<< стр. 7(всего 28)СОДЕРЖАНИЕ >>
tion of p. (b) Calculate d /dp, the rate of
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
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
EinsteinвЂ™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 MaxwellвЂ™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 MaxwellвЂ™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 MaxwellвЂ™s theory of electromagnetic radia- time, repeating each cycle. How would an elec-
tion. MaxwellвЂ™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 MaxwellвЂ™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) MaxwellвЂ™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 EinsteinвЂ™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
EinsteinвЂ™s questions about MaxwellвЂ™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
Оё Оё
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 п¬Ѓnite, 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, п¬‚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 п¬‚ash п¬Ѓrst. (c) The observer in the bottom car sees both
taneous to all other observers. This is important
п¬‚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. EinsteinвЂ™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 EinsteinвЂ™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 вЂ˜preferredвЂ™ 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 вЂ˜properвЂ™ does not denote anything superior
clock is moving is greater than that measured in
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
Оі
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
ОІ
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 inп¬Ѓnity.
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 вЂ“ 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 EinsteinвЂ™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 EarthвЂ™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 EinsteinвЂ™s 1905 paper. H. Lorentz
had proposed it as a way around the results of the
Time dilation applies to biological clocks. A
MichelsonвЂ“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. LetвЂ™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, AвЂ™s
sword will miss B, but BвЂ™s sword will cut A. However, from Fig 7.7. The effect of length changes parallel to the direc-
BвЂ™s point of view, as shown in the right п¬Ѓgure, 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 п¬Ѓrst 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. BвЂ™s sword cuts AвЂ™s head while gle stationary clock. In the top half of the frame the front of
AвЂ™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
Вўt
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
ВўtВї Вўt
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 ВўtВї
the frame in which it is at rest, is called the proper
length, L0 . This can be measured in the usual way,
v Вўt
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 Вўt
A will see BвЂ™s stick shrink, and B will see AвЂ™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 ВўtВї
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
Вї
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
Вї
waves, for which the shift depends on which is
moving. We can show that the result for electro-
a1 b
v
magnetic waves doesnвЂ™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
LetвЂ™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 sourceвЂ™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, Вї
ВўtВї
divided by the number of waves. That is
11 v c 2 Вўt
cВўtВї vВўtВї
Вї
N ВўtВї

11 v c2
The frequency is then given by
31 1v c 2 2 4 1 2
c
Вї
Вї
a1 b
v
c
c N
(7.4)
c v ВўtВї
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)
Вї
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
Вї 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)
Вї
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
Вї
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, doesnвЂ™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
в€†x'
ct
ct'

в€†y
v=c в€†y' x'
A в€†x

x
x'
(в€†x)2 + (в€†y)2 = (в€†x')2 + (в€†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 observerвЂ™s
rest frame. This means that t is the proper time
interval. In the observerвЂ™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 вЂ˜countвЂ™ 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
вЂ˜zeroth coordinateвЂ™; 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 вЂ˜fourth coordinateвЂ™.
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 MaxwellвЂ™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 вЂ˜proper lengthвЂ™ get through as viewed by the assistant? (Hint:
and вЂ˜proper timeвЂ™? Think of how the cloth appears to the
*7.5. A painterвЂ™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 starвЂ™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 friendвЂ™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
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
friendвЂ™s clock, how long does your friendвЂ™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 EinsteinвЂ™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-
EinsteinвЂ™s theory (e.g. Fig. 8.1). sion in interpreting the Greek results, which had
been given in вЂњstadiaвЂќ. Columbus believed the
вЂњsmall EarthвЂќ 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 вЂњcurvedвЂќ. 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.
EinsteinвЂ™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
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 personвЂ™s mass and a is
accelerating reference frame introduces pseudo- the personвЂ™s acceleration. The forces are the per-
forces in the direction opposite to the true sonвЂ™s weight, mg, and the upward force of the
acceleration of the reference frame. For exam- scale on the personвЂ™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 NewtonвЂ™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 вЂ“ 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 вЂњweight-
lessвЂќ. 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 вЂњmassвЂќ 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 EinsteinвЂ™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
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 п¬Ѓre a laser beam at a
applies strictly to objects that are so small that
reп¬‚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 EinsteinвЂ™s results.

EinsteinвЂ™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 MercuryвЂ™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 MercuryвЂ™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 MercuryвЂ™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, Оё
but not a crucial test, since Einstein explained
something that had been observed. A crucial test
involves predicting things that havenвЂ™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 EinsteinвЂ™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 SunвЂ™s shape was in error, but
observer thinks that the star is straight back along the
some experiments suggest that there is some flat-
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, EinsteinвЂ™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 werenвЂ™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 EinsteinвЂ™s prediction. The recognition
the gravitational redshift (Fig. 8.9). It is really a con-
of the magnitude of EinsteinвЂ™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,
changing its energy. In order to estimate the grav-
The bending applies equally to electromagnetic
itational potential energy of a photon ( GMm/r) we
assign an вЂњeffective massвЂќ, E/c2, and since E hc/ ,
radio waves is that the EarthвЂ™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 EinsteinвЂ™s predictions to greater
the mass its wavelength increases.
accuracy than the eclipse experiments.
О»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
О»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

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