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Phenomenological linear theory of gravitation
ANNALS
OF
PHYSICS:
1,
196-212
(1957)
Phenomenological
Linear
Part II. Interaction F. J. Department
of Gravitation’
with the Maxwell
BELINFANTE
of Physics,
Theory
AND Purdue
J. C.
University,
Field’
SWIHART” Lafayette,
Indiana
The theory of Part I is applied to the interaction of gravity, matter, and electromagnetism. Maxwell’s equations are valid with an apparent electric and magnetic susceptibility of space due to the gravitational field. One effect of this susceptibility is a refraction of light rays by a gravitational field, from which the bending of light rays passing the sun can be derived. The result is the same as found in Part I for the deviation of particles traveling at the speed of light. The equivalence principle is shown to be valid in our linear theory as in Einstein’s theory, notwithstanding the velocity dependence of the gravitational acceleration of free particles. For bound particles, the dependence of the gravitational forces on the relative velocities of parts of a system with respect to the system’s center of gravity is neutralized by gravitational effects on the binding forces. We have shown this only for a special case, and the calculation is performed only in electrostatic approximation. As the errors thus introduced cancel in Einstein’s theory, it is assumed that they cancel also here. The parallelism with the corresponding derivations in Einstein’s theory are apparent at each step of the calculation, and remains, with altered numerical coefficients in the formulas, for any values assigned to the constants of our linear theory of gravitation. It is, however, possible that in higher approximations our linear theory would violate the equivalence principle. This question is worth further theoretical investigation, and, if such effects theoretically would follow from our theory, EiitvGs’s experiments on the equivalence principle should be pushed from 10-S to IO-” in order to discover such effects and to make an experimental decision between the linear and the generalrelativistic theories of gravit,ation. 1. INTRODUCTION
In our preceding paper (1)4 we developed a linear theory of gravitation and its interaction with other fields. We shall now apply these methods to the interaction of this field of gravity with the Maxwell field and with charged particles 1 This work was supported in part by the National Science Foundation. 2 Based in part on chapter 3 of a thesis by J. C. Swihart submitted in partial of the requirements for a Ph.D. degree at Purdue University (1955). 3 Now at International Business Machines Corporation, Poughkeepsie, New 4 We call this paper “Part I,” and refer to its formulas by (I, . . .). 196
fulfillment York.
THEORY
OF
GRAVITATION
197
(Sections 2 and 3). The gravitational field has the effect of an apparent electric and magnetic susceptibility of space. Thus the field around the sun acts like a lens, which mill bend electromagnetic waves passing the sun (Section 4).5 The bending of such light rays is found to be equal t#o the deflection of photons calculated in Part I (1). We then tackle the problem of whether the principle of equivalence of heavy and inert mass is valid according to our theory (Sections 5 and 6). In view of the velocity dependence of the gravitational acceleration found in Part I, the question arises as to whether the fast motion of nucleons inside compound nuclei does not affect the gravitational acceleration of such nuclei as compared to the hydrogen nucleus. (This problem exists in Einstein’s theory as well as in our linear theory of gravitation.) As the problem it,self is too complicated for a rigorous t.reat,ment,6 we investigate the principle involved by considering a simple model, consisting of two oppositely charged particles, possibly with different masses, rapidly rotating in circles around their center of mass which is just starting to fall. We take into account gravitational effects both on the generaCon of the electric field by the one particle, and on the accelerating effect of this field on the other particle. For simplifying the calculation, we have inexcusably ignored retardational (and corresponding radiative and magnetic) effects except those directly due to the gravitational acceleration of our two-particle system. It is then shown that, in this approximation, neglecting also terms quadratic in the gravitational field, t’he starting acceleration of our t’wo-particle system is simply g, and not the larger value which would correspond to the velocities of t,he individual particles. [This g was the gravitational acceleration for a particle at rest (I).] This makes us confident that t,he principle of equivalence is not violated by our theory; at least not in the approximation considered in this paper, neglecting effects of the order of lo-*. In Section 5, we discuss the possibility of an experimental test between the linear and the general-relativist’ic theories of gravitation, by considering second-order gravitational effects. 2. THE
LAGRANGIBK
For the electromagnetic field, we use a first-order Lagrangian (-3, 4) so that the field strengths H,, = -HP, are to be submitted to variations independent of the variations of the potentials A, . (We shall find HIuy # d,A, - &A, anyhow, due t,o gravit,ational effect’s.) Here we do not consider ‘(Fermi” terms’ in 5 The methods Itsed are similar t,o those used by Moshinsky (2) for calculating the bending of light rays according to Birkhoff’s theory. 6 In Einstein’s theory, the problem finds its general solution in the fact that in a freely falling local frame of reference the center of mass of a compound nucleus as well as of a hydrogen nucleus must have zero acceleration when both are initially at rest in such a coordinate q-stem. Since in our linear theory of gravitation the general relativity principle does not hold, this simplification of the problem does not exist in our theory. 7 Terms which vanish on account, of the Lorentz condition.
198
BELINFANTE
the Lagrangian, So, we use
which
AND
SWIHART
are not needed as long as we do not quantize L, = (4?r)-‘[~H”“H,,
the fields.
- fm”A,]
(1)
as the electromagnetic Lagrangian density. To this Lagrangian are to be added the gravitational, matter, and interaction Lagrangians of Part I, in particular the ones described by Eqs. (I, 2), (I, 1 l), and (I, 23) with (I, 24a-b) and (I, 21). The only difference is that we shall simplify (I, 2) at once in accordance with (I, 25) and (I, 26), and that the kinetic and total momentum of the jth particle are no longer equal, but are now related by the definition *ip E Ph where pi,, and Xj, are functions
k%/hUXi),
of the parameter
3. FUNDAMENTAL
(2) aj along the world line.
EQUATIONS
In our Lagrangian, the functions aj’(sj), pj,(sj), Xj“(sj), &(x, t) = h,(x, t), A,,(x, t), and H”(x, t) = -H”“(x, t) are now to be variedindependently. We shall use the notation of Part I, dropping again from time to time the index j. Variation of a”(s) and of p#(s) gives Eqs. (I, 29) and (I, 30) and therefore (I, 31) without change. Variation of X’(s) yields this time* dp,/ds where
vprr
= ~a” V,A”(X)
=
+ $$x? V,[s%,,(X)]
- [E/clV,A”(X).
dA,(X)/ds
d?r,/dT
= [E/c]Bp”(X)o”
Here, we introduced
(3)
By
= [ V”A,(X)I
and by Eq. (I, 30), Eq. (3) with invariant” form
+ Kmc’b V&““(X),
dX”/ds
= (u”/s’)[ V”A,(X)I
the help of (2) can be written
+ >&s’ap?r“ D,,&,(X)
(4) in the “gauge-
+ Kmc’s’b V&““(X).
(5)
the abbreviation B,, = &A”
which satisfies the differential
- d”A, ,
(6)
equation
ahBllv+ a,& + a,B, = 0.
(7)
Note that Eq. (5) differs from Eq. (I, 32) for drP/dT only by the first term on the right in Eq. (5), which we therefore interpret as the electromagnetic “New* When not printing the index j, B = -e, if e = 4.8 X lo-lo statcoulomb.
we shall
write
B for
ej.
For
instance,
for
electrons,
THEORY
tonian force” force” f, by9
OF
1YY
GRAVITATION
F, on the charge & related to the electromagnetic F, = ffib ds/dT
Consequently, from (5) follows Variation of h,,” yields (c4/8?rGs) [ 0 h’” - (g”” q h:]
= [~/c]Bp”v”.
an equation
+ Cj[J/4c(a”?r”
“Minkowskian (8)
of motion of the type of (I, 93).
+ a”?r”) + Kbmc2g”“]s’6&
+ (8*)-‘[HX”Hx’
-
$~g”“HPUH,,]
-
X[t])
(9)
= 0,
with the particle quantities understood to be taken at the particle time t, so at s[t] in the notation of (I, 19). Finally, variation of A, , and of HP” = -H”‘, yields a”Hp” = 47rj’,
(10)
B,” = H,,” - >~H,“h~’ + H,xh,X - H”hh,‘.
(11)
Here we put j’(X,
t) 3 Cj
(C/C)
V”[t]
63(X
- X[t])
(12)
and we used Eq. (I, 31), while v[t] means v(s[t]). Equations (7) and (10) are Maxwell’s equations, with Blo = E, , B23 = B, , HI0 = @’ = D, , Hz, = Hz3 = H, , J’ = p, j’ = i./c, etc. Equation (11) then describes the dielectric and magnetic properties of a gravitational field. For the S(pherically) S(ymmetric) S(tatic) gravitational field with isotropic tensor ellipsoid discussedin Part 1, with ho0= -h: = x, hb" = -ho" = 0, and hkl = S,& Eq. (11) simplifies to B = HI1 + $6 (x + +>I,
E = IN1 - 3$(x + $>I.
(13a-b)
Hence, such a field behaves as a medium with dielectric constant c and permeability p given by
e = 11 - 4$(x + 4>1-’ =P = 1 +%(x+4) e m /.i x 1 + 5A~‘/2r = 1 + >$Fh,
>
(14a-b) (14c)
where we used (I, 50) and (I, 83a). Consequently, the Coulomb potential at a small distance T’ from a point charge Ze placed in such a gravitational field is given by + = Ze/tr’ 9 This
f,, obviously
is a fourvector;
= [I - f$(x + 4)]Ze/r’, not
F, or 0’.
(15)
200
BELINFANTE
and the index of refraction
AND
of the gravitational
n 1 (qp 4. BESDING
OF
SV’IHART
LIGHT
field is given by
E 1 + !i(x
RAYS
IN
THP:
+ 4).
SUN’S
(16)
GRAVITATIOSAL
FIELD
Equations (16) and (I, 50) tell us that spheres of radius r around thf sun’s center will be spheres of constant index of refraction n z
1 + FB’/2r
x
1 + 5GM/2c’r
> 1
(17)
outside the sun. Thence, t.he region around the sutl acts as a converging sphericai lens. The smallnessof (n - 1) enables us to calculate the deflection of a light ray passing through this lens in the following simple way. Let R be the distance of nearest approach (at 0) to hhe sun’s center M. (See Fig. 1.) At any given point P of the ray, the actual distance to M is r = R set 0, and the distance braveled from 0 is s = R tan 0( >< 0). As the ray travels another short distance PQ = ds = R se? 0~3, the distance from M increases by dr = R set’ 0 sin %de,
(18)
passing into a region with an index of refraction increased by dn = -f
i5k’r-”
dr = -,1 &WR-’
sin 3 d0.
(19)
Consider the gravitational “lens” to be made up out of infinitely many infinitely thin concentric layers of thickness dr each with its constant indes of refraction n(r), so that the ray will be refracted according to Snell’s law when passing from one such a layer into the next, Let d+ be the resulting deflection of the ray in one such a refraction with angle of incidence = ($$r - 0) and angle of refraction = ($+$r - 0 + d$) ; SO (n + dn) sin ($7
FIG. 1. Refraction
of light
- 0 + d$) = n sin ($5~ - 0).
ray
in gravitational
field
near
sun.
THEORY
OF
201
GRAVITATION
Thence, (dn) cos 0 + n sin 0 d# = 0.
cm
Since 72E 1, this yields for the total angle of deflection 6 of the light ray in passing the sun (21)
in agreement with the deflection at the speed of light. 5. THE EQUIVALENCE ON THE LINEARITY
found in (I, 99) for a particle
PRINCIPLE-EXPERIMENTAL OF THE THEORY
In Part I we found that the acceleration field depends on its velocity by a0 = g[l + (F/2 -
CHECK OF
GRAVITATION
of a free particle
l)v’/c”]
passing the sun
- 5g*v v/c”.
in a gravitabional (22)
[See Eq. (I, 94).] This prompts the question of whether according to our t.heory the fast-moving nucleons inside a compound nucleus (with velocities ZJ2 0.1 c), and therefore apparently the entire compound nucleus, would not fall faster than a single proton, contrary to the experimentally established fact that the gravitational acceleration of hydrogen and it,s chemical compounds is not different from that of other substances (5). Equation (22),with 5 = 4, however, as a good approximation is also valid in Einstein’s theory of gravitation, for which the equivalence principle holds rigorously, inferring that at a given point and time the gravitational acceleration must be the same for all objects6 In Einstein’s theory, therefore, the acceleration of particles bound together by nongravitational forces is not. simply equal to the sum of (22) and their accelerat.ion due to t.hose binding forces alone, but the effect of the latter forces apparently is affect,ed by gravit,y in such a manner as to compensate some of the velocit’y dependence of (22). The question then arises as to whether the same holds for the acceleration of bound particles according to our present theory. Only then can we conclude that our theory predicts the same gravitational acceleration for one substance as for another. The general problem posed is therefore the following. Consider the mutual interaction of gravity, particles, and arbitrary fields. The fields shall interact with gravity according to Section 4 of Part I. Now consider a system of particles and fields which in absence of gravity would stay together within a small region,
202
BELINFANTE
AND
SWIHART
in which the gravitational field st,rength may be treated as uniform, with h,, linear functions of position. Next, calculate the acceleration of t.he center of mass of this system when gravity is taken inbo account, not, only in the effect of the direct interaction between gravity and matter and of gravitational influence on the action of the other fields on the particles, but also in the effect of gravity on the amount of such other fields generated by the particles. It should then be proved that, as a function of the velocity v of the center of mass,the acceleration of the latter shall be given by Eq. (22), independent of the velocities of individual parts of the system; and this should hold with the experimentally claimed accuracy, which was at least 1: 10’ for a comparison of the gravitational accelerations of water and of copper (5), and therefore may be expected to be at least 1: lo7 for a comparison of hydrogen and copper. We have not tried to deal with this problem in its full generality and complexity, but in the following section we have merely considered the simple special caseof two charged particles kept together by electromagnetic forces and initially traveling in circles around their common center of gravity at rest. Moreover, we have simplified our calculations by ignoring radiative and magnetic effects except one whir,h is caused directly by the gravitational acceleration we want to find. This oversimplification is, of course, not really justified; we have, however, endeavored to make these simplifications in various parts of the calculation systematically in such a way that the errors may be hoped to cancel each other. A more correct discussion would not work with instantaneous electrostatic interactions, but with the average of retarded and advanced electromagnetic interactions. (Retarded fields alone would not admit stationary orbits of the particles around each other in absence of gravity.) We personally feel confident that a rigorous treatment of these radiative and magnetic effects, which are considerable for individual particle velocities - 0.1 c, will not introduce a dependence of the center-of-mass acceleration on the velocities of individual particles, which we find to be absent in our approximation; but we grant that this belief ought to be substant,iated by explicit calculation. At the sametime, one should then consider elliptical orbits, and not only circular ones as we do in the next section. Finally, the accuracy of our calculations may be barely sufficient, as we shall systematically neglect terms quadratic in the gravitational field, and therefore neglect the product of the acceleration g and the gravitational potential h, while on the earth, due to the sun, h - lo-*. We therefore have not proved that in our theory the equivalence principle would not be violated by terms of the relative order of this h - lo-*. If such effects would appear multiplied by v2/c2, they may be near 1: 10”. Kow, if the existence of such a violation of the equivalence principle in our theory could be proved by more detailed theoretical considerations, and if at the same time Eotvos’s famous experiments could be pushed a few places further, there might be a criterion on the correctness or incorrectness
THEORY
OF
203
GRAVITATION
of the linear theory of gravitation. Such a check would be of utmost fundamental significance, as it would free us from relying on essentially metaphysical arguments in our attitude towards the linear or towards the generally relativistic theory of gravitation. 6. VALIDITY
OF THE
EQUIVALESCE
PRIXCIPLE
IT\’ A SPECIAL
CASE
We shall now discuss the influence of gravity, according to our linear theory, on a pair of opposit’e charges el and e2 (rest massesml and m2) traveling initially in circles around their common center of mass. In absence of gravity, this motion may be described, in Greek letters, by dij/dTj = Vj = W X tj; dti/dTj
= 0;
(Cd*fj) = (Vj’tj) dVj/dTj = -w2tj; fz = -NE1 ;
N = ml/m2 > 0;
(j = 1, 2),
d lt;j12/d7j = 0;
(23a) (23b-c)
= 0;
(23d-e)
( Vj12 = 0’ / & 12.
cm-g)
721
= al -
f;2
= (1 + N)G .
(24a-c)
The actual motions xj(t) differ somewhat from the circular motions
v/c2)[1 + h(1 - 5z?/2c2b,2)], (25)
[see(I, 93)], with a, given by (22), and with h = X-‘/R w h, + x.g/c2,
(2’3
where Ic’ c GM/c2; h, = Y/R, ; and g c c2Vh cz -GMR,/R,3; R is the radius vector from the center of the source of gravitation, and has the value R, for the origin of the (x, t) frame of reference, which we choose at t = 0 in the center of mass of our pair of particles. Clearly g is the ordinary gravitational acceleration. For particle 1, we denote a, v, x, b, , etc., by a1 , vl , x1 , bl , etc., and the force F, on it is given by (8) or FI = elE,(xl) + elvl X B&)/c,
(27)
204
BELINFA?lTE
AND
S\VIH.ZRT
where Et is the electric field generat,ed by ez, etc. This field Ez is calculated from Maxwell’s equations in the gravitaGona1 field, which yield E&4
= -Vv,%(xJ
~1divl Ez(x~) + Ed.
- dAz(xl, t)/cdt,
W9
V1cl = 4-ae&(rz1),
(29)
with r21 = xl - x2 . ITor high-speed circular motion, magnetic and radiative effects may considerably influence the relation between radius and velocity of the rotational motion of the charges around each other; but we know that, in the absence of gravitation, a circular mot.ion of the charges can also be described crudely by considering merely an electrostatic interaction by e1E2’(.5) = elez .51 /~zI/-~, if we are satisfied with replacing the false relation
= 0,
B2’@1)
the actual dependence
between
(30) w and /
with bj = (1 - w21~jl’/C’)1’z.
@lb)
For the sake of simplicity, we want to take a similar point of view in discussing the actual motion in the gravitational field. We therefore neglect the v X B term in (27); but, then, for consistency, one should also neglect effects of retardation, and the terms with Fv2/c2b02 and with F-v v/c’ in Eq. (25). This equation then, with use of (26), simplifies to a = a, i-
born-lF[l + (1 - s)(ho +
gWc’)l.
(32)
Below, as a check on the “consistency” of such a treatment, we will compare our results for the case 5 = 4 with those obtained from Einst.ein’s theory in its linear approximation” [compare (I, 7) and (I, 26a)] by there making simplifications similar to (30)-(32). In order to achieve consistency, then, it turns out to be necessary in our present theory not to ignore completely the term - aA2(xr,t)/ cat in (28), but to take int,o account one contribution to (28), as might well have been expected anyhow. Let the vertical acceleration of the plane of circulation of particle 2 be given by b,g, where b?is a numerical coefficient; then, our main 10 This actually, by (I, 7), corresponds to a case with b # 0 # Q of our theory. In fact, our theory can easily be generalized to include such a case by dropping the “twofold isoit by the single isotropy condition (I, 56) tropy condition” b = q = 0 (I, 57) and replacing with K’ replaced by K. This leads to the same final results as obtained above. Then, Einstein’s theory in its linear approximation corresponds to 5 = 4, K = 0. In this approximation, though, Einstein’s theory corresponds to ‘LX = 5 for the advance of the perihelion. [Compare Eq. (I, 124), and reference 18 of Part I corrected as indicated there.]
THEORY
OF
205
GRAVITATIOS
problem is to solve for bz . Therefore, we should not neglect the fact that at a time t the velocity of this plane (and t’herefore part of the velocity of particle 2 itself) has increased by an amount bgt. Neglecting retardation, and “NEQITGF” (= Neglecting Effects Quadratic In The Gravitat’ional Yield), we conclude that A?(xi,t) = ezbgt/crdl ; so M -e2b2g/c2r21 .
--dilp(Xl,t)/Cdf
(33)
We insert (28) with (33) in (29), and for E in (29) we insert (l&j NEQITGF we thus get -[l + ?5s@o + g+4c2mwx1j We multiply
with
(edk/c”)g . VI( 1/rel) - (S/2c2)g. V1~z(xI) = &rez&(rel).
(26).
(34
t’his by [1 - >/25ho - 5g. (xi + x9)/4c2 + beg.rZ,/2c2], and put p2(x1) = [l + (5 + 202)g.r~1/4c4]~.e(x1).
(35)
Thus we get -v&&4
+ (Wjg
. hd~~)
= -I*eJl These equations
(33)
- !;iS(h,
obviously
+ g.xZ/c2)]&(r2J
are solved, NEQITGF,
cp~(xd = [I - liS& %(x1) where he = electrostatic”
I
Fi x eiE”(x1) z ele? lro1j-3rr21[1 -(5
. Vl(l/rzl).
by (37)
+ g.x2/i2)]e2/r21,
= [l - fy3hz -
= h, + g.xJ?. approsimation
+ (e&/c’)g
From
(5 + 20Jg.r21/4c2]e2/rZ1 ,
(38)
(28), (X3), (X3), t,hen, in our LLalmost
i$%hz
+ 26s)g.rzl/46’]
+ eleZ /r?1/P’(5 - 2bn)g/d$.
(39)
We now must solve the equation of motion (32) with (22) and (39) substituted in it for a, and for F. A method of solving this equation for t,he motion xl(t) of particle 1 is suggested by the following treatment of t,he analogous problem in Einstein’s theoryof gravitation, executed “in linear approximation” NEQITGF.” In Einstein’s theory, the simplest treatment, utilizes a local, freely falling, frame of reference’l (t, ~1, accelerated with respect, t,o the system lx, t 1 in which the center of the source of gravitation is at rest, and connected with it by such a nonlinear coordinate transformation that the corresponding transformation of tensors relates the Lorentz metric g,, of the freely falling (T, 7) system locally to the grV met’ric which describes the actual Static Spherically Symmetric exI1 The
locsl
time
7 in this
section
is unrelated
to the time
paxmeter
7 uSed in Part
1.
206
BELIKFANTE
AND
SWIHART
ternal gravitational field in the (x, t} system. The latter metric, for maximum analogy t’o our present linear theory, can be given in the “isot,ropic” form postulated in Eqs. (I, 54) and (I, 55), which made h,, E gUV- g,, take diagonal form with hn = hzz = h33. In the “linear” approximation to Einstein’s theory, h,, then takes the value” hm = hll = ha* = h33 x z/%/r x Zh, with h given by Eq. (I, 83), or approximately by Eq. (26) in the considered here. Notice that (40) is the value predicted for h,, by (I, 55), (I, 58), (I, 60) if we insert 5 = 4, [ = $5, K’ z K = 0 (I, 7) and (I, 26). A coordinate transformation performing the job of locally g,J&r) into this g,,Jx,t), with the required accuracy NEQITGF, is x = r[l - h, -
(40) small region Eqs. (I, 44), according to transforming given by
g*t/c*1 + % g( ~~l”/c”+ 2),
(W
t = 7[1 + h, + gQ$/c2].
(410
In the {h, r] system, then, the laws of physics according to the equivalence principle locally take a form as if no gravity would exist. For instance, a free particle would move uniformly wiyith regard to the freely falling system (f;, 7) according to ( = VT. By (41) this is transformed,
NEQITGF,
x = vt + >$a,
(42)
into the accelerated motion t2,
v = V(1 - Zh,),
(43a-b)
with respect to the frame of reference fx, tj fixed on the earth.13 In (43a), a, is the velocity-dependent acceleration given by (22) with 5 = 4. We then can obtain the falling motion of our two circling charges with respect, to the fixed (x, t) system, by transforming by means of (41) the uniform circular motion (23)-(24) which must prevail in the {t, T) system. The latter motion we describe in the electrostatic approximation, in which (30) is the electric centripetal force on particle 1 in the {h, T} system, yielding the relation (31) between angular velocity w and orbital radii ]r;,l and j&l. With this motion, the origin of the {& T) system represents the center of mass of the rotator. By (41), this origin has at t = 0 the velocity dx/dt = 0 in the fixed fx, tf system, so that we are here treating merely the case of a rotator which in the (x, t] system starts to fall with initial velocity zero. Treatment of a rotator with an initial velocity would require replacement. of (41) by more complicated formulas. 12See, for instance, 13Compare footnote
C. M#ller (6) where nz is written for our k’ = k. 18 of Part I. We neglect here the effects of the earth’s
daily
rotation.
THEORY
OF
207
GRAVITATIOX
The instantaneous acceleration of the rotator is now found by first calculating the accelerations al = d’x,/dt~ and a2 = d2xz/dt2’ of the two charges, and then averaging each over one period of rotation and taking the results instantaneously at the time t1 = tt = T = 0 at which the fixed and the freely falling local frame were related by (41). The result of this calculation is that one finds g, and not a, , for these instantaneous average values of al and a2, showing the fact mentioned earlier that according to the equivalence principle the initial acceleration of the entire rotator has the same value g as the initial acceleration of a single particle, irrespective of the internal motion going on inside the rotator. In this treatment of Einstein’s theory, by the general relativity and equivalence principles, the electric field E with respect to the (x, t) laboratory system in our electrostatic approximation follows from the field (30) which we supposed to prevail with respect to the local (c, 7) system. For later comparison it is useful to write down the expression resulting for E. We derive it from (30) by means of the tensor transformation which corresponds to the coordinate transformation (41). This easily yields WC)
= ez /b~l-~[h
- El(g. h/c’)
+ g(G. h/c”)].
(44)
Now, for one common value of t, the difference between the 7 values for the two part’icles according to (41) is linear in the gravitational field. Therefore, in rzl = XI - xz calculated from (41x) we may neglect 71 - ~2 . Thence we derive, using (24a-b), and NEQITGF, /r2J%
= (rzl .r2J3”r21 = /k211-3{r21[1 + f&
so that’ (44) may also be written Ez(xl)
= e2 jrzl]-3rzl[l
-
+
g.c.5
+
r;2)/2c”l
+
g (p$
-
[~2~“>/2cZ},
(45)
as
2h, -
2g.x2/c2
-
3g.rz1/2c2] + e,g/2c2)rzll.
(46)
We now return to our present problem of solving Eqs. (32) for particles 1 and 2 with a0 given by (22) and with the forces given by Eq. (39) and a similar one for Fz . Here, big and beg represent the vertical accelerations of the two charges as far as due to the fall of the rotator. The difference with the Einsteinian case is that there we knew bl = bz = 1 a priori from the equivalence principle applied to our rotator of vanishing initial velocity; but in our present case we shall have to solve for the unknown bl and bt by solution of the equations of motion (32). For 5 = 4 and bl = bz = 1, we may expect our formulas to become identical with those of the Einsteinian case linearized by NEQITGF. We see already that in this case Eq. (39) amounts to el X Eq. (46). This is why at least for this case of 5 = 4 the approximation (39) for the electric field-in which by (33) we took into account the electromagnetic induction term at particle 1 only in as far as
208
BELIXFANTE
AND
SWIHART
caused by gravitational acceleration of particle 2-is the proper approximation when the point of view is taken that the static field (30) prevails in the local system {c, T) . It is then reasonable to assume that the approximation (39) for the field is “consistent” with the “electrostatic” point of view taken by us, even if 5 # 4. In the Einsteinian case, NEQITGF, the equations of motion (32) with (22), and with (46) [or with (39) for 5 = 4, bl = bp = 11, were solved by obtaining the dependence of xi and of x2 on t which followed from the circular motion (23)-(24) with (30)-(31) in the local system, by means of the coordinate transformation (41) from (r, T) to (x, t) . Similarly, we shall try to solve the equation of motion (32), with (22) and with (39), even in the case of 5 # 4 and bl and b2 as yet unknown, by finding a dependence of x1 and of x2 on t from the circular motion (23)-(24) with (30)-(31) by some coordinate transformation from (F, 7) to (x, t} . This new coordinate transformation is a generalization of (41), which may be devoid of the a priori physical meaning of this transformation in the Einsteinian case,14 but which yet is a convenient mathematical trick for solving our equations of motion. As a trial solution we choose for this transformation =
Xj
tj[l
- a(& +
g’rj/C2)]
+ >‘$Kg
/~j12/C2
+
f/s@jg
Tj2,
t = Tj[l + T(h0 + g’F;j/C’)J,
(47Xj)
C47tj)
where the arbitrary numerical coefficients a, y, K, p1 , & are to be determined in such a way that (23)-(24) with (30)-(31) transformed by (47) indeed will solve our equations of motion. From the Einsteinian case we know already [see Eqs. (41)] that for 5 = 4 the transformation (47) gives a solution for cx = y = K = p1 = p2 = bl = b2 = 1. We shall first investigate what conditions we should impose upon these constants in order to ensure that the orbit of particle 2 shall have indeed the vertical accelerat.ion bzg assumed already in the derivation of (33). With hj = ho + g.Xj/'C" E h, + g.&/c2, and by (23), we find from (47) the velocity of particle j : Vj(t)
=
[dt,/dTj]-l(dXj,/dTj]
= [I + rhj + yTjg-Vj/C’]-‘{ = Vj[l
-
((Y + r)hj
(1 - ahj)Vj
- yTjg.Vj/Z]
- atjg.Vj/cS
- cjag.Vj/c’
+ Tj@jgj
(48j)
+ gpj Tj ;
14 The main difference with Einstein’s theory is that there 5 and 7 have a physical meaning as coordinates in a local freely falling frame of reference, while in our theory c and + are meaningless auxiliary variables used in making a one-to-one correspondence between the actual helical motions x,(t) and thought circular motions C~(T).
THEORY
thence,
OF
209
GRAVITATIOK
NEQITGF, we obtain for its acceleration =
aj(t)
(dt/dTj]-‘(dVj(t)/dTj)
= -
-
+ Vj[-2(a
(CX+ 2r)hj + Y)g’Vj/c*
- CXg’fj/C’
- 2Trjg.Vj/C’]
(49j>
+ YTj~2getj/C41 + gPj.
Just as in the Einsteinian case, so also in our theory, the transformation solves the equations of motion only “locally,” that is, at 7 RS t z 0. So we insert this in (49) and find aj
=
W’[l -
-tj
(a + 27)&
-
(201 + 2r)
g’fj/C2] (5Oj)
- Vj (2a
+
27)g.Vj/C2
+
g@j
.
In order to separate the vertical acceleration of the entire orbital planes from the rotation itseIf, we average (501) over one period. The terms linear in tj then vanish. The average of the terms quadratic in tj or Vj is (a + +Y>(g/C”){~’ ltj12 - IVjl”}
= 0
(51)
by (23g). Thus only the term g@j in (5Oj) represents the acceleration of particle j’s orbital plane. For j = 2, this is just the acceleration which we called b,g above Eq. (33). Similarly, for j = I, it is i&g. Therefore, in (47) for consistency we must choose /3j
=
bj
.
(52)
By (5Oj), me expressed aj in terms of the “unperturbed” variables Vj and rj . Therefore, let us do the same with the electrostatic force Fi calculated in (39). First, from (47x1,2) with g(q - 7~) cz 0, we obtain
Irn lM3rz1= (r2r. rfl)-3’121 = (t21(-~&?I[l + 2&, + (h! - 3X)( 1 - N) g*i$k2]
(53)
+ Il;zrj-I[l + N]-‘(1 - N) Kg/2c”, * where we used (24a-b). Inserting this in Eq. (39), and using (24a-b), (47x1 $, and (52), we find, NEQITGF, FI = ele2/&I[-~CZI [1 i- (2~ - %@L +
(&I
-
3K
-
$53
-
+ ele21tl (-‘(g/Zc*)[(l
b2)g't,/2C2
+
(4X
-
3K -
35s
+
b,)g+/2C']
(54)
+ N)--‘(I - N)K + %S - b,].
Our next task is to determine the values of the constants (Y,y, K, bl , b2for which
210
BELINFANTE
AND
SWIHART
the trial solution obtained from (23)-(24) by (471) for particle 1 will solve its equation of motion (32). Therefore, in the left-hand member of (32) we insert (501) for al . In its right-hand member, we insert for a,, the expression (22), and for F the expression (54). The symbol 6, occurring in Eqs. (25)-(32) represented for the j”” particle the quantity (1 - ~j2/c2)1’2 [see (I, SS)]; but since in our electrostatic approximation we have already replaced b, by 1 when multiplied by F [see above Eq. (32)], it will certainly suffice to replace b, in such terms by bj E (1 Instead
vj2,‘c2)“2 = (1 - w2 l~jj2,‘&1’z.
of at once using Eq. (31a), let us temporarily WI2 3 -ee.bm-’
Ih/-’
put (55)
lt~-“.
Further, we may put x w t and v = V wherever they appear multiplied by g. Thus we obtain for the right-hand member of (32), by (24a-b) and (23g), - &w;[l
+ (1 + 2cu - 35/2)/z, + (2a -
3~/2
- 5/4
+ (1 + 2a - 3~,‘2 - 55/4
- b2/2)g.c,/c2
+ bz/2)g. ~2/c2] - VIFg. V&” - 62+
+ g( 1 + ( ~~1~*/2C2)[u2(5 - 2) - w12(3/2
K)
-
(56)
- bz-
dN(3/2
K)]).
This expression then should be equal to (501). Sufficient for this is the separate equality of the coefficients of & , of &ho , of &g. &/c”, of &g *&/c*, of Vlg. V1/c2, of g, of g1t112/2c2,and of gJa,1*w1*N/2c2.This gives eight equations to be satisfied by the constants cr, y, K, bl , 62, and w. The first equation gives W2
= w&
(57)
and thus gives us (31a). Using this result in the other equations for eliminating w entirely, we further obtain for particle 1 the seven relations 3fx + 2-i
- b2/2
40 -I- 2-y -
3~/2
2l.Y
3~/2 + bJ2
-
2cY-I- 2-f
= 33/2
- 1,
(5W
= 53/4
- 1,
(58bJ
=
F/4,
=
$7
(5th) Wd)
bl = 1, K
-
ii
+bz
b2
=
5/2
=
T/2.
(58ed -
2,
(5%) wkl)
Similarly treating the equation of motion of particle 2 one finds additional equations with bl and b2 interchanged.
THEORY
All these equations
OF
(58) are compatible
N = K = s/a -
1;
211
C:RAVIT.~TION
and are mliquely
solved by
y = bl = bz = 1.
(59)
Alost, important to us are the results (Se,) and (S8e2); therefore it is useful to remark that these equations arc nwcssar!/ conditions, since in (50) and (56) the terms with the vertical vector g as a factor but not depending on position necessarily must he qua1 separately. Thus, by (Z-(24), (:quations (53) and (.54) then also yield the ISinsteinian results (45) and (44). Soticr that here, for bound part)icles, just as in Eq. (22) for free particles, from the three constants of our theory only the constant 3 is of any influence on the result. This may at first sight seem surprising, as one might have feared an illfluencseof the cwnst,ant,K on the equivalence principle, since K is the coeffiricnt of an intrrac;ion which seemst,o be absent in Einstein’s theory. As explained in Swtion :5of I’art I, however, these terms with K are in first approsimat.ion simply a special \vay of wit)ing an int)erac%on with the tracaeof t,he energy-density tensor; in sewnd approsimatioll, they were a way of smuggling into t.he theory interacationswhicahwould not have been linear if cspressed by means of the energydensity tellsor. The amount of this interaction then was regulated by t,he “Kepler (wldition” (I, 79) or (I, 80), which we used in Section 9 of Part I in deriving the ecIuation of motion (I, 93) which here, as Eq. (25) with (Z), was the starting point of our derivation of t’he validity of the equivalencaeprinciple. This then may explain why our swond type of interac+tion appearing wit,h a coefficient, K in the
Lagrangian does not, lead to c’onflic+s with the ~v~ui~dr~lce prinviplc, K it,self does riots seem to appear ill our results.
alit1 \vhy
The authors we indchted to the Sational S~ic~nw I~o~~~rtlation for fin;lnci:ll these investigations during t,he ye:m I XC-1 956.
suplmrt
RECEIVED:
Dccemkr
of
17, l!IN.
I. F. J. BELINFANTE ALYD J. C’. SWHART, Annuls of Physics 1, 168 (1957). 2. M. MOSHINSKY, Phys. Z&xl. 80, 514 (1950). 3. F. J. BELINYAYIT, Physicn 7,119 (1940). 4. F. J. BELISFANTE, Phys. Rev. 74, $79 (19-M). 5. R. v. E:iiwiis, D. PEB.
(1955). 6. C’. M@LLER, “The Theory of Relativity,” mcl Sew York 1952.
11. 327, l+lq. (86). Osfortl Univ. l’rws,
I,ondon
OF
PHYSICS:
1,
196-212
(1957)
Phenomenological
Linear
Part II. Interaction F. J. Department
of Gravitation’
with the Maxwell
BELINFANTE
of Physics,
Theory
AND Purdue
J. C.
University,
Field’
SWIHART” Lafayette,
Indiana
The theory of Part I is applied to the interaction of gravity, matter, and electromagnetism. Maxwell’s equations are valid with an apparent electric and magnetic susceptibility of space due to the gravitational field. One effect of this susceptibility is a refraction of light rays by a gravitational field, from which the bending of light rays passing the sun can be derived. The result is the same as found in Part I for the deviation of particles traveling at the speed of light. The equivalence principle is shown to be valid in our linear theory as in Einstein’s theory, notwithstanding the velocity dependence of the gravitational acceleration of free particles. For bound particles, the dependence of the gravitational forces on the relative velocities of parts of a system with respect to the system’s center of gravity is neutralized by gravitational effects on the binding forces. We have shown this only for a special case, and the calculation is performed only in electrostatic approximation. As the errors thus introduced cancel in Einstein’s theory, it is assumed that they cancel also here. The parallelism with the corresponding derivations in Einstein’s theory are apparent at each step of the calculation, and remains, with altered numerical coefficients in the formulas, for any values assigned to the constants of our linear theory of gravitation. It is, however, possible that in higher approximations our linear theory would violate the equivalence principle. This question is worth further theoretical investigation, and, if such effects theoretically would follow from our theory, EiitvGs’s experiments on the equivalence principle should be pushed from 10-S to IO-” in order to discover such effects and to make an experimental decision between the linear and the generalrelativistic theories of gravit,ation. 1. INTRODUCTION
In our preceding paper (1)4 we developed a linear theory of gravitation and its interaction with other fields. We shall now apply these methods to the interaction of this field of gravity with the Maxwell field and with charged particles 1 This work was supported in part by the National Science Foundation. 2 Based in part on chapter 3 of a thesis by J. C. Swihart submitted in partial of the requirements for a Ph.D. degree at Purdue University (1955). 3 Now at International Business Machines Corporation, Poughkeepsie, New 4 We call this paper “Part I,” and refer to its formulas by (I, . . .). 196
fulfillment York.
THEORY
OF
GRAVITATION
197
(Sections 2 and 3). The gravitational field has the effect of an apparent electric and magnetic susceptibility of space. Thus the field around the sun acts like a lens, which mill bend electromagnetic waves passing the sun (Section 4).5 The bending of such light rays is found to be equal t#o the deflection of photons calculated in Part I (1). We then tackle the problem of whether the principle of equivalence of heavy and inert mass is valid according to our theory (Sections 5 and 6). In view of the velocity dependence of the gravitational acceleration found in Part I, the question arises as to whether the fast motion of nucleons inside compound nuclei does not affect the gravitational acceleration of such nuclei as compared to the hydrogen nucleus. (This problem exists in Einstein’s theory as well as in our linear theory of gravitation.) As the problem it,self is too complicated for a rigorous t.reat,ment,6 we investigate the principle involved by considering a simple model, consisting of two oppositely charged particles, possibly with different masses, rapidly rotating in circles around their center of mass which is just starting to fall. We take into account gravitational effects both on the generaCon of the electric field by the one particle, and on the accelerating effect of this field on the other particle. For simplifying the calculation, we have inexcusably ignored retardational (and corresponding radiative and magnetic) effects except those directly due to the gravitational acceleration of our two-particle system. It is then shown that, in this approximation, neglecting also terms quadratic in the gravitational field, t’he starting acceleration of our t’wo-particle system is simply g, and not the larger value which would correspond to the velocities of t,he individual particles. [This g was the gravitational acceleration for a particle at rest (I).] This makes us confident that t,he principle of equivalence is not violated by our theory; at least not in the approximation considered in this paper, neglecting effects of the order of lo-*. In Section 5, we discuss the possibility of an experimental test between the linear and the general-relativist’ic theories of gravitation, by considering second-order gravitational effects. 2. THE
LAGRANGIBK
For the electromagnetic field, we use a first-order Lagrangian (-3, 4) so that the field strengths H,, = -HP, are to be submitted to variations independent of the variations of the potentials A, . (We shall find HIuy # d,A, - &A, anyhow, due t,o gravit,ational effect’s.) Here we do not consider ‘(Fermi” terms’ in 5 The methods Itsed are similar t,o those used by Moshinsky (2) for calculating the bending of light rays according to Birkhoff’s theory. 6 In Einstein’s theory, the problem finds its general solution in the fact that in a freely falling local frame of reference the center of mass of a compound nucleus as well as of a hydrogen nucleus must have zero acceleration when both are initially at rest in such a coordinate q-stem. Since in our linear theory of gravitation the general relativity principle does not hold, this simplification of the problem does not exist in our theory. 7 Terms which vanish on account, of the Lorentz condition.
198
BELINFANTE
the Lagrangian, So, we use
which
AND
SWIHART
are not needed as long as we do not quantize L, = (4?r)-‘[~H”“H,,
the fields.
- fm”A,]
(1)
as the electromagnetic Lagrangian density. To this Lagrangian are to be added the gravitational, matter, and interaction Lagrangians of Part I, in particular the ones described by Eqs. (I, 2), (I, 1 l), and (I, 23) with (I, 24a-b) and (I, 21). The only difference is that we shall simplify (I, 2) at once in accordance with (I, 25) and (I, 26), and that the kinetic and total momentum of the jth particle are no longer equal, but are now related by the definition *ip E Ph where pi,, and Xj, are functions
k%/hUXi),
of the parameter
3. FUNDAMENTAL
(2) aj along the world line.
EQUATIONS
In our Lagrangian, the functions aj’(sj), pj,(sj), Xj“(sj), &(x, t) = h,(x, t), A,,(x, t), and H”(x, t) = -H”“(x, t) are now to be variedindependently. We shall use the notation of Part I, dropping again from time to time the index j. Variation of a”(s) and of p#(s) gives Eqs. (I, 29) and (I, 30) and therefore (I, 31) without change. Variation of X’(s) yields this time* dp,/ds where
vprr
= ~a” V,A”(X)
=
+ $$x? V,[s%,,(X)]
- [E/clV,A”(X).
dA,(X)/ds
d?r,/dT
= [E/c]Bp”(X)o”
Here, we introduced
(3)
By
= [ V”A,(X)I
and by Eq. (I, 30), Eq. (3) with invariant” form
+ Kmc’b V&““(X),
dX”/ds
= (u”/s’)[ V”A,(X)I
the help of (2) can be written
+ >&s’ap?r“ D,,&,(X)
(4) in the “gauge-
+ Kmc’s’b V&““(X).
(5)
the abbreviation B,, = &A”
which satisfies the differential
- d”A, ,
(6)
equation
ahBllv+ a,& + a,B, = 0.
(7)
Note that Eq. (5) differs from Eq. (I, 32) for drP/dT only by the first term on the right in Eq. (5), which we therefore interpret as the electromagnetic “New* When not printing the index j, B = -e, if e = 4.8 X lo-lo statcoulomb.
we shall
write
B for
ej.
For
instance,
for
electrons,
THEORY
tonian force” force” f, by9
OF
1YY
GRAVITATION
F, on the charge & related to the electromagnetic F, = ffib ds/dT
Consequently, from (5) follows Variation of h,,” yields (c4/8?rGs) [ 0 h’” - (g”” q h:]
= [~/c]Bp”v”.
an equation
+ Cj[J/4c(a”?r”
“Minkowskian (8)
of motion of the type of (I, 93).
+ a”?r”) + Kbmc2g”“]s’6&
+ (8*)-‘[HX”Hx’
-
$~g”“HPUH,,]
-
X[t])
(9)
= 0,
with the particle quantities understood to be taken at the particle time t, so at s[t] in the notation of (I, 19). Finally, variation of A, , and of HP” = -H”‘, yields a”Hp” = 47rj’,
(10)
B,” = H,,” - >~H,“h~’ + H,xh,X - H”hh,‘.
(11)
Here we put j’(X,
t) 3 Cj
(C/C)
V”[t]
63(X
- X[t])
(12)
and we used Eq. (I, 31), while v[t] means v(s[t]). Equations (7) and (10) are Maxwell’s equations, with Blo = E, , B23 = B, , HI0 = @’ = D, , Hz, = Hz3 = H, , J’ = p, j’ = i./c, etc. Equation (11) then describes the dielectric and magnetic properties of a gravitational field. For the S(pherically) S(ymmetric) S(tatic) gravitational field with isotropic tensor ellipsoid discussedin Part 1, with ho0= -h: = x, hb" = -ho" = 0, and hkl = S,& Eq. (11) simplifies to B = HI1 + $6 (x + +>I,
E = IN1 - 3$(x + $>I.
(13a-b)
Hence, such a field behaves as a medium with dielectric constant c and permeability p given by
e = 11 - 4$(x + 4>1-’ =P = 1 +%(x+4) e m /.i x 1 + 5A~‘/2r = 1 + >$Fh,
>
(14a-b) (14c)
where we used (I, 50) and (I, 83a). Consequently, the Coulomb potential at a small distance T’ from a point charge Ze placed in such a gravitational field is given by + = Ze/tr’ 9 This
f,, obviously
is a fourvector;
= [I - f$(x + 4)]Ze/r’, not
F, or 0’.
(15)
200
BELINFANTE
and the index of refraction
AND
of the gravitational
n 1 (qp 4. BESDING
OF
SV’IHART
LIGHT
field is given by
E 1 + !i(x
RAYS
IN
THP:
+ 4).
SUN’S
(16)
GRAVITATIOSAL
FIELD
Equations (16) and (I, 50) tell us that spheres of radius r around thf sun’s center will be spheres of constant index of refraction n z
1 + FB’/2r
x
1 + 5GM/2c’r
> 1
(17)
outside the sun. Thence, t.he region around the sutl acts as a converging sphericai lens. The smallnessof (n - 1) enables us to calculate the deflection of a light ray passing through this lens in the following simple way. Let R be the distance of nearest approach (at 0) to hhe sun’s center M. (See Fig. 1.) At any given point P of the ray, the actual distance to M is r = R set 0, and the distance braveled from 0 is s = R tan 0( >< 0). As the ray travels another short distance PQ = ds = R se? 0~3, the distance from M increases by dr = R set’ 0 sin %de,
(18)
passing into a region with an index of refraction increased by dn = -f
i5k’r-”
dr = -,1 &WR-’
sin 3 d0.
(19)
Consider the gravitational “lens” to be made up out of infinitely many infinitely thin concentric layers of thickness dr each with its constant indes of refraction n(r), so that the ray will be refracted according to Snell’s law when passing from one such a layer into the next, Let d+ be the resulting deflection of the ray in one such a refraction with angle of incidence = ($$r - 0) and angle of refraction = ($+$r - 0 + d$) ; SO (n + dn) sin ($7
FIG. 1. Refraction
of light
- 0 + d$) = n sin ($5~ - 0).
ray
in gravitational
field
near
sun.
THEORY
OF
201
GRAVITATION
Thence, (dn) cos 0 + n sin 0 d# = 0.
cm
Since 72E 1, this yields for the total angle of deflection 6 of the light ray in passing the sun (21)
in agreement with the deflection at the speed of light. 5. THE EQUIVALENCE ON THE LINEARITY
found in (I, 99) for a particle
PRINCIPLE-EXPERIMENTAL OF THE THEORY
In Part I we found that the acceleration field depends on its velocity by a0 = g[l + (F/2 -
CHECK OF
GRAVITATION
of a free particle
l)v’/c”]
passing the sun
- 5g*v v/c”.
in a gravitabional (22)
[See Eq. (I, 94).] This prompts the question of whether according to our t.heory the fast-moving nucleons inside a compound nucleus (with velocities ZJ2 0.1 c), and therefore apparently the entire compound nucleus, would not fall faster than a single proton, contrary to the experimentally established fact that the gravitational acceleration of hydrogen and it,s chemical compounds is not different from that of other substances (5). Equation (22),with 5 = 4, however, as a good approximation is also valid in Einstein’s theory of gravitation, for which the equivalence principle holds rigorously, inferring that at a given point and time the gravitational acceleration must be the same for all objects6 In Einstein’s theory, therefore, the acceleration of particles bound together by nongravitational forces is not. simply equal to the sum of (22) and their accelerat.ion due to t.hose binding forces alone, but the effect of the latter forces apparently is affect,ed by gravit,y in such a manner as to compensate some of the velocit’y dependence of (22). The question then arises as to whether the same holds for the acceleration of bound particles according to our present theory. Only then can we conclude that our theory predicts the same gravitational acceleration for one substance as for another. The general problem posed is therefore the following. Consider the mutual interaction of gravity, particles, and arbitrary fields. The fields shall interact with gravity according to Section 4 of Part I. Now consider a system of particles and fields which in absence of gravity would stay together within a small region,
202
BELINFANTE
AND
SWIHART
in which the gravitational field st,rength may be treated as uniform, with h,, linear functions of position. Next, calculate the acceleration of t.he center of mass of this system when gravity is taken inbo account, not, only in the effect of the direct interaction between gravity and matter and of gravitational influence on the action of the other fields on the particles, but also in the effect of gravity on the amount of such other fields generated by the particles. It should then be proved that, as a function of the velocity v of the center of mass,the acceleration of the latter shall be given by Eq. (22), independent of the velocities of individual parts of the system; and this should hold with the experimentally claimed accuracy, which was at least 1: 10’ for a comparison of the gravitational accelerations of water and of copper (5), and therefore may be expected to be at least 1: lo7 for a comparison of hydrogen and copper. We have not tried to deal with this problem in its full generality and complexity, but in the following section we have merely considered the simple special caseof two charged particles kept together by electromagnetic forces and initially traveling in circles around their common center of gravity at rest. Moreover, we have simplified our calculations by ignoring radiative and magnetic effects except one whir,h is caused directly by the gravitational acceleration we want to find. This oversimplification is, of course, not really justified; we have, however, endeavored to make these simplifications in various parts of the calculation systematically in such a way that the errors may be hoped to cancel each other. A more correct discussion would not work with instantaneous electrostatic interactions, but with the average of retarded and advanced electromagnetic interactions. (Retarded fields alone would not admit stationary orbits of the particles around each other in absence of gravity.) We personally feel confident that a rigorous treatment of these radiative and magnetic effects, which are considerable for individual particle velocities - 0.1 c, will not introduce a dependence of the center-of-mass acceleration on the velocities of individual particles, which we find to be absent in our approximation; but we grant that this belief ought to be substant,iated by explicit calculation. At the sametime, one should then consider elliptical orbits, and not only circular ones as we do in the next section. Finally, the accuracy of our calculations may be barely sufficient, as we shall systematically neglect terms quadratic in the gravitational field, and therefore neglect the product of the acceleration g and the gravitational potential h, while on the earth, due to the sun, h - lo-*. We therefore have not proved that in our theory the equivalence principle would not be violated by terms of the relative order of this h - lo-*. If such effects would appear multiplied by v2/c2, they may be near 1: 10”. Kow, if the existence of such a violation of the equivalence principle in our theory could be proved by more detailed theoretical considerations, and if at the same time Eotvos’s famous experiments could be pushed a few places further, there might be a criterion on the correctness or incorrectness
THEORY
OF
203
GRAVITATION
of the linear theory of gravitation. Such a check would be of utmost fundamental significance, as it would free us from relying on essentially metaphysical arguments in our attitude towards the linear or towards the generally relativistic theory of gravitation. 6. VALIDITY
OF THE
EQUIVALESCE
PRIXCIPLE
IT\’ A SPECIAL
CASE
We shall now discuss the influence of gravity, according to our linear theory, on a pair of opposit’e charges el and e2 (rest massesml and m2) traveling initially in circles around their common center of mass. In absence of gravity, this motion may be described, in Greek letters, by dij/dTj = Vj = W X tj; dti/dTj
= 0;
(Cd*fj) = (Vj’tj) dVj/dTj = -w2tj; fz = -NE1 ;
N = ml/m2 > 0;
(j = 1, 2),
d lt;j12/d7j = 0;
(23a) (23b-c)
= 0;
(23d-e)
( Vj12 = 0’ / & 12.
cm-g)
721
= al -
f;2
= (1 + N)G .
(24a-c)
The actual motions xj(t) differ somewhat from the circular motions
v/c2)[1 + h(1 - 5z?/2c2b,2)], (25)
[see(I, 93)], with a, given by (22), and with h = X-‘/R w h, + x.g/c2,
(2’3
where Ic’ c GM/c2; h, = Y/R, ; and g c c2Vh cz -GMR,/R,3; R is the radius vector from the center of the source of gravitation, and has the value R, for the origin of the (x, t) frame of reference, which we choose at t = 0 in the center of mass of our pair of particles. Clearly g is the ordinary gravitational acceleration. For particle 1, we denote a, v, x, b, , etc., by a1 , vl , x1 , bl , etc., and the force F, on it is given by (8) or FI = elE,(xl) + elvl X B&)/c,
(27)
204
BELINFA?lTE
AND
S\VIH.ZRT
where Et is the electric field generat,ed by ez, etc. This field Ez is calculated from Maxwell’s equations in the gravitaGona1 field, which yield E&4
= -Vv,%(xJ
~1divl Ez(x~) + Ed.
- dAz(xl, t)/cdt,
W9
V1cl = 4-ae&(rz1),
(29)
with r21 = xl - x2 . ITor high-speed circular motion, magnetic and radiative effects may considerably influence the relation between radius and velocity of the rotational motion of the charges around each other; but we know that, in the absence of gravitation, a circular mot.ion of the charges can also be described crudely by considering merely an electrostatic interaction by e1E2’(.5) = elez .51 /~zI/-~, if we are satisfied with replacing the false relation
= 0,
B2’@1)
the actual dependence
between
(30) w and /
with bj = (1 - w21~jl’/C’)1’z.
@lb)
For the sake of simplicity, we want to take a similar point of view in discussing the actual motion in the gravitational field. We therefore neglect the v X B term in (27); but, then, for consistency, one should also neglect effects of retardation, and the terms with Fv2/c2b02 and with F-v v/c’ in Eq. (25). This equation then, with use of (26), simplifies to a = a, i-
born-lF[l + (1 - s)(ho +
gWc’)l.
(32)
Below, as a check on the “consistency” of such a treatment, we will compare our results for the case 5 = 4 with those obtained from Einst.ein’s theory in its linear approximation” [compare (I, 7) and (I, 26a)] by there making simplifications similar to (30)-(32). In order to achieve consistency, then, it turns out to be necessary in our present theory not to ignore completely the term - aA2(xr,t)/ cat in (28), but to take int,o account one contribution to (28), as might well have been expected anyhow. Let the vertical acceleration of the plane of circulation of particle 2 be given by b,g, where b?is a numerical coefficient; then, our main 10 This actually, by (I, 7), corresponds to a case with b # 0 # Q of our theory. In fact, our theory can easily be generalized to include such a case by dropping the “twofold isoit by the single isotropy condition (I, 56) tropy condition” b = q = 0 (I, 57) and replacing with K’ replaced by K. This leads to the same final results as obtained above. Then, Einstein’s theory in its linear approximation corresponds to 5 = 4, K = 0. In this approximation, though, Einstein’s theory corresponds to ‘LX = 5 for the advance of the perihelion. [Compare Eq. (I, 124), and reference 18 of Part I corrected as indicated there.]
THEORY
OF
205
GRAVITATIOS
problem is to solve for bz . Therefore, we should not neglect the fact that at a time t the velocity of this plane (and t’herefore part of the velocity of particle 2 itself) has increased by an amount bgt. Neglecting retardation, and “NEQITGF” (= Neglecting Effects Quadratic In The Gravitat’ional Yield), we conclude that A?(xi,t) = ezbgt/crdl ; so M -e2b2g/c2r21 .
--dilp(Xl,t)/Cdf
(33)
We insert (28) with (33) in (29), and for E in (29) we insert (l&j NEQITGF we thus get -[l + ?5s@o + g+4c2mwx1j We multiply
with
(edk/c”)g . VI( 1/rel) - (S/2c2)g. V1~z(xI) = &rez&(rel).
(26).
(34
t’his by [1 - >/25ho - 5g. (xi + x9)/4c2 + beg.rZ,/2c2], and put p2(x1) = [l + (5 + 202)g.r~1/4c4]~.e(x1).
(35)
Thus we get -v&&4
+ (Wjg
. hd~~)
= -I*eJl These equations
(33)
- !;iS(h,
obviously
+ g.xZ/c2)]&(r2J
are solved, NEQITGF,
cp~(xd = [I - liS& %(x1) where he = electrostatic”
I
Fi x eiE”(x1) z ele? lro1j-3rr21[1 -(5
. Vl(l/rzl).
by (37)
+ g.x2/i2)]e2/r21,
= [l - fy3hz -
= h, + g.xJ?. approsimation
+ (e&/c’)g
From
(5 + 20Jg.r21/4c2]e2/rZ1 ,
(38)
(28), (X3), (X3), t,hen, in our LLalmost
i$%hz
+ 26s)g.rzl/46’]
+ eleZ /r?1/P’(5 - 2bn)g/d$.
(39)
We now must solve the equation of motion (32) with (22) and (39) substituted in it for a, and for F. A method of solving this equation for t,he motion xl(t) of particle 1 is suggested by the following treatment of t,he analogous problem in Einstein’s theoryof gravitation, executed “in linear approximation” NEQITGF.” In Einstein’s theory, the simplest treatment, utilizes a local, freely falling, frame of reference’l (t, ~1, accelerated with respect, t,o the system lx, t 1 in which the center of the source of gravitation is at rest, and connected with it by such a nonlinear coordinate transformation that the corresponding transformation of tensors relates the Lorentz metric g,, of the freely falling (T, 7) system locally to the grV met’ric which describes the actual Static Spherically Symmetric exI1 The
locsl
time
7 in this
section
is unrelated
to the time
paxmeter
7 uSed in Part
1.
206
BELIKFANTE
AND
SWIHART
ternal gravitational field in the (x, t} system. The latter metric, for maximum analogy t’o our present linear theory, can be given in the “isot,ropic” form postulated in Eqs. (I, 54) and (I, 55), which made h,, E gUV- g,, take diagonal form with hn = hzz = h33. In the “linear” approximation to Einstein’s theory, h,, then takes the value” hm = hll = ha* = h33 x z/%/r x Zh, with h given by Eq. (I, 83), or approximately by Eq. (26) in the considered here. Notice that (40) is the value predicted for h,, by (I, 55), (I, 58), (I, 60) if we insert 5 = 4, [ = $5, K’ z K = 0 (I, 7) and (I, 26). A coordinate transformation performing the job of locally g,J&r) into this g,,Jx,t), with the required accuracy NEQITGF, is x = r[l - h, -
(40) small region Eqs. (I, 44), according to transforming given by
g*t/c*1 + % g( ~~l”/c”+ 2),
(W
t = 7[1 + h, + gQ$/c2].
(410
In the {h, r] system, then, the laws of physics according to the equivalence principle locally take a form as if no gravity would exist. For instance, a free particle would move uniformly wiyith regard to the freely falling system (f;, 7) according to ( = VT. By (41) this is transformed,
NEQITGF,
x = vt + >$a,
(42)
into the accelerated motion t2,
v = V(1 - Zh,),
(43a-b)
with respect to the frame of reference fx, tj fixed on the earth.13 In (43a), a, is the velocity-dependent acceleration given by (22) with 5 = 4. We then can obtain the falling motion of our two circling charges with respect, to the fixed (x, t) system, by transforming by means of (41) the uniform circular motion (23)-(24) which must prevail in the {t, T) system. The latter motion we describe in the electrostatic approximation, in which (30) is the electric centripetal force on particle 1 in the {h, T} system, yielding the relation (31) between angular velocity w and orbital radii ]r;,l and j&l. With this motion, the origin of the {& T) system represents the center of mass of the rotator. By (41), this origin has at t = 0 the velocity dx/dt = 0 in the fixed fx, tf system, so that we are here treating merely the case of a rotator which in the (x, t] system starts to fall with initial velocity zero. Treatment of a rotator with an initial velocity would require replacement. of (41) by more complicated formulas. 12See, for instance, 13Compare footnote
C. M#ller (6) where nz is written for our k’ = k. 18 of Part I. We neglect here the effects of the earth’s
daily
rotation.
THEORY
OF
207
GRAVITATIOX
The instantaneous acceleration of the rotator is now found by first calculating the accelerations al = d’x,/dt~ and a2 = d2xz/dt2’ of the two charges, and then averaging each over one period of rotation and taking the results instantaneously at the time t1 = tt = T = 0 at which the fixed and the freely falling local frame were related by (41). The result of this calculation is that one finds g, and not a, , for these instantaneous average values of al and a2, showing the fact mentioned earlier that according to the equivalence principle the initial acceleration of the entire rotator has the same value g as the initial acceleration of a single particle, irrespective of the internal motion going on inside the rotator. In this treatment of Einstein’s theory, by the general relativity and equivalence principles, the electric field E with respect to the (x, t) laboratory system in our electrostatic approximation follows from the field (30) which we supposed to prevail with respect to the local (c, 7) system. For later comparison it is useful to write down the expression resulting for E. We derive it from (30) by means of the tensor transformation which corresponds to the coordinate transformation (41). This easily yields WC)
= ez /b~l-~[h
- El(g. h/c’)
+ g(G. h/c”)].
(44)
Now, for one common value of t, the difference between the 7 values for the two part’icles according to (41) is linear in the gravitational field. Therefore, in rzl = XI - xz calculated from (41x) we may neglect 71 - ~2 . Thence we derive, using (24a-b), and NEQITGF, /r2J%
= (rzl .r2J3”r21 = /k211-3{r21[1 + f&
so that’ (44) may also be written Ez(xl)
= e2 jrzl]-3rzl[l
-
+
g.c.5
+
r;2)/2c”l
+
g (p$
-
[~2~“>/2cZ},
(45)
as
2h, -
2g.x2/c2
-
3g.rz1/2c2] + e,g/2c2)rzll.
(46)
We now return to our present problem of solving Eqs. (32) for particles 1 and 2 with a0 given by (22) and with the forces given by Eq. (39) and a similar one for Fz . Here, big and beg represent the vertical accelerations of the two charges as far as due to the fall of the rotator. The difference with the Einsteinian case is that there we knew bl = bz = 1 a priori from the equivalence principle applied to our rotator of vanishing initial velocity; but in our present case we shall have to solve for the unknown bl and bt by solution of the equations of motion (32). For 5 = 4 and bl = bz = 1, we may expect our formulas to become identical with those of the Einsteinian case linearized by NEQITGF. We see already that in this case Eq. (39) amounts to el X Eq. (46). This is why at least for this case of 5 = 4 the approximation (39) for the electric field-in which by (33) we took into account the electromagnetic induction term at particle 1 only in as far as
208
BELIXFANTE
AND
SWIHART
caused by gravitational acceleration of particle 2-is the proper approximation when the point of view is taken that the static field (30) prevails in the local system {c, T) . It is then reasonable to assume that the approximation (39) for the field is “consistent” with the “electrostatic” point of view taken by us, even if 5 # 4. In the Einsteinian case, NEQITGF, the equations of motion (32) with (22), and with (46) [or with (39) for 5 = 4, bl = bp = 11, were solved by obtaining the dependence of xi and of x2 on t which followed from the circular motion (23)-(24) with (30)-(31) in the local system, by means of the coordinate transformation (41) from (r, T) to (x, t) . Similarly, we shall try to solve the equation of motion (32), with (22) and with (39), even in the case of 5 # 4 and bl and b2 as yet unknown, by finding a dependence of x1 and of x2 on t from the circular motion (23)-(24) with (30)-(31) by some coordinate transformation from (F, 7) to (x, t} . This new coordinate transformation is a generalization of (41), which may be devoid of the a priori physical meaning of this transformation in the Einsteinian case,14 but which yet is a convenient mathematical trick for solving our equations of motion. As a trial solution we choose for this transformation =
Xj
tj[l
- a(& +
g’rj/C2)]
+ >‘$Kg
/~j12/C2
+
f/s@jg
Tj2,
t = Tj[l + T(h0 + g’F;j/C’)J,
(47Xj)
C47tj)
where the arbitrary numerical coefficients a, y, K, p1 , & are to be determined in such a way that (23)-(24) with (30)-(31) transformed by (47) indeed will solve our equations of motion. From the Einsteinian case we know already [see Eqs. (41)] that for 5 = 4 the transformation (47) gives a solution for cx = y = K = p1 = p2 = bl = b2 = 1. We shall first investigate what conditions we should impose upon these constants in order to ensure that the orbit of particle 2 shall have indeed the vertical accelerat.ion bzg assumed already in the derivation of (33). With hj = ho + g.Xj/'C" E h, + g.&/c2, and by (23), we find from (47) the velocity of particle j : Vj(t)
=
[dt,/dTj]-l(dXj,/dTj]
= [I + rhj + yTjg-Vj/C’]-‘{ = Vj[l
-
((Y + r)hj
(1 - ahj)Vj
- yTjg.Vj/Z]
- atjg.Vj/cS
- cjag.Vj/c’
+ Tj@jgj
(48j)
+ gpj Tj ;
14 The main difference with Einstein’s theory is that there 5 and 7 have a physical meaning as coordinates in a local freely falling frame of reference, while in our theory c and + are meaningless auxiliary variables used in making a one-to-one correspondence between the actual helical motions x,(t) and thought circular motions C~(T).
THEORY
thence,
OF
209
GRAVITATIOK
NEQITGF, we obtain for its acceleration =
aj(t)
(dt/dTj]-‘(dVj(t)/dTj)
= -
-
+ Vj[-2(a
(CX+ 2r)hj + Y)g’Vj/c*
- CXg’fj/C’
- 2Trjg.Vj/C’]
(49j>
+ YTj~2getj/C41 + gPj.
Just as in the Einsteinian case, so also in our theory, the transformation solves the equations of motion only “locally,” that is, at 7 RS t z 0. So we insert this in (49) and find aj
=
W’[l -
-tj
(a + 27)&
-
(201 + 2r)
g’fj/C2] (5Oj)
- Vj (2a
+
27)g.Vj/C2
+
g@j
.
In order to separate the vertical acceleration of the entire orbital planes from the rotation itseIf, we average (501) over one period. The terms linear in tj then vanish. The average of the terms quadratic in tj or Vj is (a + +Y>(g/C”){~’ ltj12 - IVjl”}
= 0
(51)
by (23g). Thus only the term g@j in (5Oj) represents the acceleration of particle j’s orbital plane. For j = 2, this is just the acceleration which we called b,g above Eq. (33). Similarly, for j = I, it is i&g. Therefore, in (47) for consistency we must choose /3j
=
bj
.
(52)
By (5Oj), me expressed aj in terms of the “unperturbed” variables Vj and rj . Therefore, let us do the same with the electrostatic force Fi calculated in (39). First, from (47x1,2) with g(q - 7~) cz 0, we obtain
Irn lM3rz1= (r2r. rfl)-3’121 = (t21(-~&?I[l + 2&, + (h! - 3X)( 1 - N) g*i$k2]
(53)
+ Il;zrj-I[l + N]-‘(1 - N) Kg/2c”, * where we used (24a-b). Inserting this in Eq. (39), and using (24a-b), (47x1 $, and (52), we find, NEQITGF, FI = ele2/&I[-~CZI [1 i- (2~ - %@L +
(&I
-
3K
-
$53
-
+ ele21tl (-‘(g/Zc*)[(l
b2)g't,/2C2
+
(4X
-
3K -
35s
+
b,)g+/2C']
(54)
+ N)--‘(I - N)K + %S - b,].
Our next task is to determine the values of the constants (Y,y, K, bl , b2for which
210
BELINFANTE
AND
SWIHART
the trial solution obtained from (23)-(24) by (471) for particle 1 will solve its equation of motion (32). Therefore, in the left-hand member of (32) we insert (501) for al . In its right-hand member, we insert for a,, the expression (22), and for F the expression (54). The symbol 6, occurring in Eqs. (25)-(32) represented for the j”” particle the quantity (1 - ~j2/c2)1’2 [see (I, SS)]; but since in our electrostatic approximation we have already replaced b, by 1 when multiplied by F [see above Eq. (32)], it will certainly suffice to replace b, in such terms by bj E (1 Instead
vj2,‘c2)“2 = (1 - w2 l~jj2,‘&1’z.
of at once using Eq. (31a), let us temporarily WI2 3 -ee.bm-’
Ih/-’
put (55)
lt~-“.
Further, we may put x w t and v = V wherever they appear multiplied by g. Thus we obtain for the right-hand member of (32), by (24a-b) and (23g), - &w;[l
+ (1 + 2cu - 35/2)/z, + (2a -
3~/2
- 5/4
+ (1 + 2a - 3~,‘2 - 55/4
- b2/2)g.c,/c2
+ bz/2)g. ~2/c2] - VIFg. V&” - 62+
+ g( 1 + ( ~~1~*/2C2)[u2(5 - 2) - w12(3/2
K)
-
(56)
- bz-
dN(3/2
K)]).
This expression then should be equal to (501). Sufficient for this is the separate equality of the coefficients of & , of &ho , of &g. &/c”, of &g *&/c*, of Vlg. V1/c2, of g, of g1t112/2c2,and of gJa,1*w1*N/2c2.This gives eight equations to be satisfied by the constants cr, y, K, bl , 62, and w. The first equation gives W2
= w&
(57)
and thus gives us (31a). Using this result in the other equations for eliminating w entirely, we further obtain for particle 1 the seven relations 3fx + 2-i
- b2/2
40 -I- 2-y -
3~/2
2l.Y
3~/2 + bJ2
-
2cY-I- 2-f
= 33/2
- 1,
(5W
= 53/4
- 1,
(58bJ
=
F/4,
=
$7
(5th) Wd)
bl = 1, K
-
ii
+bz
b2
=
5/2
=
T/2.
(58ed -
2,
(5%) wkl)
Similarly treating the equation of motion of particle 2 one finds additional equations with bl and b2 interchanged.
THEORY
All these equations
OF
(58) are compatible
N = K = s/a -
1;
211
C:RAVIT.~TION
and are mliquely
solved by
y = bl = bz = 1.
(59)
Alost, important to us are the results (Se,) and (S8e2); therefore it is useful to remark that these equations arc nwcssar!/ conditions, since in (50) and (56) the terms with the vertical vector g as a factor but not depending on position necessarily must he qua1 separately. Thus, by (Z-(24), (:
Lagrangian does not, lead to c’onflic+s with the ~v~ui~dr~lce prinviplc, K it,self does riots seem to appear ill our results.
alit1 \vhy
The authors we indchted to the Sational S~ic~nw I~o~~~rtlation for fin;lnci:ll these investigations during t,he ye:m I XC-1 956.
suplmrt
RECEIVED:
Dccemkr
of
17, l!IN.
I. F. J. BELINFANTE ALYD J. C’. SWHART, Annuls of Physics 1, 168 (1957). 2. M. MOSHINSKY, Phys. Z&xl. 80, 514 (1950). 3. F. J. BELINYAYIT, Physicn 7,119 (1940). 4. F. J. BELISFANTE, Phys. Rev. 74, $79 (19-M). 5. R. v. E:iiwiis, D. PEB.
(1955). 6. C’. M@LLER, “The Theory of Relativity,” mcl Sew York 1952.
11. 327, l+lq. (86). Osfortl Univ. l’rws,
I,ondon