computes the distance by multiplying the

speed v by the time of flight of the light beam. He measures a length L = vt0 for the distance covered by the

spaceship during the flight of the light beam.

The observer on earth would compute the distance as L0 = vt, where t is the dilated time she measures for the flight of the light beam.

L0 is called the rest length, the length measured by the observer at rest relative to the distance being measured.

All other (moving) observers will measure a shorter or more contracted length than the rest length.

“Moving sticks are shorter.”

This is the length-contraction formula.

L 1

L0

where 1

1 v2

c 2

A spaceship traveling at 1.8 x 108 m/s (0.6 c) covers a distance of 900 km as

measured by an observer on earth. What is the distance traveled in this time as

measured by the spaceship’s pilot?

a) 540 kmb) 720 kmc) 1125 kmd) 1500 km

1

1 v2

c 2

1

10.6c

c

21.25

L 1

L0

900 km

1.25720 km

b) .

How much time does it take to cover this distance as measured by the observer on

earth?

a) 4 x 10-3 sb) 5 x 10-3 sc) 7 x 10-3 sd) 8 x 10-3 s

b) .

L0 vt

t L0

v

9 105 m

1.8108 m/s5 10 3 s 5 ms

How much time does it take to cover this distance as measured by the spaceship’s

pilot?

a) 4 x 10-3 sb) 5 x 10-3 sc) 7 x 10-3 sd) 8 x 10-3 s

a) .

L vt0

t0 L

v

7.2 105 m

1.8 108 m/s4 10 3 s 4 ms

If Adele travels to a distant star and back, and her twin sister Bertha remains on

earth, will one twin be older than the other when they are reunited on earth?

a) Adele will be younger.b) Bertha will be younger.c) They will still be the

same age.

a) Adele will be younger.

If Adele travels to a distant star and back, and her twin sister Bertha remains on

earth, will one twin be older than the other when they are reunited on earth?

Since Adele must accelerate, she is not in an inertial reference frame. Therefore, Bertha’s viewpoint is the one that reflects reality: Bertha measures the proper time, Adele’s clock runs slow so Adele will be younger upon her return.

Newton’s Laws and Mass-Energy Equivalence

Accepting Einstein’s postulates requires some major changes in how we think about space and time.

For example, does Newton’s second law of motion still apply when objects are moving at large velocities?

F = ma = p / t In order to maintain conservation of

momentum, Einstein redefined momentum asp = mv

Mechanical energy also must take on a new meaning:

KE = mc2 - mc2

The quantity mc2 = E0 is called the rest energy.

Accelerating an object increases its energy above its rest energy.

Therefore its total energy would be E = mc2.

The rest energy term E0 = mc2 indicates a startling equivalence between mass and energy.

Einstein’s famous equation correctly predicts the amount of energy released in nuclear reactions. Rest-mass energy is

transformed into kinetic energy in nuclear fission and fusion.

A Bunsen burner adds 1000 J of heat energy to a beaker of water. What is the

increase in the mass of the water?

a) 1000 kgb) 1110 kgc) 1.11 x 10-14

kgd) 1.11 x 10-18

kg

E 1000 J

c 3108 m/s

m ?

c) .

E mc 2

m E

c 2

1000 J

3108 m/s 2

1000 J

9 1018 m2/s2

1.1110 14 kg

General Relativity What happens if our frame of reference is

accelerating? Imagine that we are in a

moving elevator, for example.

If the elevator is moving with constant velocity, no experiment that we can do inside the elevator could establish whether or not we are moving.If the elevator is accelerating, a bathroom scale would register a greater weight.

The change in the scale reading could be used to determine if the elevator is accelerating. If the elevator is accelerating upward, the reading will be

higher than normal. If the elevator is accelerating

downward, the reading will be lower than normal.

If the elevator accelerates downward with an acceleration g (free fall), the scale will read zero - apparent weightlessness.

Similarly, a dropped ball will reach the floor with an apparent acceleration that is different from g = 9.8 m/s2.

All these experiments can be interpreted in terms of an apparent acceleration of gravity that differs from g = 9.8 m/s2.

We cannot distinguish these effects from what would happen if the actual acceleration due to gravity were being increased or decreased.

This leads to Einstein’s basic postulate of general relativity, the principle of equivalence: It is impossible to distinguish an acceleration of a

frame of reference from the effects of gravity. In the absence of gravity (such as in outer space),

an acceleration upward of 9.8 m/s2 would be indistinguishable from a gravitational acceleration of 9.8 m/s2.

If a ball is thrown horizontally, its trajectory is the same as a ball thrown on earth. The ball “falls” toward

the floor of the elevator. We can predict its

motion by the same methods used to describe projectile motion.

The upward acceleration of the elevator’s reference frame is equivalent to the presence of a downward gravitational acceleration of the same magnitude.

If the acceleration were equal to 9.8 m/s2, mechanical experiments done in the elevator and on the surface of the earth would have identical results. This can be used to produce an “artificial gravity” effect on

a space station. The centripetal acceleration

of a rotating space station due to a constant rotational velocity would produce a “gravity” with “down” pointing radially outward and “up” pointing toward the center of rotation.

Does a light beam bend in a strong gravitational field?

If a beam of light were used instead of a thrown ball, would it still follow the projectile path in the accelerating elevator? If the acceleration were

large enough, it would. According to the principle

of equivalence, we cannot distinguish an accelerating frame of reference from a frame of reference in a gravitational field.

Does a light beam bend in a strong gravitational field?

If a beam of light were used instead of a thrown ball, would it still follow the projectile path in the accelerating elevator? This prediction has been

confirmed by the bending of starlight that passes near the sun during an eclipse.

Special relativity tells us that different observers moving with respect to one another will disagree on measurements of time and length.

Moving clocks run slow.

Moving sticks are shortened.

General relativity also tells us that an accelerated clock runs more slowly than a nonaccelerated clock. By the principle of equivalence, a clock in a strong

gravitational field should run more slowly. This is called the gravitational red shift. If the period (the time for one cycle) of a light wave

is increased, the frequency is decreased.

Gravity affects how we measure space as well as time. In Euclidean, or ordinary, geometry, two parallel

lines never meet. In non-Euclidean

geometry two parallel lines can meet.

For example, consider lines of longitude on a globe.

Parallel lines eventually intersect due to the curving of the surface of the globe.

Similarly, space is curved near a very strong gravitational field. This represents how things might be pulled into

the center of the field.

Since light rays are bent by strong gravitational fields, they can be pulled into the center of the field as well as particles having some mass.

This figure is a two-dimensional representation of a black hole. Black holes are thought to be very massive

collapsed stars, which generate an extremely strong gravitational field.

Space is very curved in their vicinity.

This figure is a two-dimensional representation of a black hole. The field is so strong that light rays are bent into

the center and do not reemerge. Light gets in but cannot get out. Many astronomical observations suggest the

presence of black holes.

Einstein’s theories of special and general relativity have had an enormous impact on

our concepts of space and time.

Special relativity deals with reference frames that, although moving at speeds near the speed of light, are still inertial (non-accelerating).

General relativity deals with accelerated reference frames. The predictions of these theories have been well confirmed.

For example, the energy released in nuclear reactions is a result of mass-energy equivalence.

Also, astronomical observations of the bending of starlight is evidence of the principle of equivalence between gravity and acceleration.

These ideas excite the imagination and are still very active areas of research.