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
QA and QB respectively.
C~ ~O ~ +
Q5 ~ C~
QA ~5 ,~.
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
respectively, then the attractive force of gravity obeys the following rule
Gravitational force ~ mA B
of A on B 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,
l ~ est
~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
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
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.
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.
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.
~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.
- Both U and L are at
_d grav. rest relative to
accel. the earth's center
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,
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
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
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
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 \
This shift was first observed in 1919 by Sir Arthur Eddington and constitutes the first confirmation of the Principle of Equivalence.
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.
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.
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).