__Chapter__ __14__

__Including__ __Gravity:__ __The__ __Move__ __to__
__the__ __General__ __Theory__

The Special Theory of Relativity initiated by Einstein in 1905 proved to be a very successful theory. Its success in predicting effects that were subsequently empirically observed is truly astounding. Today the effects of time dilation, length contraction, the relativistic Doppler effect, and many others are commonly accepted as having a firm phenomenal base. Yet Einstein, almost from the inception of the Special Theory, realized that the Special Theory has a significant blemish. Gravity, the other long range force in classical physics, was excluded as an equal partner with electromagnetism and the modified (relativistic) mechanics. Both electromagnetism and relativistic mechanics were explicitly consistent with the postulates of the Special Theory. Gravity is a long range force with some similarities to the long range electrical force (in that both forces fall off as l/r , where r is the distance from the point where the force is measured to the place where the source resides). However, the effects of gravity are notably different from the effects of electromagnetism.

Probably the most significant differences arise from the difference between the character of gravitational "charge" usually called gravitational mass) and electric "charge". It is well known that on the classical level electric charge comes in two forms: positive and negative. Coulomb's law of force between two electric charges reads as follows

(Force on charge A ~ __QA__ __QB__ due to charge B) 2

In the diagram below (Fig. 14.1) we shown the possibilities for the
__direction__ of the force between a pair of electric charges whose
magnitudes are

14-2

QA and QB respectively.

C~ ~O ~ +

Q5 ~ C~

QA ~5 ,~.

~,~ Flg.14

Thus,if the electric charges are __opposite__ there is an __attractive__
electric force, while if the electric charges are alike there is a
__repulsive__ electric force. Since there are these __two__ types of
charge we can arrange the following combinations. Suppose we take an electric
charge of +l unit and combine it with and electric charge of -1 unit. The
experimentally verified __additive__ property of electric charge then
implies that the __net__ electric charge is (+l units + (-1 unit)) = 0. Thus
with electric charges it is possible, and actually quite common, to find
electrically __neutral__ objects. If we place another (positively or
negatively) charged object in the vicinity of the neutral object, since the
electric force is proportional to the product of the charges that are
interacting we see immediately that

Electric force of a Q Q (O)(Q

l_ . .. neutral charged charged) _ ~ charged OD~ect on a ~ 2 = 2 = u

Consequently it is possible with __electric__ charges to place an
electrically neutral object in the environment of another charge and have
__no__ electric force act. From Newton's Second Law, if there is no net
force on the object under consideration there is __no__
__acceleration.__

We mention these properties of electrically charged (and neutral!) objects just to bring out the contrast with gravity. Newton's law of gravitation states that if mA and mB are the masses of objects A and B

14-3

respectively, then the __attractive__ force of gravity obeys the following
rule

Gravitational force ~ __mA__ __B__

of A on B 2

~ nnB

Fig. 14.2

Notice the similarity between the electric and gravitational forces. Both fall
off as l/r and thus are long range forces. Both depend on the __product__ of
their respective "charges". However, it is the properties of the "gravitational
charges" that are significantly different. The essential point is that there is
only one sign of "gravitational charge", the positive sign. For gravitationally
interacting particles the force is __always__ __attractive.__ For two
electric charges of like sign (positive __or__ negative) the electric force
is always __repulsive__. In addition, and this is a most important point,
there is no analog in gravity of a neutral electric charge in electromagnetism.
For gravity __all__ particles have the same, positive __sign__ of
gravitational charge and thus __all__ particles will have an attractive
force exerted on them whenever they are placed in the vicinity of another
object with non-zero mass. Thus every particle with mass __must__ fall in a
gravitational field unless it is prevented from doing so by some __other__
non-gravitational force. From Newton's Second Law we conclude that an intrinsic
property of gravity consists of the __necessity__ __of__
__acceleration.__ In the case of the electric force we did not conclude that
there was an inevitability of acceleration because of the possibility of
constructing the __neutral__ objects, which are immune from the electric
force.

This necessity of gravitational acceleration presented Einstein with a dilemma
because the Special Theory was founded on the supposition that __all__ the
laws of nature and the results of all experiments do not depend on which member
of the class of __inertial__ frames of reference is utilized to study
nature. But the concept of an inertial frame is founded on the assumption that
it is operationally possible to prepare a particle initially at rest with
__no__ other forces acting on it. In such a frame an initially
unaccelerating particle will remain unaccelerating. The presence of electric
forces does not present a problem because we need only to utilize electrically
__neutral__ particles if there is an electric force present. The neutral
particle will not be affected (no electric force will be exerted). However,
since there is no gravitationally neutral object and since the preparation of
the initial state of motion must take place __somewhere__ in the universe,
we see that there will always be a gravitational force present (due to any
number of gravitationally attracting bodies - the earth, the sun, the planets,
the galaxy, etc.) Consequently,there will always be present forces which will
destroy the "once unaccelerated, always unaccelerated" property of inertial
frames.

It is clear that these inertial frames are no longer as natural when
gravitational forces are present. The best one can do is to adapt, as closely
as possible, to the situation. Since gravity seems to imply acceleration
perhaps it is best to try to adapt by accelerating __with__ the particle
that has the force exerted on it. Let us consider the familiar example of our
earth. On the surface of the earth let us locate some probe of the
gravitational environment. In Fig. 14.3 we show a possible situation,

14-5

l ~ est

~J

~o.r~h ~ 14. 3

Suppose we drop the test mass. From Newton's law of gravity there is a net force on the test mass given by

F = TEST EARTH

TEST r

where r is the straight line distance from the point to the __center__ of
the earth. Now we utilize Newton's Second Law. The direction of the
acceleration is clear: the earth pulls the test mass in a direction toward the
earth's center. What does the __magnitude__ of the acceleration depend upon?
To see this we use the Second Law:

a TEST EARTH

mTEST TEST r2

or

EARTH

TEST 2

Note that the acceleration is __independent__ of the __mass__ of the test
object. Consequently, not only does every test object accelerate toward the
center of the earth, in addition every test object has the __same__ value of
the acceleration once dropped from the same initial position. The value of the
acceleration of one and all test objects is determined by the mass of the
source (the earth in this example) and the distance between the test

object's location and the center of the source.

Given this property of the gravitational acceleration it seems clear that a
frame of reference which is adapted as well as possible to the accelerative
character of gravity will be one that simply follows along beside the freely
falling test object. We see then, that in a significant sense, the
__freely__ __falling__ frames of reference are the most natural
candidates to replace the inertial frames of the Special Theory. However, this
realization presents us with a number of additional conceptual problems. In
devising a consistent theory which (a) includes the Special Theory in some
manner, and (b) includes gravitation we seem to be forced to somehow generalize
at least one of the basic tenants of the Special Theory. If we simply replace
the inertial frames of the special theory with the freely falling frames it is,
on the face of it, not at all clear whether the resulting postulates will be
consistent with experience. We can not just change the 'rules of the game"
without a thorough re-examination of the empirical basis for the "new rules".
Einstein thought about gravity and how to include it with the effects of the
Special Theory from around 1908 to 1915, although he did not publish any of his
thoughts on this subject between 1908 and 1911.

I. The Principle of Equivalence

In 1911, Einstein published a paper in which he put forward the "Principle of
Equivalence" and suggested an experimental test of the principle. This
principle is important for two reasons. First it points to the heart of the
problem of identifying more clearly the role of acceleration in the description
of gravitational fields over small regions. This we comment on below.
Second,the principle is a statement which has truly __new__ (in 1911)
physical content, in that it predicts that the presence of a gravitational
field affects the rate at which clocks tick. We will also discuss this aspect
in

some detail below.

What is the Principle of Equivalence? It states that if observations are
performed within "small" regions of space then there is a physical equivalence
between effects observed when the region "sits" in a permanent gravitational
field and when the region __accelerates__ in a __gravity__ __free__
region. Let us illustrate this via the following Fig. 14.4.

As an example, let the observer in Fig. 14.4(a) drop some object and observe its trajectory. All observations are confined to the little room. The dropped object falls to the floor where it remains after a "thud". This outcome of the "experiment" is totally as expected. Consider next the situation illustrated in Fig. 14.4(b). There, some external force pulls the little room containing our observer "up". We can in principle arrange it so that the rate of acceleration is the same numerical value as that experienced by an object close to the surface of the earth. Let us assume this has been done. The question is: what will our observer find for the trajectory of the object he drops? To see the answer we consider three snapshots of the observer, the little room, and the object.

14-8

Q

Tl~UD

STh ~ 4, ~

From the vantage point of the observer it is apparent that the object falls to the floor, accelerating as it goes. If he made a detailed study of the motion he would find that the acceleration is constant, just like the situation on the surface of the earth.

What do we conclude from this "thought experiment"? That, from the observations
available to the observer confined to the small room, he can not determine
whether he is residing in a "real" gravitational field or whether he is in
gravity-free space in an accelerating frame of reference. Actually, Einstein
claimed something even further. He stated that __any__ experiment performed
in the little room would be insensitive to the acceleration of the little room.
In this form Einstein's principle of equivalence is quite a sweeping claim,
applying to all forms of matter.

The validity of this principle would have the following implication. In our
discussion of the Newton gravitational force law we pointed out how the
gravitational acceleration of a freely falling object is independent of its
mass. Further,the acceleration is the __same__ for any particle dropped from
the same position relative to some gravitating source. The principle of
equivalence says that acceleration __by__ __itself__ is not an adequate
indicator of the presence of a "true" or "permanent" gravitational force field.
For

we can "transform away" the force of gravity by hopping on a frame of reference which is freely falling with the object. How, then, can we tell whether we are "really" in a gravitational field? The answer, within the context of Newtonian gravitation, is easily provided. To see how to tell, we consider the situation shown in Fig. 14.6.

*4~o*

In the above figure we have indicated in part (a) the "dotted" lines
corresponding to the trajectories of freely falling objects dropped from above
the gravitating object. Now we construct a connecting line between the
__two__ adjacent objects. Sit on one of them and observe the position of the
other one. Follow the time-dependence of the connecting line as the two freely
falling objects (and the observer) plummet toward the surface of the
gravitating object. From the figure we see that the distance from object A to
object B decreases as A and B fall toward the surface. Suppose we performed the
same kinds of observations in a gravity-free region. The situation is shown in
Fig. 14.7.

14-10

t t

~cceler~io~ d~rec ~

A ~ ~,9 t4q

Because each of the two objects A and B experience the __same__ acceleration
the distance of the connecting line stays the same. Thus to distinguish between
a "real" Newtonian gravitational field and an accelerated frame of reference in
gravity-free space we need to measure the __relative__ __acceleration__
between neighboring particles, where the separation between them is
__larger__ than was the case in the "little room" thought experiment. In
essence, then, the reality of the gravitational field is closely tied to the
fact that the gravitational force depends on position. By probing the
environment at two sufficiently separated positions we gain the needed
information because the force is __different__ at different points. This
method of determining the existence of a permanent gravitational field will
surface again in Chapter 15 when we discuss Einstein's geometric theory of
gravitation.

__Gravitational__ __Blueshift__ __Effect__

The principal new __physical__ result of the Principle of Equivalence
concerns the rate at which a clock runs in a gravitational field. We give a
derivation of this basic result by using the principle of equivalence
explicitly along with some basic results from Newtonian mechanics and
gravity.

Let us consider two points, one near or on the surface of the earth (L) and the other above the first (U) at some distance d.

14-11

U ~

- Both U and L are at

_d grav. rest relative to

accel. the earth's center

L ~,

EARTH Fig. 14.8

For the points U (for "upper") and L (for "lower") there is a force of gravity, exerted by the massive earth, which would pull any particle with mass toward the earth's center if the particle is free to fall. Let us place an observer at each point and give each an identical atomic clock. Now, from the Principle of Equivalence this situation is physically equivalent to a system which is accelerating in gravity-free space. We show this state of

affairs in the following diagram U

Note that both have the __same__ constant acceleration g. We take this
second vantage point and perform the following "thought experiment". Suppose
there is a source of light present at the point U. We wish to send a beam of
electromagnetic radiation from U to L and have the observer at L receive it and
determine its received frequency. That is, the radiation sent by U will be of a
known wavelength and therefore of a known frequency (color). What will L
measure for the "color" of the radiation? Will it be the same as sent or will
it be shifted to a higher or to a lower frequency? We emphasize that the
question we are posing is equivalent to a question of what effect, if any, does
a gravitational field have on the rate at which a clock ticks. The frequency of
the sent and received radiation is closely related to the period of
oscillation. Specifically if f is the frequency,

14-12

then T = f is the period.

Now both U and L have constant acceleration. This means that their velocities are continuously increasing. How does this increase depend on the value of the acceleration and on how fast either U or L was going at the beginning of any interval of observation? To see this recall the definition of acceleration:

* V2 a = acceleration = t2 - tl

If a = constant, as is the case in the present situation, then a(t2 - tl) = V2 - Vl

related to the velocity of either observer since the beginning of the time interval of observation plus the (constant) acceleration times the time interval.

In our example, since light travels at speed c and the distance it has to travel to go between U and L is d we see that it will take a time

Since U and L are moving there is a correction to the above time due to "length

contraction" of the order ( c ) where vrel is the relative velocity of U and L. This correction is very small for experiments performed near the earth's surface.

Suppose light was emitted at U when U had an upward velocity v. When this light beam arrives at L, the point L has an upward velocity of v I when = VU I h + gtUL

v Iwhen = VU I h + gd

where we have used a = g _- 9.8 m/s , the acceleration of gravity at the earth's surface. Thus there is an effective relative velocity difference due to this finite delay required for light to travel from U to L. Thus

rel L (when U Iwhen received) sent

= __gd__

Now recall from our discussion of Chapter 12 that when there is a
__relative__ velocity difference between the sender and the receiver of
radiation, there will be a Doppler shift in the frequency. In Chapter 12 we
showed that the relation is

received sent where

=\1 1 + v

rel

in units with c = 1. We assume that v el ~ c. Thus, in our system of units this read v 1 ~ 1. The above expression for ~ is equal to the following to a very good approximation:

1 1 - v rel ~ 1 - v

u 1 + v rel

~ rel

Consequently

T d __~__ v l) T b U

at L where

V 1 = __gd__ = gd > O

Since vrel is positive we see that Treceived Tsent by at L U

represent the period of an electromagnetic wave, emitted from an atom which
underwent a transition from a higher to a lower state. That is, T represents
the period of the "natural" vibrations of an "atomic clock". Thus to the
observer at L the interval between the ticks of U's atomic clock is measured to
be smaller. The higher __accelerated__ clock runs __faster__ than the
lower accelerated clock.

Now we make use of the Principle of Equivalence. This principle states that the
accelerated observers in gravity free space are indistinguishable from two
observers U and L who sit at rest relative to the earth, with U a distance d
above L. We conclude from this equivalence that a clock placed at rest at U
will run __faster__ than an equivalent clock placed at rest at L. This
effect is called the "__gravitational__ __blueshift__". The name arises
from the results that the frequency (f = T) is higher than the frequency of L's
clock. L infers from this that U's clock is running faster.

We can turn the experiment around and emit an electromagnetic wave of a given
frequency from the point L and have it travel upward to U. The same kind of
argument that we constructed above yields the following result: The observer at
U measures the frequency emitted by L's clock to be lower than the frequency of
his identical atomic clock. That is, the period (T = f--) for the clock lower
in the gravitational field is __longer__ than the period of the clock higher
in the field. Thus a light source of a given color would be observed to have
shifted to the __redder__ end of the spectrum. Light climbing ~ in a
gravitational field is __red-shifted__. Light __falling__ in a
gravitational field is __blue-shifted__.

This is the principal physical result of the Principle of Equivalence.

It shows that there are __new__ effects that can be expected when a
gravitational field is present. This principle first appeared in print in 1911
in a paper in which Einstein,at one time, proposed it. gave a penetrating
analysis of its implications,and suggested several applications which could be
experimentally tested. The Principle of Equivalence has been tested in a famous
experiment performed first by R. V. Pound and G. A. Rebka in 1960 and then
again with increased accuracy by Pound and J. L. Snider in 1965. These
experimenters used a source of electromagnetic radiation (gamma rays from a
source of radioactive Cobalt) and let the radiation fall down a shaft. The
detected frequency was found to agree, in the most precise experiment done by
Pound and Snider, with the prediction of the Principle of Equivalence to within
a few percent. This is considered strong support for the principle.

Another famous experiment testing this Principle was proposed by Einstein in his 1911 paper. He concluded that, there should be an influence of the gravitational field of the sun on the path of a ray of starlight that grazes near the edge of the sun. We illustrate the effect in Fig. 14.10.

S~ont ~ \l

5~r ~J \

S~Ln

This shift was first observed in 1919 by Sir Arthur Eddington and constitutes the first confirmation of the Principle of Equivalence.

14-16

__The__ __Move__ __to__ __the__ __General__ __Theory__

.

What does this principle tell us about the problem of combining gravity and the Special Theory? In Chapter 13, we reorganized the set of events in accordance with the postulates of the Special Theory of Relativity. The resulting Minkowski space-time has a different causal structure from that of Newtonian space-time. In what follows we take the Principle of Equivalence to be verified by experiment and consider the question of the consistency of adding gravity to the structure.

We will construct an argument, due originally to A. Schild, which shows that
Minkowski spacetime is incompatible with the Principle of Equivalence. The
demonstration begins with two observers A and B who are at rest relative to the
center of some gravitating source 5. Let A be hovering __above__ B. We
illustrate this situation in Fig. 14.11.

B

Fi9. ~4-~

The fact that A, B and the source S are relatively at rest implies that there
is __no__ special relativistic time dilation effect between clocks at A, B,
or 5. The Principle of Equivalence implies that there is __some__ time
dilation effect even when the clocks at A and B are relatively at rest. The
clock at B ticks more slowly than the clock at A. We now ask whether this
result is consistent with the assumption of Minkowski space-time.

Let us represent the present set--up in a spacetime diagram shown in Fig. 14.12. Note that the worldlines of S, A, and B are straight and parallel,

14-17 indicating that there is no relative motion.

*~orlll;he o4*

I

B ~ ~-g.l ~

The "thought experiment" consists in having B send two signals from some source of electromagnetic radiation to A. The radiation travels from B's worldline to A's worldline through a region of space in which there is a static, unchanging gravitational field, due to the source 5, which is some heavy object. Perhaps the radiation that dissent from B to A will be affected by the gravitational field. However, since the environment is unchanging in time, two signals sent from B at different times will have to contend with qualitatively identical gravitational environments. Thus, while we allow the possibility that the spacetime trajectories of two light signals will be affected by the presence of the field, we expect that there is no qualitative difference between them. If there were no gravitational effect on the light signals, the spacetime trajectory of the light signals would be straight lines. The effect of the gravitational field is to "bend" the space-time trajectories (see Fig. 14.13).