The Principle of Equivalence - Semantic Scholar

ANNALS

OF

PHYSICS:

!%?, 169491

The

(1963)

Principle F.

Department

of Physics and Astronomy, and Argonne National

of Equivalence* ROHRLICII State University of Iowa, Iowa Laboratory, Argonne, Illinois

City,

Iowa

Various statements identified with the principle of equivalence are not acceptable because they are either not generally valid or are simply the definition of an inertial coordinate system. The accepted statement refers, roughly speaking, to the equivalence of inertial mass with passive and active gravitational mass. The clarification of this problem is greatly eased by a detailed study of the static homogeneous gravitational field which precedes the discussion of the equivalence principle. The role of electromagnetic phenomena, and, in particular, the presence of a charge in such a field is analyzed in detail. I. PURPOSE

OF

THIS

STUDY

Unfortunately, laboratory experiments are not usually performed in failing elevators.’ They are carried out in reference frames which are not inertial, but which are supported in a static gravitational field. Nevertheless, it is taken for granted that the behavior of physical systems studied in this supported frame isapart from a simple correction due to the presence of the gravitational field, characterized by the constant g-identical with that in an inertial system. When pressed on this matter a physicist will refer to the principle of equivalence. However, this principle is by itself a matter of considerable controversy. It is studied experimentally2 while the theorists disagree concerning its formulation and meaning. Two standard references (3, 4) disagree about the equivalence of accelerated observers and gravitational fields, while a third (5) does not seem to believe that there is such a principle at all. The purpose of this study is a clarification of these matters (Sections VIII and IX). To this end it will be helpful to have a detailed calculation before us which deals with the most important gravitational field which is usually referred to in connection with the principle of equivalence, viz. the static homogeneous * Supported in part by the National Science Foundation. I According to general relativity, the special theory of relativity is valid only locally in freely falling reference frames. * E8tviis’s classical experiment has recently been repeated by R. H. Dicke (1) and other tests of this principle are proposed (2). 3 Throughout this paper the abbreviation SHGF will stand for “static homogeneous gravitational field.” 169

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gravitational field3 (SHGF). That this simplest of all fields is not completely understood can easily be seen by raising the question of the radiation from a uniformly accelerated charge. Does it radiate? And if it does, would this not contradict the principle of equivalence? While the first question was answered recently in the affirmative (6), the second question seems to be controversial; it is closely related to a correct statement of the principle of equivalence. Starting with an introductory section on uniform acceleration in special relativity, (Section II) which will be significant later on, the SHGF is defined in Section III, followed by the corresponding description of free fall (Section IV) and the associated local geodesic coordinate system (Section V) which permits one to relate uniform acceleration in an inertial system to free fall in a noninertial system. The relevance of conformal transformations in this respect is studied in Section VI. The formulation of physical laws in a noninertial system (a frame supported in an SHGF) is the problem of Section VII. It will yield the answer to the above question on radiation. The results of this study lead to a clarification of the principle of equivalence; its various forms can thus be evaluated (Section VIII). The results are discussed in the last section (Section IX). II.

Consider an inertial

UNIFORM

ACCELERATION

frame of reference, I. In this frame the motion of a point

particle P is described by its position as a function of time r(t). At each instant t the particle will have a velocity v(t) = dr/dt, an acceleration a(t) = dv/dt, as well as higher derivatives of position, b(t) = da/d& etc. At a given instant of time, to , an instantaneous inertial rest frame, II, , can be defined by the requirement that the velocity v of P, when referred to ItO , should vanish at t = to . After having chosen a particular to we shall denote It, simply

by I’, and we shall indicate all quantities referred to I’ by a prime. Uniform acceleration is a special type of motion characterized by the condition b’(t’)

= 0,

(2.1)

independent of the choice of to to which I’ refers. If, as is necessarily the case in practice, uniformly accelerated motion takes place only over a finite time interval tl < t < t2, then (2.1) is to hold for tl’ < t’ < t2’. The defining condition (2.1) can be expressed in terms of the quantities referred to I and the relative velocity v(6) of I’ relative to I at the instant to . To do this, it is essential to express first r’, t’, and the derivatives dr’/dt’, etc. in terms of r, t and its derivatives by means of a Lorentz transformation with relative velocity U, and after this is done, to identify I’ with the instantaneous rest system by u = v(to) . The condition of uniform acceleration, (2.1)) is then (7)

b + 3r2av-a = 0

(2.2)

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where y = ( 1 - D’)-~“. Uniformly accelerated motion takes place for all times for which (2.2) holds. Equations (2.1) and (2.2) are equivalent. The most general motion r(t) in 1 with uniform acceleration is given by the general solution of (2.2). To find this solution we note first that (2.2) can be written (2.2)’

$ (-y3a> = 0 and can therefore

be integrated

immediately,

(y 2 1)) to yield

T3a = g

(2.3)

where g is a constant vector (independent of t) . A simple calculation shows that Eq. (2.3) can be put into the form

d(v) dt

This equation

y(y2 -

can be written

1)v X (v X a>/2

as an equation P = myv,

= g.

of motion, using

F, = ,t,zg

(m always denotes the rest mass), provided one introduces the “pseudoforce”

Fw = -(Y

+ 1)~ X o

(2.4)

where o=

-(y-

l)vXa/v’

(2.5)

is the angular velocity vector associated with the Thomas precession. Thus, the equation of motion of uniform acceleration is dp/dt = F, + F, .

(2.6)

The force F, is a constant force, independent of time and position. The pseudoforce F, vanishes whenever v and a are parallel. In particular, for rectilinea? motion (motion with initial velocity parallel to g) where both v and a are parallel to g throughout the motion, d(mrv) /dt = mg

(rectilinear case).

(2.7)

This particular case is known as hyperbolic motion, since the integral of (2.7) is a hyperbola in Minkowski space. When the initial velocity is not parallel to g the motion will no longer be rectilinear and F, # 0. The instantaneous inertial rest frames It, and Itotiln will then differ in the direction of their velocities relative to I. Since the composite of two successive Lorentz transformations without rotation and with relative

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velocities not parallel to each other is equivalent to one Lorentz transformation with rotation, a Thomas precession effect necessarily arises. This effect is described exactly by F,,, which has the structure (but not the sign) of a Coriolis force associated with an instantaneous angular velocity o. In the nonrelativistic limit Eq. (2.4) becomes F,=

-2mvXo.

(2.4) NR

There is no term, however, analogous to a centrifugal acceleration. Since (2.6) and (2.3) are equivalent, it is obvious that in the limit when no force is acting, F, = 0, the equation of motion (2.6) leads to the expected solution a = 0 uniquely. The above definition of uniform acceleration can easily be put into the covariant language of special relativity. For charged particles the condition (2.1) is equivalent (8) to the vanishing of the Abraham four-vector of “radiation reaction,” I’” = rO(du“/dr

- axa%‘)

= 0

(2.8)

where vp is the velocity four-vector,4 a” = dv’/dr, and r. is N of the time it takes a light wave to travel the distance of one “classical electron radius” ( e2/mC2) . For any other particle the corresponding charge and rest mass is to be used. The equation da’/dr

= aha%‘,

(2.9)

where we use the quasi-Euclidean metric with positive signature, can be regarded as the covariant definition of uniform acceleration in special relativity. From (2.9) and the orthogonality of a’ and v’ follows at once that the invariant electromagnetic radiation rate (R is a constant (9), CR = rnqaha’ The equation

(2.9) can therefore

= const.

be integrated

2 0

(2.10)

with respect to r, yielding

v’ = aLeAr + fire+ X = da>,

apar’ = 0,

If (R = 0, i.e., no radiation the velocity is constant,

is emitted,

P,p” = 0,

(2.11) 2cy,p’ =

the motion is necessarily

(R = 0 t) v” = a’ + p*

(10).

-1.

(2.12)

uniform,

i.e.,

(2.13)

4 The space components of a four-vector, W, say, are of course not to be identified with the components of the three-vector v. The latter will never be used, so that no confusion can arise. In the case of VP: ZP = (y, yv).

PRINCIPLE

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EQUIVALENCE

If @ > 0, (2.11) can be integrated once more to yield i

= yc + A-‘( aPeAr- p”e+).

(2.14)

The three constant vectors (Y’, @“,and 7” are restricted by (2.12) and are determined by the initial conditions

x’(0) = X-‘(ap - p”) + Yb v”(0) = ap+ 6”.

(2.15)

The constant X is the magnitude of the acceleration in the instantaneous rest system and is determined by the external force according to (2.3). The hyperbolic nature of the world line and the noncovariant relations discussed earlier can easily be seenfrom the explicit solution (2.14). All the above considerations refer to flat space (Minkowski space) and to the absence of gravitational fields. We shall now turn to the description of a particular kind of gravitational field. III.

THE

STATIC

HOMOGENEOUS

GRAVITATIONAL

FIELD

Before we proceed with the study of this field it must be emphasized that a very general class of gravitational fields can exist (with suitable asymptotic conditions) which have vanishing curvature tensor in a Jinite domain, D, to a certain approximation. Consequently, there exists a coordinate system in which this field appears as an SHGF in D in that approximation. A measurement which does not exceed a certain accuracy will therefore yield results which are indistinguishable from those in an SHGF, characterizing the paucity of information obtained (seefootnote 2). This point should be kept in mind during the following study of the SHGF. Intuitively, one expects that uniformly accelerated motion takes place in free fall in an SHGF. However, this is not generally the case. Uniform acceleration was defined in special relativity and the only reasonable requirement one can make is that the freely falling observer in an SHGF, i.e., the one for whom (at least locally) special relativity holds, should see an object in uniform acceleration when this object is supported in the SHGF. In Section V this expectation will indeed be proven. Conversely, uniform acceleration of a freely falling object as seen by an observer supported in an SHGF will be found only for a special choice of the coordinate system. Definition: A static homogeneous gravitational field (SHGF) is defined by a timelike line element, using a metric with positive signature, dT2 = -Adx’

- Bdy’

- Cdz’+

Ddt’

(3.1)

where A, B, C, and D are functions of z only (the field is parallel to the z-axis)

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ROHRLICH

and the curvature tensor R xXpr= 0.

(3.2)

It is now a matter of computation to determine the functions A to D from the differential equation (3.2). If we denote derivatives with respect to z of the functions A to D by a prime, the only nonvanishing Christoffel symbols are D’ I?;, = 2C’

r;, = - -A’ 2C’ D’

rao = ri, = 20’ 0

r,“, = - -B’ 2C’

I

A’

r31= ri, = 52’

I?;, = -C’ 2c

r32= r;, = 2

B’ -. 2B

(3.3)

Substitution of these expressions into (3.2) yields the following set of differential equations which exhausts the 20 linearly independent components of the curvature tensor: A’B’ = 0, B’D’ = 0, D’A’ = 0 (3.4) (E = A, B, orD).

(3.5)

Equations (3.4) state that at least two of the three coefficients A, B, and D must be constant. In order to make progress at this point we look at the nonrelativistic limit of an SHGF. The gravitational potential 4 for a constant force in the negative z-direction is 4NR = gz such that

FNR = -WLVC$NR = -mgk

(3.6)

where k is the unit vector in the z-direction and g is a constant. If the gravitational field is weak, so that the gPydiffer from the Minkowski values ~]r~ (signature +2) only by small terms (4 2, 4eg2

E, = t1 -

bPj2

- u”l” +

(7.12) (%P)2’

For u2 = 1 + 2gx one finds the weak field result E, = e/r”. The first correction to it is given for u2 = 1 + 2gz + a(gx)2 by

The potential lomb” potential

A, follows

from ref. 6 in the same way and yields for the “Cou1 + (gp)2 + u”

4

=

-Ao

=

eg r 0 in Rosen’s notation or z + t > 0 in ours (cf. p. 503 of ref.6). The time-symmetry argument whether applied on the fields or on the energy-momentum tensor is therefore misleading; it yields no radiation in either references (6) or (16). If the retardation condition is not imposed one has a mixture of half retarded and half advanced fields (expressed by the identical formulas for the fields), corresponding, however, to two charges, one being the time reverse mirror image of the other in a Minkowski diagram. In that case there is indeed no radiation, as was explained in ref. 6.

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ROHRLICH

mass equivalent of any form of energyI is also both active and passive gravitational mass. In general relativity (A) is satisfied by the geodesic postulate “all test particles move along geodesics.” It is also valid for a cloud of test particles (5). In this sense it is not a local statement. (B) is satisfied by the dual role played by the energy-momentum tensor in the gravitational field equations. While Tpy plays the role of a field source (active gravitational mass) in

R,ua - 3gpvR = KT~~, it plays the role of passive matter whose motion tional field (passive gravitational mass) in

is determined

by the gravita-

T,m - 3g,nT = (l/~)R,w . Both statements, (A) and (B), are valid for true gravitational fields, i.e., irrespective of the value of the curvature tensor (including zero), independent of the strength of the field. Statement (A) is the basis for a geometrization of gravitational interaction. Statement (B) is far from trivial as soon as Newton’s third law is abandoned as is the case in a theory containing finite propagation velocities of interaction. We shall later adopt (A) and (B) as the full statement of the principle of equivalence. It is a generalization of the assertion of the equivalence of inertial and gravitational mass. An equation of motion of a test particle is an equation involving the second (and no higher) time derivative of position of that particle: although this is not essential we shall here assume that one can solve this equation for the second derivative, yielding d2X” -= dr2

p

x,:

(8.1)

. -> 7

f(

If we have a set of particles of various masses and compositions, all with the same initial conditions, it follows from (A) that f” is independent of these masses and compositions. Therefore, over a region over which the field is sufficiently uniform, an observer moving (“falling”) with these particles will see no effect of the gravitational field. The field can be transformed away locally. The existence of such a comoving observer is physically obvious. Mathematically, this means that there must exist a coordinate transformation S + S’ such that in the new coordinate system (8.1) reads d2x’” d72= 12 This is the precise meaning made as a local statement.

of the vague

o

(8.2)

* word

“matter”

in (B) when

this

statement

is

PRINCIPLE

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185

The freely falling observer, characterized by the coordinate system S’, sees no “gravitational field strength,” j” = 0. Since by assumption no other forces are present, S’ is an inertial observer. In this way it follows that a theory of gravitation which contains both Newtonian gravitation theory and special relativity in suitable limits and which is consistent with (A) must involve the following specafiation of an inertial system: (C,) A system whose origin is freely falling and which is nonrotating in a gravitational field is locally an inertial system and all the laws of special relativity are locally valid in it. This statement can also be expressed in the form. (C,) There is no physical experiment by which an observer can distinguish locally between his own free, nonrotating fall in a gravitational field and field free space. A third form of the same statement is obtained by considering the inverse transformation, i.e., the transformation leading from (8.2) to (8.1)) and using (A). The above statements (C) then can be phrased as (C,) An acceleration field is locally equivalent to a gravitational field.13 All three statements (C) depend on the existence of the transformation S * S’. In general relativity this is assured by the assumption that the underlying space has a symmetric linear connection. Thus, in this respect a space much more general than a Riemann space (assumed in general relativity) would do as well. In such a space there always exists a transformation which makes the connection I’“,, = 0 locally. It is then only necessary to make I’“,, a factor in j“ of (8.1) to obtain the desired equation of motion. In this case it is the geodesic equation (8.3) If one postulates covariance and a symmetric linear connection, the geodesic postulate follows uniquely from (A)14 and the existence of S c--) S’. General relativity also permits a precise definition of “locally” in (C) . It means “over a space-time domain D of order (6~)~ in which the curvature tensor vanishes everywhere in the sense that R,,,$ix”6x8 - 0,” i.e., “to the extent that there is no true gravitational field.” For apparent gravitational fields (R,A,, = 0) “local” means “everywhere.” I3 This statement means that two noninertial systems are locally identical. “An inertial frame in which there is a gravitational field present” is meaningless and a self-contradiction. Fock (4) identifies (Ca) with the principle of equivalence. His argument that ((23) is valid only for nonrelativistic physics to first order in G is incorrect. “Local” does not mean “to the approximation that the field is a Meller SHGF,” but refers only to the approximate vanishing of the curvature tensor. Even if the field approximates an SHGF it need not, have the Mprller metric, as was shown in Section III. I4 For a spinning particle, (A) need not be valid and neither is (8.3).

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ROHRLICH

,It is now clear that all statements (C) are simply recipes for constructing inertial systems in a world of gravitation. They define an inertial system and thus give meaning to the special theory of relativity. The definition with respect to fixed stars which is so often given is logically unsatisfactory and, being a nonlocal definition, has no place in a theory which does not fully incorporate Mach’s principle and which is based on differential equations. One or the other of the forms of the statement (C) have been preposed as statements of the principle of equivalence. According to the present point of view this is not acceptable for two reasons: (a) The statement (C) does not state an equivalence but is a definition (operational, in a certain sense) of an inertial system and can thus not be a principle. This definition is used to deduce the laws of special relativity from experiments in noninertial laboratories. The only question that could be asked is: What assures that two differently moving inertial observers (one freely falling radially and one circling the earth, say) would both see locally the same laws of special relativity? This is assured by the principle of covariance (which can thereby be tested) and has nothing to do with the principle of equivalence or the definition of inertial systems. (b) The statement (C) is by its nature meaningful only “to the approximation that there is no true gravitational field (RKx,l = 0)” and therefore physically not acceptable as principle about gravitational interactions. On the other hand, we must keep in mind that none of the definitions (C) would be possible without the validity of the principle of equivalence (part A) at least for flat space. Thus, an experimental test of (C) confirms that in JEat space the equation of motion of a test particle in a gravitational field is independent of mass. Only an experiment in which those features of the mtric enter which assure a non-vanishing curvature, can provide a test of the full principle of equivalence. In summary, then, we have two basic, fundamental, and independent assertions, (A) and (B), which are to be regarded as “the principle of equivalence.” They refer, respectively, to the equivalence of passive gravitational mass with inertial and with active gravitational mass. They are valid for true gravitational fields and are not restricted to the flat space approximation. The various statements (C) are each a definition of an inertial system and they cannot meaningfully be regarded as principles. It must be understood, however, that no such local definitions were possible without the principle of equivalence (statement (A)) being valid at least for flat space. IX.

DISCUSSION

While the bulk of this paper is concerned with the study of a static homogeneous gravitational field (SHGF), the results derived from it are only auxiliary

PRINCIPLE

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187

to the problem, of the precise formulation and meaning of the principle of equivalence. Although an SHGF is in general a poor approximation for actual physical systems (a region immediately above the center of a large disk-shaped galaxy might be the best approximation realized in nature), this study greatly clarifies the situation since an SHGF is the most important gravitational field involved in the common statements of the principle of equivalence. Specific calculations confirm that free fall of an object as seen by an observer supported in an SHGF is not the same as support of this object in an SHGF as seen by a freely falling observer. Acceleration is not reciprocal in general relativity. Since electromagnetic phenomena are usually not, included in discussions of the principle of equivalence, it is important to clarify their role. As long as we do not have a unified field theory which might well predict interference phenomena between gravitational and electromagnetic interactions that are so far not included in general relativity, the principle of equivalence is postulated to hold also for charged test particles. l5 The electrodynamics of special relativity is consequently valid in the local inertial frames of the statements (C). These statements therefore imply, in particular, that an observer falling freely in an SHGF will (a) see a similarly falling charge as purely electrostatic and nonradiating, and (b) see a charge supported in an SHGF radiate according to the laws of special relativity when applied to hyperbolic motion. Since these two situations are not related by a Lorentz transformation (under which the radiation rate is invariant), (a) and (b) do not contradict each other. The question is much more subtle for the observer supported in an SHGF. Since he is noninertial his Maxwell equations are only formally identical with those of special relativity. That they predict radiation from a freely falling charge and no radiation from a supported charge is not obvious. That this is actually the case is proven explicitly in Section VII. If one argues on the basis of (C,) that this situation involves an accelerated charge which should always radiate, the argument is erroneous, because the fact that a charge is accelerated does not necessarily imply that it radiates, unless the acceleration takes place relative to an inertial observer. A noninertial observer uses different clocks and yardsticks. Thus, even though the charge is accelerated, it follows that, because the observer is also accelerated, the co-accelerated observer sees no radiation. Since radiation is not a generally covariant concept the question whether the charge really radiates is meaningless unless it is referred to a particular coordinate system. Finally, since the Schwarzschild metric, locally, for small G, and nonI5 By definition of “test particle” one must ignore here the effect of the particles own field on its motion (electromagnetic as well as gravitational). But the electromagnetic self-energy is included as part of its mass which is not supposed to enter its equation of motion.

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ROHRLICH

relativistically, is identical with the SHGF metric, the above conclusion also holds for a charge at rest as seen by an observer in a Schwarzschild field. A “local” definition of radiation (9) is here essential. These considerations are important in that they confirm the validity of the principle of equivalence also for charged test particles. Experimental tests which confirm the validity of special relativity in various freely falling situations also confirm the principle of covariance and the definition of inertial systems (C,, say). They confirm the principle of equivalence only weakly, viz. for flat space. Strong tests must show that part (A) of that principle also holds when space is curved. For example, in a static field the freely falling observer must not only see an SHGF, i.e., the world lines of objects supported in the static field must be long enough so that they can be distinguished from a congruence of parallel straight lines. The experiments by Eiitvijs and by Dicke (1,Z) can provide such a test, since the earth is rotating and the sun’s gravitational field is changing during the experiment. A confirmation of the principle of equivalence is the simultaneous observation of the orbiting of a nonspinning space capsule (which assures free fall in a field of nonvanishing curvature) and the observation of force-free space inside this capsule, irrespective of the mass, chemical composition, temperature, polarization, etc. of the objects in it. The presence of electromagnetic forces are to be accounted for by special relativity, to arbitrary accuracy, restricted only by the flatness of space within the capsule. The latter can be determined by the observer on earth. The fact that all this holds for all space capsules confirms that the laws of nature are independent of space and time (at least within certain limits), i.e., it confirms the general principle of relativity as expressed by the principle of covariance.le Similar tests are provided by changing the bodies used in the Eotvos-Dicke experiment not only with respect to chemical composition, but also with respect to types of energy (kinetic energy of heat motion, potential energy of aligned nuclei, etc.) since the equivalent of every type of energy has to obey the equivalence principle. Such tests were recently proposed by Morgan and Peres (17). However, their argument concerning the velocity dependence of the forces in freely falling objects as seen by an observer supported in a Meller SHGF have nothing to do with these tests. In fact, the arbitrariness of u(z) can be used to eliminate the velocity dependence. There is no reason to prefer Mprller’s choice over any other U(Z). Of course, any test of the geodesic equation as the equation of motion of freely falling test particles is a direct test of the equivalence principle. However, such a test involves usually also other features of the theory of relativity. The principle of equivalence (part (A)) is satisfied when light rays follow null the

16 We fixed

also emphasize stars.

that

this

typical

inertial

observer

is accelerated

with

respect

to

PRINCIPLE

OF

EQUIVALENCE

189

geodesics. Their bending depends on the curvature of space, i.e., on the particular type of true gravitational field (despite the fact that within the experimental accuracy to which the equation can be confirmed only the first power of G enters!) and is not predicted quantitatively (nor excluded qualitatively) by this principle (18, 19). An analogous situation is the perihel precession of planetary motion (which does measure terms in G2, however). It can also be seen that no form of the SHGF can ever predict the same light bending as the Schwarzschild metric, even in first approximation in G, because the SHGF approximates that metric only nonrelativistically which is insufficient for the world line of light. The gravitational shift of spectral lines is both the weakest and the strongest of the three Einstein effects. The usual explanation given, which is Einstein’s original one (20) and which precedes the general theory of relativity, is not an explanation at all, but is at best a plausibility argument. It says that, if there were a theory which contained (in suitable limits) Newtonian gravitation theory, special relativity, and quantum mechanics, then the principle of equivalence (part (A)) will-to first order in g-predict that shift of spectral lines. All three theories mentioned enter into the argument, but each of these contradicts the other two. Since we do not have a theory which incorporates all three of them, this effect is strictly speaking unexplained. In this respect the red shift effect is the strongest of the three Einstein effects. The red shift was observed in two essentially different ways: by a comparison of a clock (radiating atom) on a star (sun) with a terrestrial clock, and by a cornparison of two clocks at different terrestrial gravitational potentials (21). The essential difference between these experiments lies in the fact that in the second experiment (and not in the first) the gravitational field at both clocks can be eliminated simultaneously by a suitable motion of the observer (coordinate transformation); it involves an SHGF within the accuracy of the experiment and is therefore a very weak confirmation of the equivalence principle. In fact, it is correctly predicted by using the definition (C,) of inertial systems in comparing any two clocks which are in different gravitational fields (true or apparent) and which use coordinate systems whose go0to first order in g agree with Minkowski space and an SHGF respectively, i.e., which are go0 = -1 and go0= -(l + Zgz), respectively. Since any theory which incorporates Newtonian gravitation and special relativity will satisfy these conditions, the red shift, experiment performed in the laboratory confirms in addition to these theories little else than the specification of inertial systems according to (C,). It does prove more than the red shift experiment on a rotating disk (22), since the latter refers to flat space and an inertial clock,17 but it proves less than the astronomical red shift observation which involves true gravitational fields. I7 The experiment earth with the emitter

is performed in the plane on the axis of the rotating

orthogonal to the disk, the observer

gravitational field of the being on the periphery.

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ROHRLICH

The comparison of a clock in a satellite and one on the earth would differ from all the previously performed red-shift experiments in that a noninertial observer compares a noninertial and an inertial clock in a true gravitational field. The principle of equivalence is fully contained in the theory of relativity. What is not contained in it is the origin of inertia. The theory assures the equality of inertial and gravitational mass but it does not explain why they are equal. A possible explanation of the latter would be afforded by Mach’s principle which is not contained in general relativity. In a theory which contains Mach’s principle, the principle of equivalence is trivial, since inertia would have a gravitational origin. If we accept the equivalence principle in the form proposed above, it is quite clear that it points to a geometrization of gravitational phenomena, as was indeed its historic role, while a geometrization of electrodynamics is not indicated; not only is the charge to mass ratio not a universal constant, but the exact equation of motion of a test charge involves the charge in a nonlinear and complicated manner Q’S). Note added in proof: After completion of the above paper an article by T. C. Bradbury (Ann. Phys. (N. Y.) 19, 323 (1962)) appeared in which the electromagnetic field of a charge supported in a Meller SHGF is computed for an equally supported observer. His results are a special case of our Eqs. (7.9) to (7.12). RECEIVED:

September 20, 1962 REFERENCES

1. R. H. DICKE, Sci. Am. 206,84 (1961); Science 129,621 (1959). 8. R. H. DICKE, Am. J. Phys. 28, 344 (1960). 8. C. M@LLER, “The Theory of Relativity.” Oxford Univ. Press, London, 1952. 4. V. FOCK, “Theory of Space-Time and Gravitation,” esp. p. 209 ff. Pergamon Press, London, 1969. 6. J. L. SYNGE, “Relativity: The General Theory,” preface, p. IX. North-Holland Publishing Company, Amsterdam; Interscience, New York, 1960. 6. T. FULTON AND F. ROHRLICH, Ann. Phys. (N. Y.) 9, 499 (1960). 7. E. L. HILL, Phys. Rev. 72, 143 (1947). 8. F. ROHRLICH, The classical electron. In “Lectures in Theoretical Physics,” Vol. II, W. E. Brittin and B. W. Downs, eds. Interscience, New York, 1960. 9. F. ROHRLICH, Nuovo Cimento 21, 811 (1961). 10. VACHASPATI AND BALI, Nuovo Cimento 21, 442 (1961). 11. A. LICHNEROWICZ, “Theories Relativistes de la Gravitation et de I’Electromagnetisme.” Masson, Paris, 1955. 1.9. C. M$LLER, Kgl. Danske Videnskab. Selskab 20, No. 19 (1943). 19. See, for example, T. FULTON, F. ROHRLICH, AND L. WITTEN, Rev. Mod. Phys. 34, 442 (1962).

PRINCIPLE

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EQUIVALENCE

14. T. FULTON, F. ROHRLICH, 16. E. T. WHITTAKER, Proc. 16. IY. 18. 19. 20. 21. 22. 23.

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