Introduction. Chapter 13

Chapter 13

The Relativistic Revolution!

• Introduction.

• Galilean (“classical”) Relativity.

• Special Theory of Relativity.

• Time Dilation

• Length Contraction

• Addition of Velocities

• Modification to the Laws of Physics

• Doppler Effect

• Space-time Diagram

• Summary.

Introduction

The period from 1895 to 1905 was arguably the most “productive” and “revolutionary” time in the history of science. Many of the things we take for granted today had their origins during those ten- years. For example:

• In papers published in 1895-1898, Wilhelm Röntgen (1845-1923) announced the discovery of a new kind of ray (the X-ray).

• In 1896, Henri Becquerel (1852-1908) discovered natural radioactivity.

• In 1897, J.J. Thomson (1856-1940) discovered the electron and measured the ratio of their charge (e) to their mass (m), i.e., e

m^.

• In 1900-1, Max Planck (1858-1947) announced his quantum hypothesis; that “vibrations” had specific and well-defined amounts of energy associated with them. So, energy was not

continuous but discrete and made up of packages he called quanta.

• In 1905, Albert Einstein (1879-1955) announced his explanation of the photoelectric effect and his special theory of relativity.

In this chapter we will consider the theory of special relativity and its repercussions.

It is difficult to classify the theory of relativity, of which the special theory is only a part, according to any of the major branches of science. Rather, it is a part of the fundamental fabric of the physical world.

Newton’s Laws and Newtonian mechanics appear to fit our everyday experiences very well but, in truth, they are limiting cases and not valid generally. Also, unlike most of what we have considered in this course, relativity did not spring primarily from observations nor as a way of

reconciling different points of view; rather, it resulted from a critical examination of well-known and widely accepted principles. For example, until the theory of relativity, no one had questioned the concept of absolute time. In The Scholium in the introductory section of his Principia, Newton described absolute time as:

“absolute, true, and mathematical time, of itself, and from its own nature flows equally without regard to anything external. ...”

So time had nothing to do with space. (In this context, by space we mean the boundless extent in which objects move and events occur; before the advent of relativity, it has three dimensions.)

Class discussion question: If time is measured by

clocks, what is space measured by? Space can be measured by rulers!

The meaning of past , present, future, seemed obvious; the present was simply an infinitesimally small interval between past and future. All

classical mechanics was based on this simple concept with no restrictions on time, nor time intervals, and it seemed to fit our everyday experiences very well. However, as we will see, relativity places strict limits on some of these concepts; for example, the interval between past and present has a finite extension in time that depends on the distance in space between an event and its observer.

Newton also speculated about absolute motion, defining it (extremely vaguely) as:

“... the translation of a body from one absolute place into another.”

He does not define what he means by absolute place, rather he leaves it as an intuitive concept.

Some 200 years after Newton’s speculations about absolute motion, the subject rose again with Maxwell’s e-m theory. As we saw in chapter 12, hiss theory predicted that the speed of light was fixed in any given medium, depending on only two constants of the medium. For example, in vacuum,

c = 1

ε µ_{o o}.

But the question was, what was this speed relative to? The Earth? The Moon? Surely, the values of ε_o and µ_o, which refer to vacuum, will be the same irrespective of where they are measured. Since the Earth and Moon move relative to each other, what was the appropriate “reference frame” in which the speed of light is “c”? At the time Maxwell

concluded that the reference frame for the

propagation of light was the “ether”. But 15 years after Maxwell’s theory, Michelson and Morley failed to confirm the presence of the ether.

There were many attempts to account for the apparent absence of an observable motion of the ether, none of which proved satisfactory, until Einstein proposed his special theory of relativity.

In this chapter we will review briefly the Galilean and Newtonian concepts of relativity and then introduce some of the results of Einstein’s theory.

The theory of relativity consists of two rather different theories:

• Special relativity (1905): involves a comparison of measurements made in different inertial reference frames moving with constant velocity relative to one another,

and

• General relativity (1916): involves a comparison of measurements made in different accelerated reference frames and gravity.

Special Relativity requires fairly straightforward mathematics and is applicable to a wide range of situations in physics. It also has a role in practical applications, e.g., in the design of particle

accelerators, the timing involved with the global positioning system (GPS). These devices would not woork if designed according to Newtonian

mechanics! General Relativity is much more mathematically intense and its applications are mostly in the area of gravitation and it is of particular importance in cosmology.

The theory of relativity was probably Einstein’s most important discovery; certainly the one that brought him popular fame. Conceptually, it

represents one of the boldest ideas in the history of science. However, when Einstein was awarded Nobel Prize in 1921, it was for his work on the Photoelectric Effect - which we will look at in the next chapter - not for his theory of relativity.

Galilean (“classical”) relativity

In Galilean (or Newtonian) relativity it is assumed that relative motion has no effect on space or time.

For example, imagine that two people have identical clocks, i.e., with identical ticking rates.

Then if one of them ran past the other (who is stationary), they would both agree that the ticking rates are still exactly the same.

Similarly, imagine a stationary person (called an

“observer”) is holding a one-meter rule, when an identical one-meter rule flies by. The observer will

measure the length of his stationary rule and the moving rule to be the exactly the same.

Now, imagine a stationary observer, O, is watching two people, A and B, on a walkway. Let us assume that the walkway is moving to the right with a

speed of 5cm/s. Person A who is standing still, i.e., at rest, on the walkway, will appear to O to be moving to the right at the same speed as the walkway, i.e., 5cm/s. But, if person B is walking at a speed of 10cm/s relative to the walkway, he will appear to O to be moving with a speed of 10cm s+5cm s=15cm s, i.e., a simple addition of speeds.

It turns out that none of these observation is true although the discrepancies, at these everyday speeds, would be so small as to be insignificant and not measurable. However, at much higher speeds we will see that the results must be modified.

A B

O

5 cm/s 10 cm/s

There is one aspect of the classical relativity that is true. As we saw in chapter 10, Galileo argued, through the mouth of Salviati in his Diologo, that if you are in a cabin below decks in a ship traveling at a uniform (constant) rate then:

“You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.”

In other words, there is no experiment that you can perform that will tell you that the ship is moving (providing the velocity is uniform, i.e, there is no acceleration). As another example, consider a child, sitting in a truck, throwing a ball vertically into the air and catching it. If an observer O and the truck

are both stationary, the child and the observer will agree that the motion of the ball obeys Galileo’s and Newton’s Laws.

O (a)

If now the truck moves to the right at a constant velocity and the child throws the ball vertically upward again, what happens? As far as the child is

concerned, the ball rises and falls

vertically, satisfying the Laws of motion, just as it did when the truck was stationary. On the other hand, to the observer the situation appears different; the ball follows a parabolic path. Nevertheless, and this is the crucial point, Newton’s Laws will account for the observed motion. Although scenarios (a) and (b) appear different, Newton’s Laws hold in both the reference frame of the observer and the reference frame of the child. So, for example, if the child was enclosed in a large box on the bed of the truck, the child could not tell if the truck was moving at a constant velocity or whether it was at rest. So, one cannot define absolute rest .

(b) O

Reference frames in which Newton’s Laws hold are referred to as inertial reference frames and it is to these reference frames that the special theory of relativity applies.

Special theory of Relativity

So, what’s wrong with the first three examples of Newtonian relativity, I gave? In order to answer those questions, we must look at the Newtonian (mis)conceptions about space and time. And that is precisely what Albert Einstein did.

The year 1905 was Einstein’s annus mirabilis; in that year he published four papers in the journal Annalen der Physik:

• “On a Heuristic Viewpoint Concerning the Production and Transformation of Light”, March 18, in which he introduced his theory of the

photoelectric effect (discussed in the next chapter).

• “On the Motion - Required by the Molecular Kinetic Theory of Heat - of Small Particles

Suspended in a Stationary Liquid”, May 11, in which he explained the random movement of very small objects as direct evidence of molecular action, so-called Brownian movement.

• “On the Electrodynamics of Moving Bodies ”, June 30, in which he introduced his theory of special relativity.

• “Does the Inertia of a Body Depend Upon Its Energy Content?”, September 27, in which he introduces mass-energy equivalence, which up to that time had been considered separate concepts. It is the paper that E=mc² first appears (as “Masse um L

V²”, where Einstein uses L for energy and V as the speed of light).

His theory of special relativity is based on two simple postulates:

“1. The laws according to which the nature of physical systems alter are independent of the manner in which these changes are referred to two coordinate systems which have a translatory motion relative to each other.

2. Every ray of light moves in the “stationary coordinate system” with the same velocity V, the velocity being independent of the condition whether this ray of light is emitted by a body at rest or in motion.”

Basically, the postulates state:

1. Physical laws and principles have the same form in all inertial frames.

2. The velocity of light in any inertial frame is independent of the velocity of that frame, i.e., the velocity of light is independent of the motion of its source.

The first generalizes Galileo’s relativity; if there were inconsistencies in the laws between frames, then which frame is the absolute frame ?

If we have to ask that question then the frames would not be equivalent. But, as we will see, in order to satisfy the first postulate some

modifications are necessary to the classical laws and equations. The second postulate essentially re- states the Michelson-Morley result. It is also required by postulate 1; if the speed of light were not the same in all inertial frames, then the frames would not be equivalent.

Using these postulates Einstein showed that there is no such thing as absolute space nor absolute time intervals. We will see that the distance between points and time intervals between events depend on the reference frame in which they are measured.

Also, events at different locations that are observed to occur simultaneously in one frame are not

necessarily observed to be simultaneous in another frame moving uniformly with respect to the first!

Imagine a spaceship is traveling from the

neighborhood of one star X towards another star Y,

at a speed v= 0 5. c. Postulate 2 tells us that an observer on the spaceship will measure the speed of light originating from X and Y to be c.

Time dilation

To show that time is not absolute but depends on the reference frame, we will measure time intervals

using a light clock. In a light clock, a pulse of light is reflected between two mirrors a distance L₀ apart. Each time the pulse is reflected from M₁ will call a “tick”. So, the time interval between “ticks” is

T L

0 2 0c

= distance =

speed .

M¹ M²

L⁰

c c

v= 0 5. c

X Y

Now imagine our clock is moving to the right, with velocity v, as shown in (b), below. We define a “tick”

in the same way. Since, the speed of light is the same in (a) and (b) - because of postulate 2 - the time interval between “ticks” of the moving clock is

T= distance = c speed 2l _.

But l > L0 because in the time it takes for the light pulse to travel from M₁ to M₂, the clock will have moved a finite distance. So, T> T₀, i.e., the time interval between the “ticks” of the moving clock (b) is greater than the time interval between the “ticks”

of the stationary clock (a). So, a moving clock appears to run more slowly than a stationary clock!

M₁ M₂

L₀

M¹ M₂

L⁰

M¹ v

(a) (b)

l l

The relationship between the time intervals is actually

T T

v c T

=

−

( )

0

2 2 0

1 γ ,

where, for convenience, we make the substitution γ =

−

( )

1 1 v c^{2 2} .

This effect is called time dilation and is true for all clocks. By definition, the time interval measured by the stationary clock, T₀, is called the proper time.

The proper time is defined as the time interval between two events measured by an observer who sees the events occur at the same point in space; in this case at mirror M¹.

Note, for “everyday” values of v, v c^{2 2} << , so 1 γ ≈ 1 and so T ⇒T₀, the “classical” result.

0 2 4 6 8 10

0 0.2 0.4 0.6 0.8 1

γ =

 

T  T₀

v c

γ =

−

( )

1 1 v c^{2 2}

Class discussion question: Astronauts in a spaceship traveling at 0.6c relative to Earth are going to watch a movie that has a run time of 90 mins and request that they not be disturbed while the movie is running. How long does mission control have to remain quiet?

In the reference frame of the astronauts the start at end of the movie occur at the same location (in the spaceship). So, the movie time of 90 mins is the proper time. As far as observers on the Earth are concerned, the start and end of the movie occur in different locations , since the spaceship is moving with respect to Earth. Thus,

T T T

= = v c

−

( )

γ ₀ ⁰ _{2 2}

1 ^,

where T₀ is the proper time and T is the corresponding time on Earth.

∴ =

−

( )

T 90

1 0 6

90 0 64 90 0 8

. 2 . .

= 112 5. mins.

Class discussion question: Ann flies her rocket past Albert at constant velocity v. Both Ann and Albert measure the interval of time it takes the nose and tail of the rocket, to pass Albert. Which of the following is true? Why?

A: Ann measures a shorter time than Albert.

B: Albert measures a shorter time than Ann.

C: Ann and Albert measure the same amount of time.

Albert

Answer: B. Ask the question ... which observer measures the proper time? Albert measures the proper time since his clock is at the same point for

both events. Since T= γT₀,

and γ ≥ 1, then T T≥ ₀, i.e., the proper time T₀ is the shortest measured time interval between two events measured from any reference frame. Thus, the time interval between two ticks is shortest in the reference frame in which the clock is at rest; so Albert measures a shorter time than Ann.

Albert

Class discussion question: You, on Earth, have two identical clocks X and Y. You keep clock X with you but send clock Y on a spaceship that makes an orbit of the Earth before returning to you.

(1) How, to you, do the ticks of clocks X compare with those of clock Y while clock Y is in orbit?

(2) When clock Y returns to you, how do the ticks of clocks X compare with those of clock Y?

(3) When clock Y returns to you how does its reading compare with clock X?

(1) How, to you, do the ticks of clocks X compare with those of clock Y while clock Y is in orbit?

Since clock X is at rest, its ticks represent the proper time. Hence, the ticks of clock Y appear to take a longer time.

(2) When clock Y returns to you, how do the ticks of clocks X compare with those of clock Y?

Since the clocks are now at the same location, they both tick at the same rate.

(3) When clock Y returns to you how does its reading compare with clock X?

Since clock Y appears to have been ticking more slowly, it reads a shorter elapsed time for the trip than clock X. However, when it returns they both tick at the same rate and so the time difference is maintained. Clock Y does not suddenly read the same as clock X!

Time dilation is a very real phenomenon; for example, it explains why it is we observe an unstable elementary particle called a muon at the Earth’s surface. (Muons are produced by the collision of cosmic rays with atoms in the upper atmosphere at height of several thousand meters about the Earth’s surface.) Muons have a lifetime of only 2 2. µs when measured in a reference frame at rest with respect to them. So, if we could travel along with a muon and assuming they travel at a speed close to the speed of light (0.99c), we would travel a distance

vT₀ = 0 99 3 10. × × ⁸×2 2 10. × ⁻⁶ ~650m before the muon decayed. Clearly, we (and the muon) would not reach the Earth’s surface before the muon decays. But, experiments show that muon’s do reach the Earth’s surface, and time dilation explains why.

The proper time in this case is 2 2. µs, so the time measured by an observer on Earth is

T T T

= = v c

−

( )

γ ₀ ⁰ _{2 2} −⁶ ₂

1

2 2 10

1 0 99 .

.

≈ 16 10× ⁻⁶s.

So, relative to the observer on Earth, the distance traveled by the muon is

vT= 0 99 3 10. × × ⁸×16 10× ⁻⁶ = 4750m, i.e., it could reach the Earth’s surface. So, relativity explains why we find muons on the Earth. Note that, as far as the muon is concerned, it travels only 650m in the same time!

So, clearly, Newton’s and Galileo’s concept of absolute time is not correct; time intervals depend on the relative motion of the observer and the clock, i.e., time is relative.

One of the interesting and perplexing

consequences of time dilation is the effect on people’s ages; an example is the twin-paradox.

Suppose Albert and Isaac were born at the same time and they each have a lifetime of 80 years; that is Albert measures his lifetime to be 80 years and Isaac measures his lifetime to be 80 years. Let us assume that Albert and Isaac spend their entire lives traveling at 0.80c relative to each other. Then, according to Albert’s

descendants, Isaac lives for T

v c

0 1 2 2

80 1 0 64 133 3

−

( )

as measured by Albert’s clock. Likewise, according to Isaac’s descendants, Albert lives for

T

v c

0 1 2 2

80 1 0 64 133 3

−

( )

as measured by Isaac’s clock.

From Albert’s point of view, Isaac ages by just one year for each of Albert’s 1.67

(

)

Albert dies after 80 of his years and Isaac dies after 133.3 years of Albert’s years but will have the appearance of a person who is 80. According to Isaac, all of this is reversed. Sounds unbelievable, but it is true!

Question: when Isaac sees himself as 60 years old, how old will Albert appear? Answer ...

1 67 60. × = 100 years.

This suggests the following question. Suppose again that Albert and Isaac are twins, born at the same time on Earth. Let Isaac board a spaceship, take a fast, return trip to a distant star. Note that this scenario is different from the previous one because now Albert and Isaac begin and end in the same reference frame.

Once back together, they must agree on who is older, because there is only a single time in any single reference frame. So, which twin will be older or will they be the same age?

We must exercise CARE! The special theory only applies to inertial frames traveling with uniform velocity. In this case, only Albert is in an inertial

frame because Isaac has to accelerate away from the Earth, travel in a half-circle (which also involves acceleration) and then slow down to land on thhe Earth; Isaac is not in an inertial frame and so the special theory of relativity does not apply to his observations. But the theory does apply to Albert’s observations. Albert sees Isaac age slowly during his trip, because Isaac is moving relative to him.

For example, if the time spent accelerating is negligible and Isaac moves with a speed of 0.75c, the previous scenario tells us that Isaac will age by one year for every 1.67 of Albert’s years. So, if Isaac’s trip takes 60 years (as measured by Albert), Isaac appears to age by 60

1 67 36. = of his (Isaac’s) years. So, when they get back together on Earth, Albert observes that he is 60 years old and Isaac is 36 years old! Back on Earth, Isaac’s observations must agree with Albert’s, so Isaac is 24 years younger than his “twin” brother!

What about this scenario? Suppose your mother leaves Earth for a planet around a neighboring star. Suppose that light takes 26 years to travel from Earth to the planet. If your mother travels at a speed of 0.999c, she will take just over 26 years (of Earth time) to get to the planet. Let as say she spends 3 years on the planet before returning to Earth, so that she is away for just over

26 3 26 55+ + ≈ Earth years.

If you were 5 years old and your mother was 30 years old when she left Earth, you would be 5 55+ =60 years old when she returns. But how old is she? Every one year on Earth represents

1

1 0 999₂ 22 4

− =

. . years

of your mother’s clock. So, according to her clock, in the 52 Earth years of travel, she ages by only

5222 4. ≈2 3. years

according to her clock. So, when she lands back on Earth, she has aged by 2 3 3 5 3. + = . years according to her clock, i.e., she is 35.3 years old but you are 60 years old!

This is how your mother could become younger than you. Yes, it’s a form of time travel but it’s only a one way trip; your mother cannot get back to the past from which she departed.

Length Contraction

It turns out that space is relative also, i.e., the distance between two points in space or the length of an object, is not absolute. Like time intervals, it depends on the relative motion of the observer.

Consider the following scenario ... a spacehship, traveling with velocity v, leaves Earth for a distant planet a distance L₀ away in the Earth’s frame. (We will assume the the planet is stationary with respect to the Earth so that L₀ remains constant.) So, as far as someone on Earth is concerned the journey takes a time t L

= ⁰v. But what about a traveler on the spaceship? To them, the spaceship is stationary and it is the Earth and planet that are moving past them, with speed v.

L₀ v

v

But, the traveler’s clock ticks more slowly than a clock on Earth. So, as far as the traveler is concerned, the journey takes less time ( )t′ than that measured by an observer of Earth (t), because of time dilation, i.e., t′ = t

γ. Because it takes less time, the traveler concludes the distance between the Earth and planet is less than L₀. Then the distance calcuated by the traveler is

L= vt′ =vtγ. But, from above we have

t L

= ⁰v, i.e., v L

= ⁰t .

∴ ^L =^L0γ =^{L0 1}−

(

)

proper length L₀ when measured by an observer at rest relative to the object, if the object moves in a direction parallel to its length, the measured length L is shorter than L₀.

This relativistic effect is called length contraction and it occurs only in the direction parallel to the motion; the length perpendicular to the motion is unchanged. It is sometimes referred to as the Lorentz-Fitzgerald contraction or sometimes simply as the Lorentz contraction.

v

L₀ L

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

v c Relative length

LL₀

L

L v c

0

2 2

= 1−

( )

There is one caveat, the positions of the ends must be measured simultaneously .

Class discussion question: A meter stick moves with speed 0.80c relative to you in a direction parallel to the stick. What do you measure for the length of the stick? How long does it take to pass you?

v= 0 80. c

The proper length is L₀ = .1 00m and so the observed length is

L =L^{0 1}−

(

)

= 0 60. m. So, the time is takes to pass you is

t = =

× ×

apparent length speed

0 60 0 80 3 00 10⁸

.

. .

=2 50 10. × ⁻⁹s (= 2 50. ns).

v= 0 80. c

Class discussion question: A square sheet of metal, with 1m sides, passes you traveling at a speed of 0.80c. Exactly what will you observe?

1m 1m

v= 0 80. c

?

Answer: The sheet will appear rectangular with an area of 0 6. m²! Remember that length contraction occurs only in the direction of travel, so the height remains the same, i.e.,

AD=BC= 1 .m However, the width is contracted, so

AB CD= =L^{0 1}−

(

)

=1 00. × 1 0 64− . =0 6. m.

Thus, the original area (1m²) is reduced to 0 6. m².

A B

D C

Class discussion question: If, in the previous problem, the metal sheet was a sheet of glass and it was held in a rigid metal frame, would you expect the length contraction to cause the glass to shatter?

No! Both the frame and the glass will undergo identical contraction. The amount of contraction is completely independent of material; the relative contraction depends only on the relative speed.

So, the Galilean and Newtonian concept of absolute space is not correct; space, or more particularly, lengths depend on the relative motion of the observer. As a result, we must refer to “Albert’s space” and “Isaac’s space” rather than a single, universal space.

As we can see from the figure above, where (a) is a two-dimensional space that is at rest relative to an observer. Space becomes “deformed” as do objects in that space, (b), when in motion relative to the observer. If the motion is along the x direction at a speed of 0.80c, the relationship between the x and y coordinates are:

∆x′ = γ∆x=0 60. ∆x and ∆y′ = ∆y.

x y

O

v= 0 80. c

′ x ′y

O′

(a) (b)

Stationary observer

In fact, in special relativity, space and time are intimately interwoven as demonstrated by the next example. Imagine three identical, synchronized clocks spaced at equal intervals on the x-axis, as shown in (a). Now let those same clocks move with

velocity v in the x-direction.

If the left-hand-most clock registers the same time as the stationary clocks, then a stationary observer will see that not only do the moving clocks appear closer together because of length contraction, but they register successively earlier times! In fact, the time

difference is proportional to position. The reason is that the “reading” on any clock takes a finite time to reach the observer and so the farther the clock from the observer, the earlier the “observed” time.

This consequence of relativity has some unusual effects on simultaneity, as I’ll show now.

v x′

(a) x (b)

Imagine a spaceship flies by in the x-direction at constant velocity. On the spaceship there are two lights, one at the front (G) and one at the rear (R).

To an occupant of the spaceship the lights flash synchronously every few seconds. The question is

... what does a stationary observer on the ground see as the spaceship flies by ? Using the previous

example, when the spaceship is at (c), the observer sees the R flash before the G flash; even though the flashes are synchronized in the frame of the

spaceship, the G flash has to travel a greater distance to the observer than the R flash and so it arrives later. At position (b), where the R and G lights are equidistant from the observer, the flashes will appear synchronized to the observer.

R G v

(a) (b) (c)

Using similar arguments, we deduce that at

position (a), the observer sees the R flash after the G flash.

There are two conclusions we can draw from these observations:

• Clocks that are synchronized in one reference frame are not synchronized in any other frame moving relative to the first.

• Two events in any reference frame are simultaneous if light signals from the events reach an observer equidistant from the events at the same time.

So, the first statement tells us that simultaneity is not an absolute either, but depends on the observer.

The second statement is the definition of simultaneity.

Addition of Velocities

Because, in special relativity, space and time are not absolute concepts, then quantities that are related to them cannot not absolute, either. For example, velocity (and speed) is defined as

v = a distance

time to travel that distance.

so we conclude that velocity (and speed) cannot be absolute quantities but will depend on the observer.

Earlier we showed that assuming classical

relativity, the speed of person B as measured by the stationary observer O, is simply

v u+ .

(If person B walks in the opposite direction, his speed would be v u− .) But, according to Einstein’s special theory of relativity that cannot be correct.

A B

O

v u

In fact, according to the special theory the observed speed of person B is

′ = + +

u v u

vu 1 c₂

.

If v and u are small - everyday values, say - then vu<<c² and so u′ →(v u+ ), i.e., the classical result. But what if v = 0 8. c and u= 0 8. c? Classically, the result would be u′ =1 6. c, which violates postulate 2. But, according to special relativity

′ = +

+ × =

+ =

u c c

c c

c

c c

0 8 0 8 1 0 8 0 8

1 6

1 0 64 0 976

2

. .

. .

.

. . .

Modification to the Laws of Physics

Einstein’s theory of special relativity affects almost every physical quantity; many classical laws and principles need correction or modification in order to comply with the two postulates. Examples that we will consider here are, the conservation of linear momentum, conservation of energy and Newton’s Second Law. Detailed analysis requires some math, so we will concentrate on the

implications of the corrections, rather than their derivation.

In chapters 10 and 12 we saw how Newton and Young introduced the principle of concervation of momentum; in particular, in Lecture VIII of his A course of lectures on Natural Philosophy and Mechanical Arts, Young wrote:

“in all case of collision, whether of elastic or inelastic bodies, the sum of momenta of all the

bodies of the system, that is of their masses ...

multiplied by ... their velocities, is the same when reduced to the same direction, after their mutual collision, as it was before their collision.”

So, Young defined momentum as p =mu, where m is the mass and u is the velocity of an object. In a

“colliding system” that is stationary relative to an observer, see (a) below, the total momentum of the colliding objects as defined by Young, before and after the collision, is conserved. The problem is

that the total momentum before and after the collision is not conserved if the “colliding system” is moving relative to the observer.

(a) Stationary system (b) System moving relative to the observer

In fact, p=mu has to be replaced by the expression

p mu

u c mu

=

−

( )

1 ^{2 2} γ ,

where u is the velocity of the object and m its proper mass or rest mass, i.e., the mass measured by someone at rest with respect to the object. In this expression, p is the relativistic momentum.

However, at everyday speeds, u<< , then γ → 1 c and p≈ mu, i.e., the classical result is recovered.

(Note that the velocity that appears in γ is the speed of the object not the speed of the “system”

relative to the observer.)

p

Classical u c Relativistic

Unfortunately, the expression p= γmu has given rise to some misunderstandings! Sometimes

γm m

u c m_rel

=

−

( )

1 ^{2 2 ,}

is called the relativistic mass, which increases as the speed of the object increases.

As we saw in chapter 10, the Newtonian concept of mass was associated with “inertia”, i.e., the

“resistance” of a body to a change in motion, the greater the mass the greater the force required to stop or to accelerate an object.

m_rel m

u c

In special relativity, some physicists equate the relativistic mass with the inertial mass. However, in Definition I at the beginning of his Principia , Newton defined mass as “the quantity of matter ” in an object, see chapter 10. So, if we accept the latter definition, relativistic mass suggests that the

quantity of matter in an object increases with speed, which doesn’t seem logical! There has been debate over the years over which is the “correct”

interpretation but most physicists today accept the idea that mass is constant (invariant) in all

reference frames. Einstein’s initial interpretation was unclear on this point, but in a letter to the author Lincoln Barnett in 1948 he wrote:

“It is not good to introduce the concept of mass

M m

= u c

− ( )

1 ^{2 2}

[i.e., relativistic mass] of a body for which no clear definition can be given. It is better to introduce no other mass than the “rest mass” m.

Instead of introducing M, it is better to mention the expression for the momentum and energy of a body in motion.”

In fact, an examination of Einstein’s notes indicate that he very rarely used relativistic mass himself;

whenever “m” appears in his equations it is always the invariant (or rest) mass. So, why do most non- scientists accept that mass increases with speed ? There are two possible answers. First, prior to 1950, most textbooks used relativistic mass and so it was in common uasge. But, when the study of particle physics became more important, the

concept of invariant mass became more significant.

So, gradually, among physicists, invariant mass became the normal convention, but among non- physicists the “traditional” interpretation remains.

Second, the difference between invariant and relativistic mass does require an understanding that is of little consequence to most people.

Indeed, in Stephen Hawking’s “Brief History of Time” and Richard Feynman’s “The Character of Physical Law”, which were written for popular consumption, relativistic mass is used. The only argument I can think of is that they didn’t want to introduce any mathematics. So, there is an

unfortunate dichotomy - physicists use invariant mass between themselves but relativistic mass when communicating with the public - that often leads to confusion! For example ...

Class discussion problem: A black hole is an object whose mass is so great that nothing, not even light can escape. According to the expression for

relativistic mass, as the speed of an object increases its mass increases. Therefore, at some speed, its mass will exceed the critical mass and it will form a black hole. Is this a possibility?

Answer: No. If we assume that a fast moving object can become a back hole we have a paradox;

because someone who is stationary relative to the object will not see it as a black hole (assuming it wasn’t one to begin with!). Clearly, if the object is not a black hole in one reference frame it cannot be a black hole in another moving relative to the first.

The way to resolve the paradox is to work only with invariant mass; then if the object is not a black hole in the stationary frame it will not be a black hole in the moving frame.

Let’s turn our attention to another classical law, Newton’s Second Law. As we saw earlier in

chapter 10, most textbooks, quote Newton’s second Law as F=ma, and it works well for everyday experiences. However, it is unacceptable in special relativity because, if we write the acceleration as

a F

= m,

it implies that an unchanging force on an object will produce a constant acceleration. Constant acceleration means that the speed of the object will continue to increase so that, eventually, the object will traveling at the speed of light and still

accelerating! Thus, postulate 2 will be violated, and so Newton’s second Law is not consistent with the special theory. Using the proper analysis, the special theory predicts that the correct (relativistic) expression for the acceleration is

a F

=

(

)

where γ =

−

( )

1

1 u c^{2 2} , as before, and u is the speed of the object and m is its rest mass. So, with an constant force, the acceleration will decrease as the speed of the object increases. The object cannot travel with a speed greater than c, since a = 0 when u= , i.e., there is no acceleration to carry it c beyond c.

Note that when u<<c, the expression becomes a F m= , i.e., the classical limit.

0 0.2 0.4 0.6 0.8 1 1.2

0 0.2 0.4 0.6 0.8 1

a

Relativistic

Classical

a F

m u c

=

(

)

a F

= m

uc

Occasionally, you will hear people say that

Einstein proved that Newton was wrong. In fact, that is definitely not the case! In chapter 10, we showed that Newton’s Second Law was actually

F dp

= dt,

i.e., an impressed force on a object would increase the object’s momentum, and the force equals the rate of change of momentum. Classically, p= mu, and that’s where F= ma, shown in most textbooks, is derived from, but if we use the expression for the relativistic momentum, p= γmu, we obtain the relativistic form of Newton’s Second Law just shown. So, Newton’s Second Law is correct!

The special theory of relativity also requires a modification of our classical expressions. In

chapter 12, we saw that in Lecture VII from his “A course of lectures” Young defined the energy of a moving object as

“... the product of the mass ... of a body, into the square of ... its velocity.”

This quantity was later called kinetic energy and written in the form

K =

( )

Again, because it involves motion, the expression has to be corrected to comply with the postulates of special relativity. The relativistic expression is

K mc

u c mc

=

−

( )

2 2 2

2

1

= (γ −1)mc .²

It may not look anything like the classical

expression for kinetic energy, but when u<< c, it can be shown that

K→

( )

We can write the expression

K = (γ −1)mc² =γmc²−mc²

where the first term depends on u (the speed) but the second is independent of u. Einstein called the first term (γmc²) the total energy (E) and the second term (mc²), the rest energy (E₀) of the object.

∴E= γmc² = K mc+ ²,

which is Einstein’s famous mass-energy equivalence equation. It says that an object has energy (= mc²) even when it is stationary!

0 1 2 3 4 5

0 0.5 1 1.5 2

Relativistic

Classical K

mc^{2 K mc}

u c

= mc

−

( )

2 2 2

2 1

K=

( )

But, is their any proof that the relativistic

expression for the kinetic energy is correct? Yes, it was confirmed by William Bertozzi in 1964. In Bertozzi’s experiments, electrons were given various kinetic energies by accelerating them across a known voltage (K = ∆ ) and their speeds q V were measured by flight times over a known distance.

1 20. 1 00. 0 80. 0 60. 0 40. 0 20.

0 1 0. 2 0. 3 0. 4 0. 5 0. u

c 2

2

Non-relativistic Relativistic

Kinetic energy (MeV)

u=c

The mass-energy equivalence expression has a number of applications. We are all aware that in many nuclear reactions, the combined mass of the initial nuclei, M_i, is greater than the combined mass of the final products, M_f. That difference is not lost, in the sense that it simply disappears.

According to Einstein’s equivalence equation it is converted to energy, ∆E, possibly kinetic energy of the products, or radiation or any combination, i.e.,

∆E= (M_i−M c_f) ².

This is, of course, the equation that underpins the production of energy from our Sun, nuclear

reactors, etc. There are also some more subtle applications. For example:

Class discussion question: When a rubber band is stretched around a package, its (stored) energy increases. What happens to its mass?

Answer: the mass increases. The equivalence equation says that

E= K mc+ ².

But, if K = 0, then if the total energy, E, increases, so must m.

Class discussion question: Two objects are identical except one is hotter than the other. Explain how they would respond to identical forces.

Answer: The hotter object has greater internal energy than the colder object; the molecules in the hotter object are vibrating more rapidly and with greater amplitude. But, the total energy is

E= K mc+ ².

So, if the objects are stationary, K = 0. Therefore, the hotter object, with greater energy, has a greater mass. Newton’s Second Law tells us, since it’s mass it greater, it’s acceleration from rest, a F

=

(

)

be less that the colder mass.

As you are probably aware, when collisions occur, or explosions or nuclear reactions, classically, momentum (p= mu) is always conserved before and after the event although kinetic energy

(K =

( )

However, in special relativity, both the relativistic momentum and the relativistic mass-energy are conserved in all events.

U − 235 neutron

Barium Krypton

3 neutrons

Nuclear reaction

Inelastic collision Elastic collision

The Doppler Effect

Here is another example from special relativity, which we will return to in chapter 15; namely, the Doppler effect. The Doppler effect occurs for both sound and light; in the case of sound it manifests itself as the change in pitch of a sound experienced by an observer as the source approaches and then recedes. It was explained quantitatively in 1842 by the Austrian scientist Christian Doppler (1803- 1853) and confirmed experimentally in 1846.

Doppler attempted to use his idea to explain the color of binary stars. However, in the case of a light source, the constancy of the speed of light requires a different analysis to that for sound.

Imagine we have two stationary observers, A and B, with a light source that is moving away from A and towards B.

Source

c v c

A B

If the frequency of the light source is ν₀, and the speed of the source is v, the frequency measured by A is

ν ν β

A = − β

 +

 

0 1 

1 (receding),

where β = v c, and that measured by B is

ν ν β

B = + β

 −

 

0 1 

1 (approaching).

So, the frequency measured by A (source receding) is less than the frequency measured by B (source approaching). For observer A, the frequency shift is towards the lower frequency end of the spectrum - referred to as a redshift - and for observer B, the frequency shift is towards the higher frequency end of the spectrum - referred to as a blueshift. As we will see in chapter 15, a redshift implies the source is receding, and a measurement of the shift,

∆ν= ν₀ −ν, provides a measurement of β (= v c) and therefore the speed of recession.

Space-time Diagram

Let’s consider one final consequence of special relativity. As we mentioned in the Introduction to this chapter, relativity places strict limits on our concept of the past, the present and the future. We will consider a simple one-dimensional case, i.e., all motion is restricted to the x −direction. Let’s imagine we start at the origin and wander, first in the + −x direction and then the − −x direction. We could represent our “wanderings” like this:

However, in 1907, the German mathematician Hermann Minkowski (1864-1909), who was one of Einstein’s teachers, introduced the concept of a space-time diagram, which included not only the space we traveled but also the passage of time.

0 +x

−x

So, a space-time diagram for our “wanderings”

might look like this, if we started at x = 0 and t = 0:

So, we would be at a distance x¹ from the starting point at time t₁ and sometime later, at time t₂, we would be a distance x₂ from the starting point. The advantage of this representation is that space and time are combined into one construction. Note, however, if we were allowed to “wander” freely in three-dimensions, then the space-time diagram would be a four-dimensional construct, which we could not draw! But, the space-time diagram actually allows us to represent the past, present and future.

Time (t)

−x +x

×P t₁

x¹

×

x²

t² Q

0

Let us take one space dimension (x) and construct a space-time diagram where the previous time axis (t) is now ct, i.e., the product of the time and the speed of light. Now, the x-scale and ct-scale have the same units, e.g.,

x ⇒ meters and ct m

s s m

⇒

( )

If the origin is considered “NOW” then the diagram can be separated into “PAST” and

“FUTURE” regions.

The lines x = ±ct represent the

motion of a pulse of light sent from

“NOW”, and since nothing can travel faster than the speed of light, there are forbidden regions, i.e., regions we cannot enter, called “ELSEWHERE”.

FUTURE x ct= x= −ct

ct

x

Not only is our existence confined to the PAST and FUTURE, but we can only cause events to occur in the green region, not in the pink region. So,

using a timer- switch, we can get a light to switch-on at point P at some future time, but we cannot get a light to switch-on at point Q as the message to “switch-on” would have to travel faster than the speed of light.

We can also use the space-time diagram to tell us if one future event can cause another event further in the future. For example:

FUTURE x ct= x= −ct

ct

x

•

•^P _Q

In (a) three future events, A, B and C, are shown.

Could event A be the cause of events B and C?

To answer that question we draw a new space-time diagram with event A being NOW. Notice that even B lies in the ELSEWHERE region and not in the FUTURE region. So, in this case, event A cannot cause event B. But event C lies in the FUTURE region and so event A can cause event C.

ct

x

• •

A B

(a)

•^C

•_A• ^B

(b)

•^C

Summary

At the end of the 19th century, most scientists believed they had learned most of what there was to knnow about physics. Newton’s Laws of motion and his Universal Law of Gravitation, Maxwell’s theoretical work on unifying electricity and

magnetism as well as developments in other areas such as heat and the kinetic theory of atoms,

explained a wide variety of phenomena. But, at the turn of the 20th century, a major revolution shook the world of physics; in 1900 Planck proposed what was to be the foundation of quantum theory and in 1905 Einstein published his special theory of relativity. Einstein captured the excitement of the period when he said:

“It was a marvellous time to be alive.”

Within a few decades, the ideas of Planck and Einstein precipitated numerous developments and theories in many areas of physics and chemistry.

Einstein made a number of important contributions to science; his special theory of relativity alone represents arguably one the greatest intellectual achievements of the 20th century. With this theory one can predict correctly observations over a range of speeds from zero to speeds approaching the speed of light. Newtonian mechanics, which had been accepted for over 200 years, and had

accounted for most everyday experiences. But Einstein showed that the traditional concepts of space and time were inappropriate. In fact, Newtonian mechanics is simply a limiting case of Einstein’s special theory. Einstein felt compelled to reconsider the “classical” ideas and with great insight, he developed his theory. Regarding his theory he wrote:

“The relativity theory arose from the necessity, from serious and deep contradictions in the old theory from which there seemed no escape. The strength of the new theory lies in the consistency

and simplicity with which it solves all these difficulties using only a few very convincing assumptions ...”

Relativity has the reputation for being “difficult ”, but in truth this description arises from its

“strangeness” rather than its inherent

mathematical complexity. Its conclusions often seem to violate our common sense. The main requirement for getting some understanding of the theory is some mental flexibility.

In a later chapter, we will explore some of the implications of Einstein’s general theory of relativity. As we have seen, the special theory is concerned with the observations of observers in non-accelerating frames, i.e., those moving with constant (uniform) speed; the general theory, however, applies to observers in accelerating frames.