6.2. Calculate the effective temperature of the Sun

change of with p. (c) Use your results to dis-

from the given solar luminosity, and radius,

and compare your answer with the value cuss the sharpness of the solar limb.

*6.10. Consider a charge Q near a neutral object. If

given in the chapter.

6.3. Assume that for some process the cross sec- the object is a conductor, charge can flow

within it. The presence of the charge Q

tion for absorption of a certain wavelength

photon is 10 16 cm2, and the density of H is induces a dipole moment in the conductor,

1 g/cm3. (a) Suppose we have a cylinder and there is a net force between the dipole

that is 1 m long and has an end area of and the charge. (a) Show that this force is

1 cm2. What is the total absorption cross attractive. (b) How does this apply to the pos-

section? How does this compare with the sible existence of the H ion?

area of the end? (b) What is the absorption 6.11. What is the thermal Doppler broadening of

coefficient (per unit length)? (c) What is the the H line in a star whose temperature is

mean free path? (d) How long a sample of 20 000 K?

material is needed to produce an optical 6.12. We observe the H line in a star to be broad-

depth of unity. ened by 0.05 nm. What is the temperature of

6.4. Suppose we have a uniform sphere (radius R ) the star?

of 1 M of hydrogen. What is the column den- 6.13. Compare the total thermal energy stored in

sity through the center of the sphere? the corona and photosphere.

6.5. How large must the optical depth through a 6.14. (a) At what wavelength does the continuous

material be for the material to absorb: (a) 1% spectrum from sunspots peak? (b) What is the

of the incident photons; (b) 10% of the inci- ratio of intensities at 550 nm in a sunspot

dent photons; (c) 50% of the incident photons; and in the normal photosphere? (c) What is

(d) 99% of the incident photons? the ratio of energy per second per surface

6.6. If we have material that emits uniformly over area given off in a sunspot and in the normal

its volume, what fraction of the photons that photosphere?

we see come from within one optical depth of 6.15. How long does it take before material at the

the surface? solar equator makes one more revolution

*6.7. Suppose we divide a material into N layers, than that at 40 latitude?

each with optical depth d /N, where is 6.16. Calculate the energy per second given off in

the total optical depth through the material the solar wind?

and d V 1. (a) Show that if radiation I0 is *6.17. (a) What is the pressure exerted on the Earth

by the solar wind? (Hint: Calculate the

incident on the material, the emergent

momentum per second on an object whose

radiation is

cross sectional area is that of the Earth.)

d )N

I I0 (1

(b) How large a sail would you need to give an

(b) Show that this reduces to I I0e (equa- object with the mass of the space shuttle an

tion 6.19) in the limit of large N. (Hint: You acceleration of 0.1 g at the distance of the

Earth from the Sun?

may want to look at various representations

of the function ex.) *6.18. To completely describe the radiative transfer

6.8. For what value of x does the error in the problem, we must take emission into account

approximation ex 1 x reach 1%? as well as absorption. The source function S is

6 THE SUN: A TYPICAL STAR 121

defined so that S d is the increase in (a) If S is a constant, solve for I vs. , assuming

an intensity I0 enters the material. (b) Discuss

intensity due to emission in passing through

a region of optical depth d . This means that your result in the limits V 1 and W 1.

the radiative transfer equation should be

written

I S

dI/d

Computer problems

6.1. Consider the situation in Fig. 6.4 with 1000 layers. 6.2. Estimate the Doppler broadening for the H lines

Draw a graph of the fraction of the initial beam from the atmospheres at the mid-range of each

emerging from each layer, for total optical depths spectral type (e.g. 05, B5, etc.). (Hint: Scale from the

(a) 0.1, (b) 1.0, (c) 10.0. Show that the fraction result in example (6.1).)

emerging from the final layer agrees with equa-

tion (6.18).

Part II

Relativity

Einstein™s theory of relativity caused us to rethink the meaning of both

space and time, concepts that had been taken for granted for centuries.

The foundation of this revolution is the special theory of relativity, which

Einstein published in 1905.The general theory of relativity, published in 1916,

is really a theory of gravitation set in the foundations of the special

theory; it also allows us to analyze the properties of frames of reference

that are accelerating.

Chapter 7

Special relativity

Einstein examined Maxwell™s equations to see

7.1 Foundations of special relativity if they obeyed this simple rule. His reasoning is

illustrated in Fig. 7.1. The solution of Maxwell™s

7.1.1 Problems with electromagnetic equations gives us waves that vary sinusoidally in

radiation both space and time. That is, the waves vary with

The problems that lead to special relativity start position, repeating each wavelength and, with

with Maxwell™s theory of electromagnetic radia- time, repeating each cycle. How would an elec-

tion. Maxwell™s equations, presented in 1873, tromagnetic wave appear to an observer moving

allow for the existence of waves of oscillating along with the wave at the speed of light? The

electric and magnetic fields. All waves known wave would appear sinusoidal in space but con-

before electromagnetic waves required a medium stant in time, since the observer is moving along

in which to travel. For example, sound waves can with the wave (crest for example). However, there

travel through air, but not through a vacuum. is no mathematical solution to Maxwell™s equa-

There is no obvious medium necessary for the tions that is constant in time, but which varies

propagation of electromagnetic waves. Physicists sinusoidally. (Remember, it is precisely the time

postulated a medium that is difficult to detect, variation of electric and magnetic fields that

called the luminiferous ether, or simply the ether. allows the propagation of the waves.) This seems

The ether supposedly fills all of space. Once we to be a contradiction.

have a medium, then we have a reference frame Two possibilities were left: (1) Maxwell™s equa-

for the motion of the waves. For example, the tions were correct in only one reference frame,

speed of sound is measured with respect to the air that of the ether, or (2) there was something

through which it is moving. An observer moving wrong with the basic concepts of space and time.

through the medium will detect a different speed The first possibility violates Einstein™s postulate

for the waves than an observer at rest in the of special relativity, so he chose the second, say-

medium. ing that, for some reason, it must be impossible

Einstein™s questions about Maxwell™s equations to move at an arbitrary speed relative to an elec-

involve the appearance of electromagnetic waves tromagnetic wave. He concluded that the speed of

to different observers, who are moving at differ- light in a vacuum is the same for all observers, indepen-

ent speeds. Einstein started with the postulate of dent of their motion. This suggests that electromag-

special relativity, that, the laws of physics, properly netic waves must be different from the familiar

stated, should be independent of the velocity of the mechanical waves. There must also be something

observer. It may be that the values of certain quan- wrong with the concept of the ether.

tities change with the motion of the observer, but When Einstein was working on this problem,

the relationships among the physical quantities experiments had already been done which cast the

do not change. existence of the ether into doubt. An experiment,

126 PART II RELATIVITY

Incoming

Starlight

t=0

(a)

Motion

t = t1 of

Telescope

(b)

(a)

t = t2

(c)

Moving Observer

θ θ

t1

Time

t2

Stationary Observer

(d)

(b)

Fig 7.1. Observers of electromagnetic waves. One

Fig 7.2. Aberration of starlight. (a) Assume that the tele-

observer is stationary and the other observer is moving

scope is moving to the right as the beam of light enters, with

with the wave. (a) The moving observer is at a crest, and the

the telescope tube lined up with the beam of light. Since the

stationary observer is at a null. (b) The moving observer is

speed of light c is ¬nite, the telescope moves as light passes

still at a crest, and the stationary observer sees a negative

through, and the light strikes the side. (b) To observe the

value. (c) The moving observer is still at the crest and the

light, we must tilt the telescope slightly.Thus, as the tele-

stationary observer is at a dip. (d) We plot what each sees as

scope moves over, the beam is always centered in the tube.

a function of time.The moving observer sees a constant

We must tip the telescope in the direction in which it is

value while the stationary observer sees a sinusoidally

moving.

varying value.

ration of starlight. (This has nothing to do with

originally done by A. A. Michelson in 1881, and in

an improved fashion by Michelson and Morley in aberrations in optical systems.) It is a slight

1887, was designed to measure the motion of the change in the angle at which light from stars

Earth through the ether by measuring the speed appears to be arriving due to the motion of the

of light in two directions perpendicular to each observer, in this case the motion of the Earth

other. No change was found. This meant that the about the Sun. (It is analogous to the change in

Earth could not be moving through the ether. If the apparent angle at which rain is falling when

the ether existed, it must be dragged along with you start to move.) The shift is always in the direc-

the Earth. However, there is another observation tion of the motion of the Earth, so it changes

which rules out the dragging of the ether by the throughout the year. The positions of some stars

Earth. It is illustrated in Fig. 7.2, and is called aber- are shifted by as much as 20 arc seconds from

7 SPECIAL RELATIVITY 127

their true positions. (This effect has been used in

the past to measure the speed of light.) In the

ether theory there is no way for aberration to be

(a)

observed if the Earth is dragging the ether.

The fact that the speed of light is independent

of the velocity of the observer contradicts our

everyday experience, in which relative velocities

are additive. Einstein began to look at the under-

lying cause for the speed appearing to be constant.

In measuring a speed, we measure a distance and

a time interval. Einstein suspected that the problem (b)

lay in our traditional concepts of space and time.

Physicists such as Newton simply assumed that

space and time were given. Einstein suggested that

they might not be absolute but might depend on

the motion of the observer. Einstein examined the

idea of an absolute time and looked at whether

time might actually be a quantity that depends on

the motion of the observer.

(c)

7.1.2 Problems with simultaneity Fig 7.3. Flashes in railroad cars and simultaneity.The top

Einstein realized that an absolute time was tied car is moving past the bottom car. (a) When the observers

at the centers of each car are closest to each other, ¬‚ashes

to the concept of absolute simultaneity. By absolute

go off at opposite ends of the lower car. (b) The motion of

simultaneity we mean that if two events appear

the top car means that the observer in that car sees the

simultaneous to one observer, they appear simul-

right ¬‚ash ¬rst. (c) The observer in the bottom car sees both

taneous to all other observers. This is important

¬‚ashes at the same time.

because telling time is actually noting the simul-

taneity of two events. For example, if we say the

train left the station at 7:00, we are saying that With this as a starting point, we now go on to

two events are simultaneous. The first event is investigate how different types of situations

the train leaving the station, and the second appear to observers with different velocities. In

event is the clock showing 7:00. If those events are special relativity, we deal only with reference

simultaneous for one observer, but not for all frames that are not accelerating with respect to

observers, then the concept of absolute time has each other or in which there are no external

no meaning. gravitational forces. Such a reference frame is

An experiment depicted in Fig. 7.3 shows that called an inertial reference frame. An inertial frame

two events can be simultaneous for one observer, might be provided by a space station far from

but not another. The two observers are at the cen- any mass and with the engines off so there is no

ters of identical railroad cars. One car is at rest acceleration. Einstein™s postulate about the laws

with respect to the station. The other is moving of physics being the same in different reference

past at some speed. When the two observers are frames only applies to inertial frames. (We know

opposite each other, two flashes go off at the ends that accelerating frames must be different,

of one car. The flashes are judged to be simulta- because they have pseudoforces, such as centrifu-

neous by the observer at rest. How are they seen gal force.) Another way of stating Einstein™s pos-

by the other observer? The figure shows that the tulate is that There is no experiment we can perform

flashes that the observer is moving toward is seen to tell us which inertial frame is moving and which is at

first. The flashes are not simultaneous for the rest. There is no ˜preferred™ inertial frame. All we can

moving observer. Simultaneity is not absolute. talk about is the relative motion of two inertial

Therefore, time is not absolute. frames.

128 PART II RELATIVITY

We want to solve for t in terms of t0:

7.2 Time dilation

t2 [1 (v/c)2] t02

Taking the square root of both sides and solving

Now that we know that time is relative, we can see

for t, we have

how a clock appears to two different observers.

One observer is at rest with respect to the clock,

t0

31 1v c 2 2 4 1 2

and the other observer sees the clock moving. The t (7.2)

time viewed in the frame in which the clock is at

The significance of this result is that the time

rest is called the proper time for that clock. The

interval measured in the frame in which the

word ˜proper™ does not denote anything superior

clock is moving is greater than that measured in

about this frame; it just happens to be the frame

the frame in which the clock is at rest. Suppose

in which the clock is at rest. We can think of

we have two identical clocks. If we keep one at

proper time as being the time interval between

rest (with respect to us) and let the other one

two events at the same place.

move, the moving clock appears to run slow. It is

A simple clock is shown in Fig. 7.4. In this clock

important to realize that the situation is per-

a light beam bounces back and forth between two

fectly symmetric. If there is an observer traveling

mirrors, separated by a distance L. We would keep

with each clock, each observer sees the other clock run-

time by counting the light bounces. The time for

ning slow. This effect is called time dilation.

the light pulse to make a round trip is

From equation (7.2), we can see that the amount

t0 2L/c (7.1)

of time dilation depends on the quantity 1/[1

(v/c)2]1/2. This quantity is generally designated

In the frame in which the clock is moving, the

light beam takes a longer path. Since the speed of and is plotted as a function of (v/c) in Fig. 7.5. Note

light is the same in both frames, the beam must that this quantity is close to unity for small veloci-

take longer to make the round trip. From the fig- ties, and only becomes large when v is very close to

ure, we see that the distance traveled is 2[L2 c. This confirms our intuition that the results of

(vt/2)2]1/2, so the time is

8

(2/c) [L2 (vt/2)2 ]1/2

t

Squaring this gives 7

2 22 2

t (4L /c ) [1 (vt/2L) ]

6

We use equation (7.1) to eliminate L, giving

t2 t02 (v2/c2) t2

5

γ

4

t/2) 2

+ (v

3

L

L

L2

2

vt/2

1

vt

0 0.2 0.4 0.6 0.8 1

(a) (b) (c)

v

β

Fig 7.4. \Light clock. (a) In the rest frame of the clock, the

c

light bounces back and forth. (b) In the laboratory frame,

Fig 7.5. The quantity vs. v/c. For v/c small, is close to

with the clock moving, the light beam travels a longer path.

(c) Calculating the extra distance traveled. unity. As v/c approaches unity, approaches in¬nity.

7 SPECIAL RELATIVITY 129

special relativity should reduce to familiar everyday one on Earth should appear younger. It is alright

results when speeds are much less than c. for two moving observers to see each other age

Time dilation is not an artifact of the light slower. However, we have a problem if we try to

clock that we have depicted in Fig. 7.4. It applies bring the twins together “ both at rest. We can

to all clocks. For example, it applies to the decay see which one is really younger and decide which

of unstable elementary particles. Particles mov- was really moving. This would seem to violate

ing close to the speed of light should appear to Einstein™s postulate. However, if the twins start

live longer than the same particles at rest. This is and end together at rest, then one twin must accel-

tested almost daily in particle accelerators around erate to get to very high speeds. That acceleration

the world. A dramatic example is in cosmic rays, produces pseudoforces which can be felt by only

which contain unstable particles which can decay one twin. This breaks the symmetry of the prob-

as they pass through the Earth™s atmosphere. If lem without any logical contradiction. (Remember,

we measure the flux of cosmic rays at high alti- a pseudoforce is really an inertial response to an

tude and near the ground, we find that many acceleration of the reference frame.)

more survive this trip than we would expect,

unless we account for the effect of time dilation.

7.3 Length contraction

Example 7.1 How fast must a particle be travel-

ing to live ten times as long as the same particle

Once the concept of time becomes suspect, the

at rest?

concept of length must also be reinvestigated.

Think of how we measure the length of an object.

SOLUTION

We measure the positions of the two ends and

We simply set

take the difference between the two positions.

1 For this procedure to have any meaning, the

31 1v c 2 2 4 1 2

10

measurements must be carried out simultane-

ously. (If I measure the position of the front of an

Squaring gives

airplane, when it is in NY, and the position of the

tail 6 hours later when it is in LA, I should not

1

31 1v c 2 2 4

100

conclude that the airplane is 5000 km long.)

Unfortunately, we have seen that observers in dif-

Solving for (v/c)2 gives

ferent inertial frames cannot agree on the simul-

(v/c)2 taneity of events separated in space.

0.99

It is therefore not surprising that lengths will

Taking the square root gives v/c 0.995. The parti-

appear different to observers in different inertial

cle must be within one-half of one percent of the

frames. In fact, physicists had been playing with

speed of light!

this idea before Einstein™s 1905 paper. H. Lorentz

had proposed it as a way around the results of the

Time dilation applies to biological clocks. A

Michelson“Morley experiment. He said that the

person traveling at a high speed will not age as

ether could be saved if the lengths of objects

fast as a person at rest. Of course, the situation

depended in a particular way on their state of

must be symmetric. Each person sees the other

motion.

age slower. This leads to a puzzle known as the

twin paradox. Two twins are on Earth. One is an In considering changes in length, we look sep-

astronaut who goes on a trip at a speed close to c. arately at lengths perpendicular to and parallel

to the direction of motion. We can first show that

The other stays on the Earth. From the point of

there can be no length changes perpendicular to

view of the one that stayed on Earth, the astro-

the direction of motion. Let™s assume that there

naut is moving and will not age as fast as the one

were such a change and that moving objects

on Earth. The astronaut will appear younger upon

shrink. We now consider an experiment. Two peo-

returning. However, the astronaut sees the one on

ple of identical height are standing, as in Fig. 7.6.

Earth moving away at high speed. Therefore the

130 PART II RELATIVITY

v

v (a)

B A B A

? v

Fig 7.6. The effect of possible length contraction perpen-

dicular to the direction of motion. Assume that objects

shrink perpendicular to the direction of motion. A and B are

the same height when both are at rest. Both hold swords

v (b)

parallel to the ground, and B moves past. If B shrinks, A™s

sword will miss B, but B™s sword will cut A. However, from Fig 7.7. The effect of length changes parallel to the direc-

B™s point of view, as shown in the right ¬gure, A is moving, tion of motion. (a) To measure the length of a stick, we must

and it is A who shrinks.That would result in an injury to B ¬rst measure its speed.We do this by measuring the time

not A. for one point on the stick to go a known distance between

two stationary clocks. In the upper frame, the front of the

stick starts at the right clock. In the lower frame it reaches

Each has a sword held out at the level of the top the left clock.The time difference is noted, and the speed is

of the head. Now person B is carried past person calculated. (b) Knowing the speed of the stick, we measure

A at a high speed. According to A, B is moving, its length by seeing how long it takes the stick to pass a sin-

and B gets shorter. B™s sword cuts A™s head while gle stationary clock. In the top half of the frame the front of

A™s sword passes safely over B. According to B, A is the stick is at the clock, and the measurement starts. In the

bottom half of the frame, the back of the stick reaches the

moving and the situation is reversed. We have a

clock and the measurement ends.

true contradiction. Each person is wounded in

their own rest frame but not in the other. The

only way out of this is to say that there can be no timing the passage of the object past one marker.

length change perpendicular to the direction of The time interval between the two measurements

motion, and no one gets hurt. (A similar argu- at the one marker t, as measured in the frame of

ment would rule out expansion perpendicular to the object, is

the direction of motion as well as contraction.)

L0 v

¢t

We can think of no such examples to rule out

changes parallel to the direction of motion. Here, In our frame the time interval is different

there is actually a change of length. Moving because of time dilation, so the interval is

objects appear to shrink. We call this effect

¢t¿ ¢t

Lorentz contraction. To see this we use Fig. 7.7 to

show how we might measure the length of a mov- We now say that the length of the object is

ing object. The length of an object, measured in

L v ¢t¿

the frame in which it is at rest, is called the proper

length, L0 . This can be measured in the usual way,

v ¢t

since its ends are not going anywhere. We now

measure its length in a frame in which it is mov- Finally, substituting t L0/v gives

ing. We can tell its speed v by having two markers

L L 0/ (7.3)

at rest in our frame, and measuring the time for

the front of the object to travel from one marker Not surprisingly, the length contraction has the

to the other. We can then measure its length by same dependence on v/c as does the time dilation.

7 SPECIAL RELATIVITY 131

As with time dilation, length contraction is of waves, N, divided by the time interval, t, as

symmetric. If A and B are carrying meter sticks measured in the source frame:

parallel to the direction of their relative motion,

N ¢t

A will see B™s stick shrink, and B will see A™s stick

shrink. There is no contradiction here since we We can use the time dilation formula, t t,

cannot compare the ends of the sticks simultane- to make this

ously for both observers. This symmetry has pro-

N ¢t¿

vided some interesting puzzles that start with

seeming contradictions, but end up with logical This can now be used to eliminate t in equation

resolutions. (See Problem 7.5.) (7.4), giving

1

¿

7.4 The Doppler shift vc

1

31 1v c 2 2 4 1 2

31 1v c 2 4

With lengths and times appearing different to dif-

ferent observers, it is also necessary to take a

3 11 v c 2 1 2 11 v c21 24

closer look at the Doppler shift, since wavelengths

31 1v c 2 4

obviously involve length, and frequencies obvi-

ously involve time. Since there is no ether, the

If we multiply the numerator and denominator

Doppler shift for electromagnetic waves can only

by (1 v/c)1/2, this simplifies to

depend on the relative motions of the source and

11 v c2

observer. This is different from the case of sound

31 1v c 2 2 4 1 2

¿

waves, for which the shift depends on which is

moving. We can show that the result for electro-

a1 b

v

magnetic waves doesn™t depend on which is mov- (7.5)

c

ing by considering separately the case of the

moving source and the moving observer. In both This is like the classical Doppler shift formula,

cases, we denote quantities measured in the rest except for the extra factor of which comes from

frame of the receiver as primed ( ). time dilation.

7.4.1 Moving source 7.4.2 Moving observer

Let™s assume the source is moving towards the We now consider the receiver moving towards the

receiver at a speed v. The source emits N waves in source. In the source™s frame, in time t, the receiver

time t , as measured by the receiver. In this time, v) t. This

will receive all waves in a length (c

the first wave travels a distance c t and the source number is the length divided by the wavelength:

travels a distance v t . The wavelength will then be

N

the distance between the source and the first wave, ¿

¢t¿

divided by the number of waves. That is

11 v c 2 ¢t

c¢t¿ v¢t¿

¿

N ¢t¿

11 v c2

The frequency is then given by

31 1v c 2 2 4 1 2

c

¿

¿

a1 b

v

c

c N

(7.4)

c v ¢t¿

This is identical to equation (7.5), proving that the

Doppler shift is independent of whether the

We would like to relate this to the frequency

source or the receiver is moving.

in the source frame, . It is given as the number

132 PART II RELATIVITY

7.4.3 General result This means that

We will now generalize the result. If we had con- (1.36) (656.28 nm)

sidered the source and observer moving apart, we

892.54 nm

would have 1 v/c in the numerator of equation

(7.5). The v in the 1 v/c is just the radial velocity, The line is shifted from the visible into the near

the component of the velocity along the line of infrared!

sight. We should therefore replace that v in equa-

tion (7.5) with vr . However, the other v in equation

(7.5), in the , comes from time dilation, which is

7.5 Space-time

independent of the direction of motion. It must

remain as the total relative speed of the source

and receiver. This means that it is possible to have Phenomena such as time dilation and length con-

a Doppler shift, even when the motion is perpen- traction are not simply illusions. They are real

dicular to the line of sight. This is simply a result effects. Our failure to appreciate this previously

of time dilation, and is not important until v is comes from a failure to appreciate the true nature

close to c. With these generalizations, equation of space and time. Classical physicists assumed

(7.5) becomes that space and time were simply there, just like a

blank piece of graph paper, and that the laws of

vr /c)

/(1 (7.6)

physics were laid down on top of them. Einstein

realized that the laws of physics were intimately

We can derive a corresponding expression for

entwined with space and time. We can understand

c/ ,

wavelength, remembering that

11 vr>c 2

the nature of this relationship by abandoning our

(7.7)

¿

normal three-dimensional world and replacing it

with the four-dimensional world of space-time.

Later in this book, we will encounter objects

moving away from us at speeds close to c. For

7.5.1 Four-vectors and Lorentz

these, the radial velocity is very close to the total

transformation

speed v. This allows us to make the simplification

In space-time we simply treat time as another

31 4 coordinate. To remind us that time is just another

31 4

¿ 2 12

way of measuring distance, we sometimes write

31 4

the time coordinate as ct, so that it has the same

dimensions as the other coordinates. In this way,

3 11 2 11 2 41 2 we could measure time in meters. What is a time

of one meter? It is the time that light takes to

which simplifies to

travel one meter. (Note that we have previously

used time as a measure of distance when we

v c 12

a b

1

(7.8)

¿

introduced the light-year.)

vc

1

An interesting aside to this has come from the

Example 7.2 Relativistic Doppler shift organizations that set international standards such

Find the wavelength at which we will observe the as the meter and the second. It used to be that such

H line if it is emitted by an object moving away units were defined independently, and c was just a

with v/c 0.3. measured quantity. The speed of light is now taken

to have a defined value, where all decimal places

SOLUTION

beyond the most accurate measured value are

From equation (7.8), we find that taken to be zero. It now gives the conversion from

meters to seconds. This means that we only need a

0.3 1 2

a b

1

¿

standard for the second or the meter, but not both.

1 0.3

In space-time we speak of four-vectors to dis-

1.36 tinguish them from ordinary three-dimensional

7 SPECIAL RELATIVITY 133

only one space coordinate, x, as well as the time

vectors. Any event is characterized by the four

coordinates (ct, x, y, z). Observers in different iner- coordinate. We can keep track of events in such a

tial frames will note different coordinates for diagram by plotting the coordinates of the event.

events, but the coordinates are related. If one By convention, we have time running vertically.

inertial system is moving with respect to another The effect of the Lorentz transformation is to

at a speed v, in the x-direction, the coordinates in rotate the axes through an angle whose tangent

is v/c. The unusual feature is that the x-axis and

the transformations between the two coordinate

t-axis rotate in opposite directions, so that the

systems are found by assuming they are linear in

the coordinates, and must give the correct results axes are no longer perpendicular to each other.

Note that v c puts both axes in the same place.

for length contraction and time dilation. The

v/c)

result is (letting It should not surprise us that something funny

happens when v c, because this is where the

ct x)

(ct

quantity becomes infinite.

x ct )

(x We know that in ordinary three-dimensional

space, a rotation changes the coordinates of an

y y

object, but the lengths of things are unchanged.

z z (7.9) That is, if we have two objects, as shown in Fig.

7.9, whose separations are given in one coordi-

The reverse transformation is given by

nate system by ( x, y, z) and ( x , y , z ) in

ct x)

(ct another, then the distance between the two, which

is the square root of the sum of the squares of the

x ct)

(x

coordinate differences, doesn™t change. That is

y y

( x)2 ( y)2 ( z)2 ( x )2 ( y )2 ( z )2 (7.11)

z z (7.10)

We say that the length is invariant under rotation.

These relationships together are called the Lorentz Since the Lorentz transformation has proper-

transformation. ties of a rotation, is there a corresponding concept

We interpret the Lorentz transformation as

y

telling us that the rules of geometry are different

y'

in space-time than they are for ordinary space. To

illustrate this point, we use a space-time diagram,

like that shown in Fig. 7.8. For simplicity, we plot

B

∆x'

ct

ct'

∆y

v=c ∆y' x'

A ∆x

x

x'

(∆x)2 + (∆y)2 = (∆x')2 + (∆y')2

Fig 7.9. Invariance of lengths under rotation.The dark line

represents the distance between points A and B.The compo-

x

nents of this length with respect to the x- and y-axes are x

Fig 7.8. Lorentz transformation on a space-time diagram. and y, respectively, and with respect to the x - and y -axes

The transformation looks like a rotation of the axes except are x and y , respectively. Independent of the compo-

that the time and space axes rotate in opposite directions. nents used, the length of the dark line is the same.

134 PART II RELATIVITY

in space-time? The answer is yes, but the invari- ct

Timelike

ant quantity is slightly different, because the

time axis rotates in the opposite direction to the

v=c

space axes. We define the space-time interval as

( s)2 (c t)2 ( x)2 ( y)2 ( z)2 (7.12)

Lightlike

This is the quantity that is invariant under a

Lorentz transformation. Note that the Lorentz

Spacelike

transformation can be derived by assuming that

x

this quantity is invariant, and that the transfor-

mations be linear in the coordinates. When this is

done, time dilation and length contraction can be

derived from the Lorentz transformation rather

than the other way around. This reinforces the

idea that length contraction and time dilation are

not artifacts of some particular measurement, but

are an integral part of the nature of space-time.

To get a feeling for the physical meaning of s,

(a)

consider an observer moving from one place to

another in time t, as measured in the observer™s

rest frame. This means that t is the proper time

interval. In the observer™s rest frame, there is no

change in position, so x y z 0. This

means that s c t. Therefore, s is just the

proper time interval (in units of length). Moreover,

since it is an invariant, for any two events and any

inertial reference frames s will always equal the

proper time interval between the two events.

We can define three types of space-time inter-

vals (Fig. 7.10a), depending on whether ( s)2 is

zero, positive or negative. Suppose our two events

are the emission and absorption of a photon. A

photon will move on the sphere whose surface is

given by, ( x)2 ( y)2 ( z)2 (c t)2. This means

that ( s)2 is zero for a photon traveling any dis-

tance in any direction. We call such intervals

lightlike. Intervals for which ( s)2 is positive are

(b)

called timelike. The positions are close enough in

space that a photon would have had more than Fig 7.10. (a) Space-time intervals. (b) Light cone.

enough time to travel from the first event to the

second. This means that the first event could the light cone (Fig. 7.10b). Events that could have

have caused the second. In the opposite case, caused the event at the origin of the cone are

when ( s)2 is negative, we call the interval space- inside the past light cone. Events that could be

like. A photon cannot traverse the distance in the caused by events at the origin of the cone are

time given. Unless a signal can be sent faster than inside the future light cone. Events that are out-

the speed of light, there is no way the one event side the light cone can have no causal connection

could have caused the other. with the event at the origin.

If we extend our space-time diagram to more It should be noted that in our discussion of a

dimensions, we call the surface defined by ( s)2 0 space-time interval we could have defined it to be

7 SPECIAL RELATIVITY 135

the negative of what it was in equation (7.12), and not the case, since we know that moving particles

must have some energy. This means that the rest

not changed any of the interpretation (apart from

energy, E0, cannot be zero. So, this gives us an

carrying through the minus sign). It is just a mat-

ter of convention to do it one way or the other, expression for the relativistic energy:

and you will find some authors who do it one way

E E0 (7.15)

and some who do it the other. As long as they are

We can then find the relativistic momentum as

internally consistent there is no problem. People

who use one convention or the other then also

cpx E0 (7.16)

differ in how they ˜count™ the time coordinate.

In the non-relativistic ( close to one) limit, the

That is, we can write (x, y, z) as (x1, x2, x3). If we use

momentum must give the classical expression, px

the space-time interval as given in equation (7.12),

mvx . From equation (7.16) this can only occur if

then we write ct as x0 and think of time as the

˜zeroth coordinate™; if we use the space-time m 0c 2

E0 (7.17)

interval as the negative of that given in equation

where m0 is the rest mass of the particle.

(7.12), then we write ct as x4 and think of time as

We can rewrite the expressions for relativistic

the ˜fourth coordinate™.

energy and momentum:

7.5.2 Energy and momentum m 0c 2

E (7.18)

The space-time coordinates of an event are not

p m0 v (7.19)

the only quantities that transform according to

the Lorentz transformation. For example, another We can also define a kinetic energy as the differ-

important four-vector involves energy and ence between the total energy and the rest

momentum. To see the analogy with (ct, x, y, z), energy:

remember that for a photon moving in the x-

1) m0c2

Ek ( (7.20)

direction, x ct. The energy and momentum of a

photon are related by E cp, so, for a photon mov- In the limit v V c, we can write

ing in the x-direction, E cpx. This suggests that

(v/c)2]1/2

[1

the energy-momentum four vector should be (E, cpx,

v2/2 c2

cpy, cpz). These should then obey the Lorentz 1

transformations:

where we have used the fact that, for x V 1,

(1 x)n nx. The kinetic energy for v V c

1

E cpx )

(E

then becomes

cpx E)

(cpx

m0c2 (1 v2/2c2

Ek 1)

cpy cpy

(1/2)m0v2

cpz cpz (7.13)

which is the classical expression.

The reverse transformation is given by

Since the energy-momentum four-vector obeys

the Lorentz transformations, it must have an invari-

E cpx)

(E

ant length associated with it. It is

cpx E)

(cpx

E2 (cpx)2 (cpy)2 (cpz)2 (7.21)

cpy cpy

To give this quantity a physical meaning, we eval-

cpz cpz (7.14)

uate it in the rest frame of some particle. In that

If we let the ( ) reference frame be one in which case, the momentum is zero and the energy is the

the particle is at rest, so that px 0, then the first rest energy. So the invariant length is simply

m0c2. Since this quantity is invariant, its value

thing we note is that E E , so, if the energy of

must be m0c2 for any observer. (We just choose to

the particle at rest, E , were equal to zero, then the

energy, E , would always be zero. This is obviously work it out in an easy frame.)

136 PART II RELATIVITY

of energy to accelerate an object with non-zero

Example 7.3 Rest energy of a proton

rest mass to the speed of light. This means that

What is the rest energy of a proton? the speed of light is a limiting speed.

Some physicists have speculated on particles

SOLUTION that can travel faster than light. These particles

By equation (7.20) have been given the name tachyons. The trick is

that these particles, if they exist, can never go

24

1010 cm/s)2

E0 (1.67 10 g)(3.0

slower than the speed of light. The speed of light

3

1.5 10 erg would seem to be a barrier for them as well, only

from above. If tachyons do exist they can interact

To form an idea of how large this is, we express the

with photons, and make their presence known.

answer in eV, to get 939 MeV (as compared, for

All experiments to look for tachyons have indi-

example, with the 13.6 eV needed to ionize a hydro-

cated that they do not exist.

gen atom).

Note that, as v/c approaches unity, approaches

infinity. This means that it takes an infinite amount

Chapter summary

In this chapter we saw how the special theory of We saw that we can no longer think of space

relativity has changed our thinking about the and time as being separate entities, but must con-

nature of space and time. sider a four-dimensional coordinate system,

We saw how the requirement that the laws of called space-time. We defined a space-time inter-

physics be the same in all inertial frames leads to val which is invariant under the Lorentz trans-

the idea that the speed of light is the same for all formation (and is equal to the proper time).

observers. This, in turn, leads us to the phenom- We saw that energy and momentum must be

ena of time dilation and length contraction. The treated like space and time. This leads to a rela-

m0c2 and the idea of a rest

tivistic energy E

phenomena are only large when the speeds

energy, m0c2.

involved are close to c.

Questions

7.1. What are the differences between sound waves it, parallel to the ground. From the point of

and electromagnetic waves? view of an observer on the cloth, the ladder

7.2. How does the speed of light being independent shrinks to less than 5 m in length and fits

of the velocity of the observer eliminate the through the hole. From the point of view of

problem that Einstein found with Maxwell™s the assistant, the hole shrinks, so it is even

equations? smaller than the ladder. Yet we know that the

7.3. What is the relationship between simultaneity ladder must get through in all reference

and absolute time? frames if it gets through in one. How does it

7.4. What do we mean by the terms ˜proper length™ get through as viewed by the assistant? (Hint:

and ˜proper time™? Think of how the cloth appears to the

*7.5. A painter™s assistant is carrying a 10 m ladder assistant.)

parallel to the ground. The assistant is moving 7.6. How is the geometry of space-time in special

at 0.99c. The painter is up on a high ladder relativity different from the geometry of three-

and drops a cloth with a 5 m diameter hole in dimensional space?

7 SPECIAL RELATIVITY 137

Problems

7.1. The angular displacement of an image (in stick appear to you? How long does your stick

radians) due to aberration is approximately appear to your friend, assuming the sticks are

v/c, as long as v V c. Use the fact that the parallel to each other.

Earth orbits the Sun once per year at a dis- 7.6. How fast does an object have to be going so

tance of 1.50 108 km to find the maximum that it is found to be 10% of its original

displacement of a star™s image due to the length?

motion of the Earth. Express your answer in 7.7. A source of radiation is moving away from

arc seconds. you at 10% of the speed of light. At what

7.2. You and your friend carry identical clocks. wavelength is the H line seen?

We define the redshift, z, as the shift in wave-

Your friend passes by in a rapidly moving 7.8.

train. As your clock ticks off 1.00 s, you see length, divided by the rest wavelength 0.

your friend™s clock tick off 0.50 s. How much On the assumption that only radial motions

are involved, find an expression for z as a

time would your friend see your clock tick off

function of v/c.

in the time it takes your friends clock to tick

off 1.00 s? 7.9. Show that equation (7.8) reduces to the classi-

cal result when v V c.

7.3. How fast must a clock be moving to appear to

run at half the rate of an identical clock at 7.10. Think about how the length of an object is

rest? determined and show that the Lorentz trans-

7.4. Some radioactive particles are traveling at formations give Lorentz contraction.

0.999c. If their lifetime is 10 20 s when they 7.11. Suppose we have two events that take place at

(ct1, x1, y1, z1) and (ct2, x2, y2, z2) in one refer-

are at rest, what is their lifetime at this

ence frame and at (ct 1, x 1, y 1, z 1) and (ct 2,

speed? How far do they travel in that time (as

x 2, y 2, z 2) in the other reference frame. The

viewed in the frame at which they are moving

at 0.999c? coordinates in the two frames are related by

7.5. You and your friend carry identical meter the Lorentz transformations. Show that the

sticks and identical clocks. Your friend goes space-time interval between the two events is

by on a fast moving train, holding the meter the same in both reference frames.

stick parallel to the direction of motion of the 7.12. Show that if tachyons exist, their rest mass

train. If in the time it takes your clock to tick must be an imaginary number if the energy

is to be real for v c. (An imaginary number

off 1.00 s you see 0.5 s tick off on your

friend™s clock, how long does your friend™s is the square root of a negative number.)

Computer problems

7.1. Make a table showing the speeds (v/c) for which 7.3. For the relativistic Doppler shift, make a graph of

time dilation is a 1%, 10%, 50%, 90%, 99%, 99.9% ( ) vs. (v/c) for (v/c) ranging from 0 to 1.

effect. 7.4. For the relativistic Doppler shift, make a graph of

7.2. Make a graph of 1/ vs. (v/c) for (v/c) ranging from (v/c) vs. ( ) for ( ) ranging from 1 to 10.

0 to 1.

Chapter 8

General relativity

General relativity is Einstein™s theory of gravita- the sphere.) In general, on any surface, the shortest

distance between two points is called a geodesic.

tion that builds on the geometric concepts of space-

time introduced by special relativity. Einstein was People on the surface of the Earth can tell

looking for a more fundamental explanation of that it is curved, and can even measure the

gravity than the empirical laws of Newton. radius, without leaving the surface. For example,

Besides coming up with a different way of think- two observers can measure the different position

ing about gravity (in terms of geometry), general of the Sun as viewed from two different places at

relativity makes a series of specific predictions of the same time. (Thus, even the ancient Greeks

observable deviations from Newtonian gravita- knew the Earth was round. When Columbus

tion, especially under strong gravitational fields. sailed the only issue being seriously debated was

These predictions provide a stringent test of how big the Earth is, since there was some confu-

Einstein™s theory (e.g. Fig. 8.1). sion in interpreting the Greek results, which had

been given in “stadia”. Columbus believed the

“small Earth” camp, explaining why he thought

8.1 Curved space-time he had reached India.)

Surveying the surface will also tell you that

A central tenet of general relativity is that the the rules of geometry are different. For example,

presence of a gravitational field alters the rules of consider the triangle in Fig. 8.2. In a plane, a tri-

geometry in space-time. The effect is to make it angle has three sides, each made up of a straight

seem as if space-time is “curved”. To see what we line. The sum of the angles is 180 . On the surface

mean by geometry in a curved space, we look at of a sphere, we replace straight lines by great cir-

geometry on the surface of a sphere, as illus- cles. A triangle should therefore be made up of

trated in Fig. 8.2. The surface is two-dimensional. parts of three great circles. In the figure we use

We need only two coordinates (say latitude and sections of two meridians and a section of the

longitude) to locate any point on the surface. equator. Each meridian intersects the equator at

However, it is curved into a three-dimensional a right angle, so the sum of those two angles is

world, and that curvature can be detected. 180 . When we add the third angle, between the

To discuss the geometry of a sphere, we must two meridians, that makes the sum of the angles

first extend our concept of a straight line. In a greater than 180 . The results of Euclidean (flat

plane, the shortest distance between two points is space) geometry no longer apply. The greater the

a straight line. On the surface of the sphere it is a curvature of the sphere, the more non-Euclidean

great circle. Examples of great circles on the the geometry appears. On the other hand, if we

Earth are the equator and the meridians. (A great stick to regions on the surface that are much

circle is the intersection of the surface of the smaller than the radius of the sphere, the geom-

sphere with a plane passing through the center of etry will be very close to Euclidean.

140 PART II RELATIVITY

Fig. 8.3. In the absence of gravity, objects move in

straight lines at constant speeds. If we throw a

ball straight up with no gravity, the world line

for the ball is a straight line. If we turn on grav-

ity, the world line looks like a parabola. We can

t

Fig 8.1. In this image of a cluster of galaxies, the light of an

even more distant galaxy is bent into an arc (Einstein ring) by

the severe curving of the geometry of space-time.This curva-

ture is caused by the large mass of an intervening galaxy. [ESO]

x

We now look at what we mean when we say (a)

that gravity curves the geometry of space-time.

This is illustrated in the space-time diagram in

t

x

(b)

Fig 8.2. Geometry on the surface of a sphere.The short-

est distance between two points is along a great circle.We Fig 8.3. Space-time diagram for a ball thrown up from the

look at the triangle bounded by the equator and two meridi- ground. (a) With no gravity, the space-time trajectory is a

ans.The meridians cross the equator at right angles, so the straight line. (b) With constant gravity, the trajectory is a

sum of the angles in the triangle is greater than 180 . parabola.

8 GENERAL RELATIVITY 141

Real Forces

say that it follows this path because the space- Gravity Acceleration

time surface on which it must stay is curved.

Ultimately, to represent fully the trajectory of the

ball we would have to consider all of the four

space-time dimensions. The effect of gravity is mg

g

then to curve that four-dimensional world into a

fifth dimension. It is hard to represent that

FS

dimension in pictures, but we can still measure

the curvature by doing careful geometric meas-

urements.

In this geometric interpretation of gravita-

tion, we need two parts to a theory. The first is to

calculate the curvature of space-time caused by

the presence of a particular arrangement of

a

masses. The second is to calculate the trajectories

FS

of particles through a given curved space-time.

Einstein™s theory of general relativity provides

both. However, the mathematical complexity

goes well beyond the level of this book.

(Supposedly, even Einstein was upset when he

realized the area of formal mathematics into

g

mg

which the theory had taken him.) However, we a

can still appreciate the underlying physical

ideas, and we can even carry out some simple cal- FS

culations that bring us close to the right

answers.

Fig 8.4. Person in an accelerating elevator.When gravity is

present it is indicated by a downward arrow, marked g.

8.2 Principle of equivalence When the elevator is accelerating it is indicated by an arrow

marked a.

The starting point for general relativity is a

statement called the principle of equivalence,

which states that a uniform gravitational field in take g as a positive number.) We now want to add

some direction is indistinguishable from a uniform up all of the forces on the person, and equate

acceleration in the opposite direction. Remember, an them to ma, where m is the person™s mass and a is

accelerating reference frame introduces pseudo- the person™s acceleration. The forces are the per-

forces in the direction opposite to the true son™s weight, mg, and the upward force of the

acceleration of the reference frame. For exam- scale on the person™s feet, FS. The acceleration is

ple, if you are driving in a car and step on the zero, so

brakes, the car has a backward acceleration.

mg FS 0

Inside the car, you have a forward acceleration

relative to the car. Solving for FS gives us

We can illustrate the principle of equivalence

FS mg

by looking at the forces on a person standing on a

scale in a elevator, as illustrated in Fig. 8.4. In the By Newton™s third law, the force the scale

first case, we have the elevator being supported so exerts on the person has the same magnitude as

there is no acceleration, but there is gravity. We the force the person exerts on the scale. Therefore,

take the acceleration of gravity to be g. (Upward FS also gives the reading of the scale. In this case it

forces and accelerations are positive; downward is simply the weight of the person “ the expected

forces and accelerations are negative, and we have result.

142 PART II RELATIVITY

We now look at the case of no gravity, but

with an upward acceleration a. The only force on

the person is FS. Applying F ma gives

FS ma

If we arrange for the acceleration so its value is

equal to g, we have

FS mg

This is the same result we had in the first case. As

far as the person in the elevator is concerned,

there is no way to tell the difference between a

gravitational field with an acceleration g down-

ward and an upward acceleration g of the refer-

ence frame.

To illustrate the point farther, we look at a

third case, in which there is gravity, but the ele-

vator is in free-fall. The forces on the person are FS

upward and mg downward, and the acceleration

is mg downward. This gives us

FS mg mg

This tells us that FS is zero. The person is “weight-

less”. The acceleration of the elevator has exactly

canceled the gravitational field. For the person

inside the elevator, there is no way to distinguish

this situation from that of a non-accelerating ele- Fig 8.5. Pseudo-force in an accelerating space station. In

vator and no gravitational field. This is the same this case the station is accelerating towards Earth (like the

weightlessness felt by astronauts in orbiting free-falling elevator) so the astronaut appears weightless.

space vehicles (Fig. 8.5). Orbiting objects are also [NASA]

in free-fall, but the horizontal component of

their velocity is so great that they never get closer

to the ground; they just follow the curvature of The principle of equivalence is really a state-

ment that inertial and gravitational masses are the

the Earth.

same for any object. If the two masses are equal

If you look carefully at the above discussion,

you will see that we have really used the concept then they do cancel in the above examples, as we

have done. This also explains why all objects have

of mass in two different ways. In one case we said

that a body of mass m, subjected to a force F, will the same acceleration in a gravitational field, a point

have an acceleration F/m. In this sense, mass is first realized by Galileo. It is not obvious on the

the ability of an object to resist the effects of an surface of the Earth, since air resistance affects

applied force. We call this resistance inertia. how objects fall. However, a hammer and a feather

When we use mass in this sense, we refer to it as fall with the same acceleration on the surface of

inertial mass. The second use of mass is as a meas- the Moon, where there is no air resistance.

ure of the ability of an object to exert and feel a It is important to remember that just because

gravitational force. In this context, we speak of we call both quantities “mass” there is no obvious

gravitational mass. In the same sense, we use elec- reason for gravitational and inertial mass to have

tric charge as a measure of an object to exert and the same numerical value. In the same way, we

feel electrical forces. (So, we should think of gravi- expect no equality between the electric charge

tational mass as being like a gravitational charge.) of an object and its inertial mass. If inertial and

8 GENERAL RELATIVITY 143

gravitational mass are the same, this tells us that

gravity must somehow be special. As we will see

in the next section, considerable effort has gone

into verifying the principle of equivalence.

8.3 Tests of general relativity

Over the years since Einstein™s publication of gen-

eral relativity, a number of exacting tests have

been carried out to test observational predictions

of the theory. Some of the tests are really only

tests of the principle of equivalence, while others

are true tests of the full theory.

A direct test of the principle of equivalence

involves the measurement of the attraction of

two different objects by some third body. A class

of such experiments are called Eotvos experiments,

after the person who devised the original experi-

ment around the turn of the 20th century. The

most accurate recent versions of the experiment

were carried out by a group at Princeton University

in the 1960s and a group at Moscow University in

the 1970s. Their findings indicate that the princi-

ple of equivalence is accurate to one part in 1011.

Fig 8.6. The 2.7 m telescope of the McDonald

The equivalence principle we have discussed

Observatory,Texas, has been used to ¬re a laser beam at a

applies strictly to objects that are so small that

re¬‚ector on the Moon, then they detect the weak return. By

we can ignore the differences from one side to

timing the round trip, the distance to the Moon is very accu-

the other in the gravitational field they feel. We

rately determined. [McDonald Observatory]

can treat them as point objects. However, there is

a stronger form of the principle of equivalence

that says that it also applies to objects with sub- gravitation. An important feature involves ellipti-

stantial gravitational binding energy, such as cal orbits. In an elliptical orbit, the distance of

planets or stars. This has been tested by closely the orbiting body from the body exerting the

measuring the motion of the Moon (Fig. 8.6). A force is changing. The orbiting body is therefore

series of mirrors have been left on the Moon by passing through regions of different space-time

the Apollo astronauts. Laser signals can be sent curvature. (See Fig. 8.7, which may help in visual-

from Earth, bounced off these small mirrors, and izing this.) The effect of the changing curvature is

then detected as very weak return signals. By tim- to cause the orbit not to close. After each orbit,

ing the round trip we can measure the distance the position of perihelion (closest approach) has

to the Moon very accurately, to within a few cen- moved around slightly.

timeters. These studies have indicated that the The effect will be greatest for orbits of highest

Earth and Moon fall towards the Sun with the eccentricity, since the widest range of curvatures

same acceleration to within seven parts in 1012. will be covered. Also, the smaller the semi-major

axis, the greater the effect. This is because the

8.3.1 Orbiting bodies gravitational field changes faster with distance

One series of tests of general relativity involves when you are closer to the object exerting the

the behavior of orbiting bodies. The paths are force. In the Solar System, both of these points

slightly different than predicted by Newtonian make the effect most pronounced for Mercury.

144 PART II RELATIVITY

at this point it appears that there is not enough

solar flattening to challenge Einstein™s results.

8.3.2 Bending electromagnetic radiation

Einstein™s chance to predict an effect that had not

been seen came in the bending of light passing by

the edge of the Sun. He said that the warping of

space-time alters the path of light as it passes

near the source of a strong gravitational field.

According to general relativity, photons follow

geodesics. The light will then appear to be com-

(a) (b)

ing from a slightly different direction. If the light

is coming from a star, the position of the star will

Fig 8.7. (a) Curved space-time for Mercury™s orbit around

appear to be slightly different than if the bending

the Sun.The closer to the Sun you get, the greater the cur-

vature of space-time. Since Mercury™s orbit is elliptical, its had not taken place, as indicated in Fig. 8.8.

distance from the Sun changes. It therefore passes through According to Einstein, the angle (in radians)

regions of different curvature. (b) This causes the orbit to through which the light passing a distance b

precess.We can keep track of the precession by noting the

from an object of mass M is given by

movement in the perihelion, designated P1, P2 and P3 for

4GM/bc2

three successive orbits. (The amount of the shift is greatly (8.1)

exaggerated.)

If we set b equal to the radius of the Sun (6.96

1010 cm) we get an angle of 8.47 10 6 rad, which

is equal to 1.74 arc seconds. This is a very small

It is closest to the Sun, and, except for Pluto, has

angle and is hard to measure.

the most eccentric orbit.

The measurement is made even more difficult

The perihelion of Mercury™s orbit advances by

by the fact that we cannot see stars close to the

some 5600 arc seconds per century. However, of

this, all but 43 arc seconds per century can be

accounted for by Newtonian effects and the per-

Apparent

Actual

turbations due to motions of other planets. The

position

position

Newtonian effects could be calculated accurately

of star

of star

and subtracted off. Einstein was able to explain

the 43 arc seconds per century exactly in his gen-

eral relativity calculations. This was considered

to be an interesting result for general relativity, θ

but not a crucial test, since Einstein explained

something that had been observed. A crucial test

involves predicting things that haven™t been Sun

observed yet.

In recent years a controversy has grown out of

this test of general relativity. A group at Princeton

in the 1960s measured the shape of the Sun and

found a slight flattening. A flattened Sun would

also have an effect on the orbit of Mercury, reduc-

ing the general relativistic effects by enough to

say that Einstein™s calculation is wrong. Further

Observer

measurements have indicated that the original

Fig 8.8. Bending of starlight passing by the Sun.The

experiment on the Sun™s shape was in error, but

observer thinks that the star is straight back along the

some experiments suggest that there is some flat-

received ray.

tening. While some of this research is continuing,

8 GENERAL RELATIVITY 145

Sun on the sky. Therefore, the test must be made delay as the spacecraft pass behind the Sun.

during a total eclipse of the Sun, when the sky is Using this technique, Einstein™s predictions have

photographed, and then the same part of sky is been confirmed to an accuracy of 0.1%.

photographed approximately six months later. There is another interesting result related to

The positions of the stars on the two photographs the bending of the paths of electromagnetic

are then compared. The first attempt to carry this waves. A massive object can bend rays so well that

it can act as a gravitational lens. Physicists have

out was by a German team trying to get to a

Russian viewing site for a 1914 eclipse. They were speculated on this possibility for some time.

thwarted by the state of war between the two Recent observations of quasars, to be discussed in

countries. The next try was in 1919, in an effort Chapter 19, have revealed a number of sources in

headed by Sir Arthur Eddington. In the intervening which double images are seen as a result of this

years, Einstein had found an error in his calcu- gravitational lens effect (e.g. Fig. 8.1).

lations, so it is probably just as well that the

8.3.3 Gravitational redshift

observations weren™t done until the theoretical

The wavelengths of photons change as they pass

prediction was finalized. The result was a confir-

through a gravitational field. This effect is called

mation of Einstein™s prediction. The recognition

the gravitational redshift (Fig. 8.9). It is really a con-

of the magnitude of Einstein™s contribution was

sequence of the principle of equivalence.

immediate, both among physicists and the gen-

We can make a plausibility argument to esti-

eral public.

mate the magnitude of the effect. We have already

The solar eclipse experiment is a hard one,

seen in the previous section that the gravitational

and the original one had a 10% uncertainty asso-

effect of some mass is to alter the trajectories of

ciated with it. More recent tries have reduced the

photons (i.e. they follow geodesics that are not

uncertainty to about 5%. Different types of exper-

straight lines). This makes it plausible that the

iments are needed for greater accuracy. A major

gravitational field can do work on the photon,

improvement can be made by using radio waves.

changing its energy. In order to estimate the grav-

The bending applies equally to electromagnetic

itational potential energy of a photon ( GMm/r) we

radiation of all wavelengths. The advantage of

assign an “effective mass”, E/c2, and since E hc/ ,

radio waves is that the Earth™s atmosphere does

this effective mass is h/c . So if a photon moves

not scatter them. We can observe any radio

source as the Sun passes in front of it and watch

Fig 8.9. Gravitational redshift. As

the position of the source change. These tests

the photon moves farther from

have confirmed Einstein™s predictions to greater

the mass its wavelength increases.

accuracy than the eclipse experiments.

»2

There is another effect related to the bending

of light. The longer path that results from the

curvature of space-time around the Sun causes a

delay in the time for a signal to pass by the Sun.

r2

Two types of observations have been done to test

this. One involves the reflection of radio waves

from Mercury and Venus as they pass behind the

»1

Sun. We know the positions of the planets very

accurately, so we know how long it should take

for the signal to make a round trip. The other

r1

type of experiment involves spacecraft that have

been sent to various parts of the Solar System,

especially Mariners 6, 7 and 9, and Viking orbiters

and landers on Mars. We simply follow the signals

M

from the spacecraft. Since we know where the

spacecraft should be, we can determine the time

146 PART II RELATIVITY