Physics Essays volume 4, number 4, 1991

Relativistic Kinematics III: A Relativistic Modification for Newton's Gravitational Force Law

Young-Sea Huang

Abstract This is the third in a series of papers on a new theory of relativity [Phys. Essays 4, 68 (1991); ibk[ 194; "Relativistic KJnematm IV,," submilted to Phys. Essays]. This paper presents, in detail, an application of the new theory of relativity to relativistically modify Newton's gravitational force law. The new relativistic gravitational force law thus obtained gives a simple correction factor, 1 - (v/c) 2, to Newton's gravitational force law; it contains Newton's gravitational force law as a low-speed limit. However, the new relativistic theory of gravitation differs from Einstein's general relativi(y of gravitation. We also discuss experimental tests of general relativi(y and wish to emphasize that these experimental tests are not reliable as originally claimed.

Key words: relativity, relativistic modification of Newton's gravitational force law, force- directed accelerations, interpretations of light frequency shift, ambiguities in the physical meaning of curved space-time, reliability of experiments in general relativity

1. INTRODUCTION The most generally accepted relativistic theory of gravitation to date is

Einstein's general relativity of gravitation. (~) In his general relativity Ein- stein introduced the principle of general covariance and the principle of equivalence. According to the principle of general covariance, laws of na- ture are to be expressed by equations that are covariant for all systems of space-time coordinates, that is, the mathematical forms of laws of nature are invariant under arbitrary space-time coordinate transformations. The principle of equivalence of Einstein's general relativity is a conceptual ex- tension from indistinguishability of the inertial mass and the gravitational mass. The principle of equivalence assumes there to be a complete physi- cal equivalence between the uniform gravitational field and the uniformly accelerated reference frame.

Einstein assumed that the presence of matter in space causes the space- time continuum to bend. That is, the space-time structure of the physical world changes from flat Euclidean space to curved Riemannian space due to the presence of matter. The effect of gravitation is completely described by the curved space-time metric of Riemannian space. He geometrized the physical word into the mathematical Riemannian space; his concept of curved space- time was then introduced accordingly. However, Einstein's concept of curved space-time coordinates can not be operationally defined without ambiguities, and its precise physical meaning is far from clear. <2),(3)

In contrast, the new theory of relativity is based (besides the light-speed

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constancy and the transverse Doppler shift) on the mere assumption of the existence of an instantaneous rest frame: for a material particle moving with respect to a reference frame, at any instant of time there exists an instantaneous reference frame in which this one particle is at rest/4) This assumption is, in fact, logically more restrictive than the principle of equiv- alence in applications describing the motion of particles in a gravitational field. Indeed, according to the new theory of relativity, the uniform gravita- tional field and the accelerated reference frame are not physically equivalent. The acceleration of a material particle due to gravity can only be virtually eliminated with respect to the chosen instantaneous rest frame of that single particle. In addition, the new theory of relativity does not accept the princi- ple of general covariance. The relativistic transformation entailed by the new theory of relativity is an instantaneous Lorentz transformation in velocity space only, or equivalently in energy-momentum space only. The gener- al relativistic equations of motions are covariant under the instantaneous Lorentz transformation and not under general coordinate transformations. The concepts of space and time adopted by the new theory of relativity are different from Einstein's concept of space-time. The new theory of relativity is not a metric theory of space-time as is Einstein's theory of relativity.

The formalism of the new theory of relativity provides a unified theo- retical scheme that relativistically modifies the classical force laws./4) We have discussed, in detail, an application of the new theory of relativity to retativistically modify the classical electromagnetic force law3 5) The new

Young-Sea Huang

relativistic electromagnetic force law thus obtained gives a simple correction factor, 1 - ( v / c ) z , to the classical dectmmagnetic force law; it contains the classical electromagnetic force law as a low-speed limit. In this paper we will apply the new theory of relativity to relativistically modify Newton's gravitational force law. Although the technique of modification is exactly the same as presented in modifying the classical electromagnetic force law, we repeat it, in detail, in order to emphasize that the new theory of relativity indeed provides a unified scheme to relativistically modify any given classical force law. The new relativistic gravitational force law thus obtained also gives a simple correction factor, 1 - ( v l c ) 2 , to Newton's gravitational force law; it contains Newton's gravitational force law as a low-speed limit. Neverthe- less, the new relativistic theory of gravitation presented here is different from Einstein's general relativity of gravitation.

This paper is organized as follows. In Sec. 2 we apply the new theory of relativity to modify Newton's gravitational force law. In Sec. 3 we describe the motion of a material particle in a central gravitational field. In Sec. 4 we study planetary motion. The results thus obtained give rise to small correction terms to Kepler's laws of planetary motion. We discuss the experiments of the precession of the perihelia of planets of the solar system. In Sec. 5 we study deflections by gravity of particles with a wide range of incident speeds. The new relativistic theory of gravitation predicts that photons are not accelerated by gravity and thus not bent by gravity. We question whether the deflection of light rays is indeed due to gravity. In Sec. 6 we discuss a modem test of general relativity - experiments of radar echo delay. In Sec. 7 we study frequency shift of light waves emitted by a light source moving in a gravitational field. We also present the conceptual differences between the new theory of relativity and Einstein's theory of relativity in their interpretations of light frequency shifts.

2. TItE NEW RELATIVISTIC GRAVITATIONAL FORCE LAW Consider a given static gravitational field N, defined in an inertial refer-

ence frame X. Here, "inertial" means that if particles do not move uniformly with respect to the frame X, these particles are presumably under the influ- ence of some interactions. The gravitational field is predefined by Newton's gravitational force law. Consider, also, a material particle of mass m moving under" the influence of this gravitational force field in the inertial frame X. As derived in the first paper of the new theory of relativity, (41 the general relativistic equations of motion of a material particle are

d2x t.t d_x L dx v d x 2 + F;~v d' t d x - O, where (1)

dz = (g~.vdxgdv v) t/2,

where F~v are defined as

1 { 3 g ~ ag,,,, r L - ~ ~ ~ a-U + axX

and where a ~v are defined by

(2)

a&v } (3) ~x o '

I, f i X = v ; a~'g~ o, if Z,~v. (4)

In our notation, ~., M, v, o = 0, 1, 2, 3. Any index that appears twice, once as a subscript and once as a superscript, is understood to be summed over.

Furthermore, Eq. (2) is reduced to

d x = f ( x ) {1 - [v ( x ) / c ] 2 }l/2dx~ (5)

where x represents the position and time coordinates of the moving particle with respect to the frame X, and the functionf(x) is, in general, related to the state of motion, that is, it depends on the velocity, the acceleration, the time derivative of acceleration, and so on, of the particle under consideration.

Rewriting the general relativistic equations of motion Eq. (1) in terms of the coordinates of the inertial reference frame X, we have

) dxX dr v dZx i dx i _ F~v dx 0 dx 0 a~o~: r~ ~-o- (i = 1, 2, 3). (6)

From Eq. (6) in the low-speed limit v << c (the speed of light), we obtain

dZ x~ / dt 2 = - c 2 I'~o . (7)

This equation, Eq. (7), is the general equation of motion for particles of which the speeds are not comparable with the speed of light. Therefore, for a particle moving in the gravitational field N, Eq. (7) should be equivalent to Newton's gravitational force law, which holds for low-speed particles,

F = m N, or (8)

d2x i / dt 2 = N i . (9)

Rewriting Eq. (5) we have

d x 2 =Ja(x) [(dx~ -- ( d x l ) 2 - ( d x 2 ) 2 - (dog3)2] . (10)

Hence, by Eqs. (2) and (10) we have

g.~ = f (x) n~,~, (11)

where rlvv are defined as

1 0 0 0 )

0 -1 0 0 (12) rluv - 0 0 -1 0 '

0 0 0 - 1

From Eqs. (3), (4), and (11) we obtain

r~o(X)--1 Or 2 ~x i , (13)

where r is defined by

f ( x ) - exp[ - r (14)

Then, from Eqs. (7), (9), and (13) we obtain

a r i : 2 N i ( x ) / c 2 . (15)

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Relativistic Kinematics III: A Relativistic Modification for Newton's Gravitational Force Law

From Eqs. (3), (4), (11), (13), and (15) we obtain

g o : r ; = r;i : r, ' i = -W/c

I7) =Ni/c2(j--/=i; i, j = 1, 2, 3) (16)

F~v = 0 (otherwise).

Since the gravitational field N does not vary with time, the function 0(x) does not explicitly depend on time, that is, ~a(x)/ax ~ = 0. Substituting F~v of Eq. (16) into the relativistic equations of motion Eq. (6), we finally obtain the new relativistic gravitational force law,

d2x/dt 2 = N(x) [1 - (v/c)2]. 07)

The new relativistic gravitational force law, Eq. (17), gives a simple cor- rection factor, 1 - ( v / c ) 2 , to Newton's gravitational force law, Eq. (9). The accelerations of particles by a given gravitational field are force-directed, that is, particle accelerations are in the same direction of that given force field. The new relativistic gravitational force law contains Newton's gravitational force law as a low-speed limit. It also shows that the accelerations of ma- terial particles by any gravitational field decrease as the speeds of particles increase. No material particles can be accelerated to the speed of light. In the limit case that v ~ c, the new relativistic theory of gravitation predicts that photons are not accelerated by gravity.

Comparing the new relativistic gravitational force law and the new rela- tivistic electromagnetic force law in the second paper in the series, ~5) we see that both give a simple correction factor, 1 - (v/c) 2, to the classical force laws. This correction factor depends only on speeds of particles, but not on the kind of force field. The second paper together with this paper clearly demonstrate that the new theory of relativity provides a unified method to relativistically modify any given classical force law.

3. MOTION OF A MATERIAL PARTICLE IN A CENTRAL GRAVITATIONAL FORCE FIELD

Suppose an object of mass M is fixed at the origin of an inertial reference frame X. By Newton's gravitational force law, a static and isotropic gravita- tional field can be defined in the inertial frame X. Consider a particle of mass m moving under the influence of this central gravitational force field. For mathematical simplicity we use the spherical polar coordinate system. For the given gravitational field, from Eq. (16) as well as the relation- ship between the spherical polar coordinates (r, O, ~) and the orthogonal Cartesian coordinates (x 1 , x 2 , x3),

x 1 = r sin 0 cos ff~,

x 2 = r sin 0 sin ~, (18)

we obtain

x 3 = r cos O,

d x = e x p --7- 1 - \ dx0 ]

z - - \ 2 1 1 / 2

- r 2 s i n Z O ( - ~ o ) ] dx O, and (19)

534

I~or = K / r 2, F~O =K/r 2, F(r = K / r 2

F~o = - ( r + K), F ~ = - ( r + K) sin 2 0

F~ = l i t + k l r 2 , I"~ = -s in 0 cos 0

I ~ = 1/ r + k / r 2, 1 - ~ =cotO

FgXv = 0 (otherwise),

where the constant K is

(20)

and G is Newton's gravitational constant. Substituting Eq. (20) into the general relativistic equations of motion Eq. (1), we obtain

d2r K (dx~ 2 K ( d r ) d'~ 2 + 7" \ dx ] +7" "d~

dZO (1 K ) dr dO dx 2 +2 + 7 d ' t da:

2 < )2 - ( r +K) --~

(dcP~ 2 - (r + K) sin 2 0 ~, 7~- ] = 0

(dq)~ 2 - - - - - s i n O c o s O \ ~ ] = 0 (22)

dZq) + 2 + + 2 cot 0 - 0 dz 2 7. ~ ~ d't:

d2x~ 2 K--- dx ~ dr dg 2 + r z d'c d ' c - O.

Now we consider the motion in the plane 0 = x/2. From Eqs. (19) and (22) we obtain

d ' r = exp ( # ) [ ( d x ~ (23)

d2r K (d.x~ 2 K ( d r ~ 2 (d~p~ 2 d'r 2 + 7. \ d't ] + -~ \ d'c l - (r + K) \-d-~ ] =0, (24)

( 1 K ) d r d~ d2~ +2 + =0, and (25) d,~2 ~ ~

d2x~ 2 K dx ~ dr d't 2 + r 2 d* d'g-O" (26)

From Eq. (25) we obtain

r 2 ~ exp =J, (27)

where J is an integration constant. From Eq. (26) we obtain

(dx~ exp(-XK/r) = D, (28)

where D is an integration constant. The constants E - Dmc 2 and L =-Jmc

K =~ GM/c 2 , (21)

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are the total energy and angular momentum, respectively, of the moving material particle in the gravitational field. (5) Substituting Eqs. (27) and (28) into Eq. (24) we obtain

dZr K ( d r ~ z K r + K j2 da:O a r2 \da. 0 ] + r2 r4 D2 - O, and (29)

d2r 2 ( d r ~ 2 K ( d r ) 2 Kr2D2 a ~ ~ \ d ~ o ] - ; T Y4 -(~+s176 (30)

Solving Eq. (30) we obtain

(drldq~) 2 =D2r4tj 2 - r e - S r4exp ( -2Kt r ) , (31)

where S is an integration constant. From Eqs. (27), (28), and (30) we obtain

(dr /dx~ 2 = 1 - j 2 / D 2 r e - - (3)r2/D 2) e x p ( - 2 K / r ) . (32)

Substituting Eqs. (27), (28), and (32) into Eq. (23), we obtain

d , 2 = (SJ2 /D 2 ) exp(-4K/r ) (dx ~ 2. (33)

Comparing Eq. (33) with Eq. (28), we find

S = 1/J 2. (34)

From Eqs. (27), (28), and (32) we obtain

r,r 2 ( V ) 2 = \docO] + r 1 (_7.~_)

= 1 - 3 - 7 -exp . (35)

Hence from Eq. (35) we have

O = [ 1 - ( v /c ) 2] -1/2exp(_K/r). (36)

Finally, we obtain the equations of motion of a particle in a plane:

dx~ = [ 1 - ( v /c ) z] -1/2exp(K/r) ' (37)

dcp/ dx ~ = j/r2 D , (38)

(dr /dx~ z = 1 - j2 /D2r2 - (1 /DZ)exp( -2K/r ) , and (39)

(dr/dq)) 2 = D a r 4 / j 2 _ r 2 _ (r4/j2) exp( -2K/r ) . (40)

Due to mathematical difficulties, the closed-form solutions of Eq. (40) cannot be obtained. In order to obtain the orbits of motion, we consider the first-order approximation, that is,

exp(-2K/r) ~ 1 - 2K/r. (41)

Solving Eq. (40) we obtain the solution,

ro r = where (42) 1 + e cos(~o - ~oo) '

ro = j 2 / K , (43)

(44)

and q)o is an integration constant. The orbit of motion represented by Eq. (42) is a circle ({ = 0), or an ellipse (e < 1), or a parabola (c = 1), or a hyperbola (e > 1). The orbit of motion obtained by the new relativistic theory of gravitation to the first-order approximation is consistent with Newton's theory of gravitation.

4. THREE RELATIVISTIC KEPLERIAN LAWS AND THE PRECESSION OF THE PERIHELIA OF INNER PLANETS OF THE SOLAR SYSTEM

Now we study planetary motion. Solving Eq. (40) to the second-order approximation, that is, exp(-2K/r) ~-, 1 - 2 K i t + 1/z (2K/r)2, we obtain

RXro r = where (45)

{1 + e cos[R (q~ - ~0) ] }'

r0 = j2/K, (46)

e - - [ l + ( D 2 - 1 ) ( l + 2 K 2 / j a ) ( j 2 / K 2 ) ] 1/2 , and (47)

R = (1 + 2K2/J 2) t/2 (48)

The orbits of Eq. (45) are more complicated than the orbits of a simple conic section of Eq. (42). Let us focus on the motion of elliptic orbits. Suppose that the major axis of an ellipse is 2a, that is, 2a -- rmax + rmin. Consequently, we have

rmax/rmin = ( 1 + e)/( 1 - e). (49)

From Eqs. (46), (47), (48), and (49) we obtain

R2ro = (1 - { 2 ) a , and (50)

a = K/(1 - D Z ) . (51)

Solving Eq. (39) we find that the period T of the elliptic motion is

(cT) 2 = 41t2a3D2 /K. (52)

Summarizing these results, we have the three relativistically modified Kep- lerian laws.

(1) The law of orbits: the path of a planet revolving about the center of a spherically symmetric force field is an ellipse with the center at one focus of the ellipse. This elliptic motion has a regression of the perihelion [refer to Eq. (45)],

Atp = 2a;[ (1 - 2K/R2ro) 1/2 _ 1]

,~ -2 rag/R 2 ro per revolution. (53)

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Relativistic Kinematics [lI: A Relativistic Modification for Newton's Gravitational Force Law

(2) The law of area.v: the areas swept by an imaginary line drawn from the center of the force field to the planet are equal in equal periods of time, that is,

l r 2 d q ) _ Jc _Jc (1 K)1 /2 2 dt 2D 2 - a

= 2m 1 - , (54)

(3)

where L =Jmc is the angular momentum.

The law of periods: the square of the period is proportional to the cube of the major axis of the elliptic orbit, that is,

T 2 = 4ItZa3D2/Kc 2 = (41t2a3/GM) (1 - K / a ) . (55)

The three relativistic Keplerian laws of planetary motion give rise to small correction terms to the three classical Keplerian laws of planetary motion.

Comparing Eqs. (42) and (45), we see that the regression of the perihelia of elliptic orbits is due to the second-order effect, that is, exp(-2K/r ) ~ 1 - 2K/r + Yz ( 2K/r) z, not due to the first-order effect, that is, exp(-2K/r ) ~, 1 -2K/r. The new relativistic theory of gravitation predicts a backward precession of the perihelia of elliptic orbits, for example, about 14 arc s per century for the precession of Mercury's perihelion. In contrast, Einstein's general relativity of gravitation predicts a forward preces- sion of the perihelia of elliptic orbits. (6) The observation of the precession of Mercury's perihelion (about 43 arc s per century for anomalous perihelion advance) is claimed to corroborate Einstein's general relativity. (6),(7)

It should be noted that the predicted values of the precession of the perihelia of the inner planets of our solar system due to the relativistic effect comprises only a minute part of the observed values. (8) For example, the total observed value of the precession of the perihelion for Mercury is about 5600 arc s per century, whereas only about 43 arc s per century is predicted due to the relativistic effect. One must subtract the geometrical effect of the general precession of equinoxes and the perturbations by other planets from the observed values to cull the minute values of anomalous precession due to the effect of relativity.

The anomalous precessions of the perihelia of the inner planets of our solar system, calculated by the same perturbative method using the same as- trophysical data, are not self-consistent. One even gets a backward precession of the perihelia for some inner planets. (9) Reanalyzing the precession of the perihelia of inner planets based the observations on the center of gravity of the whole solar system instead of the center of the Sun, Nedv&l recently claimed that the perihelia of inner planets should precess backwards, rather than forwards. (1~ If Nedv&l's recalculations are correct, then the regression of the perihelia of inner planets turns out to militate against rather than to support Einstein's general relativity of gravitation. In addition, astronometric measurements of solar oblateness by Hill et al. reveal that the Sun is not so spherically symmetric that its gravitational quadmpole moment can be neglected in the prediction of the precession of perihelia3 n) Their results combined with two published planetary radar results give values of 0.987 + 0.006 and 0.991 + 0.006 for 1/3 (2 + 2 " / - ~), which is about two standard

536

deviations removed from the prediction of Einstein's general relativity, that is, 7/3 (2+2T-1~) = 1 .(11),02) The results of Hill et al.'s experiments indicate that Einstein's general relativity is probably wrong with 95% confidence.

There is too much uncertainty involved in experimental observations of the precession of the perihelia of inner planets. For example, the uncer- taint), may be due to systematical errors of astronomical data being taken by different observers over a century, approximations in the calculation of many-body perturbations of planets, and the ignored solar oblateness. Fur- thermore, Yilmaz has argued theoretically against Einstein's general relativity in its inability to predict the planetary perturbations. (i3) Einstein's general relativity is there shown to be a relativistic generalization of Hooke's theory and is shown to be essentially a theory of a single body. It is thus in principle unable to calculate the many-body perturbations. We think that the experimental tests of the precession of the perihelia of inner planets are far from convincing.

5. DEFLECTION OF MATERIAL PARTICLES BY GRAVITY Now we investigate deflection by gravity of material particles with various

incident speeds. Suppose that a material particle of mass m approaches the center of a spherically symmetric gravitational field due to a mass M(m << M) from a very great distance (see Fig. 1). Considering the first-order approximation, we have, for this material particle, a hyperbolic path given by Eq. (42), that is,

r0 r = (56)

1 + e cos(q) - q)o) '

where ro =J2 / K and e -= [1 + ( D 2 - 1)JZ /K 2 ] 1/2. Suppose, also, that the material particle is incident from infinity with an initial speed Vo and an impact parameter b. Then we have

q)0 = cos -1 l/e, (57)

D = [1 - (Vo/C) 2]-1/2, and (58)

J = vobD/c. (59)

As the particle approaches the closet distance to the center of the gravitational field, that is, the point (rmin, ~ ) at which dr/dq) = O, then from Eq. (56) we obtain

a , = q)o. (60)

Therefore, we have for the deflection angle of the particle

O = n - 2,~,. (61)

Finally, from Eqs. (56), (57), (60), and (61) we obtain

/ t O = 2 cot-' \ - - f f T - - j GM[1 :-(-Vo/C) 2] . (62)

We see that the deflection angle of the incident particle depends on its speed. As the incident speed of the particle increases, the deflection angle of that particle decreases. In the limit case when v0 --, c, the new relativistic theory of gravitation predicts no bending of photons by gravity. In contrast, Einstein's general relativity predicts that photons are bent by gravity, although the local speed of photons remains constant.

Young-Sea Huang

rn~ve

Figure 1. The schematic diagram of the deflection of a material particle by a central gravitational force field.

So far, experimental observations of light rays passing near the Sun indicate that light is indeed bent; these observations are claimed as one of the great successes of Einstein's general relativity. (14),(15) However, accuracies of these observations are low; one of the major sources of errors in these observations is due to the solar corona. It bends radio waves much more strongly than it bent the visible light rays which Eddington observed. (15), (16) There is much uncertainty concerning the solar corona's effects. Whether the deflection of light is indeed due to gravity independently of refracting factors, due to the solar corona and the atmosphere of the Earth, has not been convincingly settled by experiments to date. Hopefully in the future we can place space stations far away from the Sun to observe the light rays passing by the Sun, but not through the solar corona. In this way, we will then have experiments to convincingly test whether the deflection of light is indeed due to gravity. Furthermore, the new relativistic theory of gravitation provides a general prediction for the deflection of particles with varying incident speeds. If the experiments of the deflection of incident particles having a wide range of incident speeds could be performed systematically, then those distributed results would no doubt be more reliable than the isolated experimental results of the deflection of light rays only.

6. RADAR ECHO DELAY From the well-known Schwarzschild metric in Einstein's general relativity,

Shapiro proposed experiments measuring the time delay, due to gravitation, of radar signals to travel to an inner planet and be reflected back to the Earth. (17) It should be noted that the same Schwarzschild metric is also used to predict the so-called three classical tests of general relativity: the precession of the perihelia of inner planets, the deflection of light rays by gravity, and the light frequency shift by gravity. Experiments on the time delay of radar signals were carried out by Shapiro et al.; the experimental results are claimed to corroborate Einstein's general relativity with high accuracy. (18),(19) In contrast, according to the new relativistic theory of gravitation, photons are not accelerated by gravity and thus not bent by gravity. Therefore, the new relativistic theory of gravitation predicts no time delay, due to gravitation, for photons that travel to the inner planets and are reflected back to the Earth.

Experimental results of radar echo delay have claimed to give one of the most precise supports for Einstein's general relativity to date. (is) It seems that this modem experimental evidence definitely confirms Einstein's general relativity of gravitation and refutes the new relativistic theory of gravitation. However, recently Logunov and Loskutov recalculated the time delay of radar signals (2~ and found that their result is different from that obtained in the original calculation./17),{2]), {22) Both Logunov et al. 's calculation and the

original calculation are based on the same Schwarzschild metric in Ein- stein's general relativily. The mere difference between these two calculations is in their methods of approximation. When there are gravitational effects on radar signals, the radar signals follow the geodesic path according to the Schwarzschild metric by Logunov et al.'s calculation as well as the original calculation. When there are no gravitational effects on radar signals, the radar signals travel in a straight line from the Earth to an inner planet as treated in Logunov et al.'s calculation. (2~ However, according to the approximation of the original calculation, the radar signals travel first in a straight line from the Earth to the point where the radar signals approach closest to the Sun and then travel in another straight line from this point to an inner planet. (21) The original calculation is less accurate than Logunov et al.'s calculation, because light should travel in a straight line when there are no gravitational effects on light according to Einstein's general relativ- ity. It should be noted that the difference between these two calculations is much larger than the quoted uncertainty of experiments of Shapiro et al. Therefore, the most precise modem tests of general relativity, the time-delay experiments, turn out to refute rather than corroborate Einstein's general relativity, if the experiments of Shapiro et al. are as accurate as claimed.

As mentioned in previous sections, the experimental tests of the precession of the perihelia of inner planets and the deflection of light rays are currently far from convincing. Is the time-delay experiment of Shapiro et al. so reliable that it could convincingly refute either Einstein's general relativity of gravi- tation or the new relativistic theory of gravitation? The solar corona is also considered as a major factor limiting the accuracy of time-delay measure- ments. The radar signals are slowing down due to refracting effects by the solar corona. (15) A substantial portion of data, which indicates the observed values of time delay being larger than the predicted value, is neglected by Shapiro et al.(z3) This leads us to question generally the reliability of the data analysis and the possibility of interpretations of the experimental results towards the preknown expectations in experimental relativity. (23) Thenfon, a clean experiment avoiding contentious complications in the interpretation of the experimental data is needed. As proposed in the previous section, to test whether the deflection of light is indeed due to gravity, we can place s- pace stations far away from the Sun and perform the time-delay experiments avoiding the refracting effect due to the solar corona and the atmosphere of the planets. In this way we will have experiments to convincingly tell whether the time delay of light signals is indeed due to gravity and not due to refracting effects causing the speed-of-light signals to slow down.

7. FREQUENCY SHIFI" OF LIGHT WAVES EMITI~D BY A LIGHT SOURCE MOVING IN A GRAVITATIONAL FIELD

7.1 Frequency Shift of Light Waves Emitted by a Light Source on a Satellite

Consider a satellite moving in a circular orbit of radius r around a planet of mass M (see Fig. 2). Suppose that the mass of the planet is much larger than the mass of the satellite, and therefore the planet can be considered at rest at the origin of an inertial reference frame X. Suppose, also, that the frequency of light waves emitted by a light source on the satellite is Vs with respect to that satellite. Then the period of light waves, the time interval between two successive crests of light waves, emitted by that light source is A ts = l/Vs with respect to its rest frame, the satellite. From Eq. (37) the period of light waves emitted by that light source, measured with respect to the inertial reference frame X, is

At = [ 1 - ( v /c ) 2] -l/2exp(K/r) Ats, (63)

537

Relativistic Kinematics III: A Relativistic Modification for N~'ton's Gravitational Force Law"

/ \

/ \ N O ~ \

/ \ \ / \

Figure 2. The schematic diagram of light waves emitted from a light source on a satellite which revolves around a planet. The frequency of light waves is measured by an observer on the planet.

where v is the speed of that satellite with respect to the frame X, and K = G M / c 2 . For an observer "o" on that planet (see Fig. 2), the period of those light waves is measured as

Ato = At[1 - ( v / c ) cos O], (64)

where 0 is the angle of emission viewed from the observer "o". This is the Doppler effect due to the longitudinal relative velocity between the light source and the observer "o". When 0 = ~ 2 the period of the light waves measured transversely by the observer "o" is

Ato = At = [1 - ( v / c ) z ] - l / 2 e x p ( K / r ) A t s .

Combining Eq. (63) and Eq. (64) the frequency of light waves emitted by a light source on a satellite, measured by the observer "o," is

Vo = [1 - ( v / c ) 2] V2exp(_K/r) [ 1 - ( v / c ) cos O]-lVs. (65)

Therefore, the light frequency shift measured by the observer is

Av/v - (Vo - v O / v ~

= [ 1 - ( v / c ) 2] x/2 exp(-K/r ) [ 1 - ( v / c ) cos O] -1 _ 1. (66)

For the circular orbit, by Eqs. (29) and (38) we have v / c = [ K / ( K + r ) ] 1/2

Hence the light frequency shift, Eq. (66), becomes

- - = 1+ exp 1 - cosO - 1 . v

(67)

For the first-order approximation the light frequency shift is

AV/V ~-, ( K~ r) 1/2 cos O + ( K~ r) (cos z 0 - 3/2). (68)

Unless O ~-, x/2, the first term of Eq. (68), which is related to the Doppler effect due to the longitudinal relative velocity between the light source and the observer, dominates the shift of light frequency.

F

r " r

@ Figure 3. The schematic diagram of light waves emitted from a light source which falls freely in a central gravitational field. The frequency of light waves is measured transversely at various heights.

7.2 Frequency Shift of Light Waves Emitted by a Light Source Freely Falling in a Central Gravitational Field

Consider a light source falling freely in a radial direction in a central gravitational field due to a planet of mass M (see Fig. 3). As before, suppose that the mass of the planet is much larger than the mass of the light source, and therefore the planet can be considered at rest at the origin of an inertial reference frame X. Suppose, also, that the frequency of light waves emitted by that light source is Vs with respect to the rest frame of that light source. That is, the period of light waves emitted by that light source is A/s = t/Vs measured with respect to its rest frame. From Eqs. (36) and (37) the period of those light waves measured with respect to the frame X is

At = D exp(2K/r) Ats, (69)

where D is a constant and K = G M / c 2 . Consequently, the frequency of light waves emitted by that light source measured transversely with respect to the frame X is

v( r ) = ~- exp Vs. (70)

The frequency of light waves v(r) measured transversely with respect to the frame X depends on the position r of that freely falling light source. Therefore, the frequency shift of light waves emitted by that freely falling light source between any two positions r~ and r2 is

( , , ) Av v ( r a ) - v ( r z ) - e x p 2K v - v(r2) 72 7~ - 1 . (7D

For a weak gravitational force field we have

- - .~a

v - 7 7 " (72)

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Young-Sea Huang

7.3 The Physical Interpretations of Light Frequency Shift Einstein's theory of relativity, special relativity and general relativity, is a

metric theory of space and time. Relativistic physical phenomena, for exam- pie, the light frequency shift, are interpreted in terms of his revolutionary space-time concept. The interpretation of the frequency shift of light waves emitted by a moving light source is as follows. The frequency of light waves emitted by the light source, measured with respect to a frame in which the light source is at rest, can be considered as the rate of the clock in its rest frame. Suppose that this light source moves with a constant velocity with respect to another reference frame. The shift of frequency of light waves emitted by that light source measured with respect to another reference frame is interpreted as the rate of that clock being changed - the moving clock physically runs slower than the same clock at rest. Furthermore, according to Einstein's general relativity, the presence of matter causes the geometry of the space-time structure of the universe to change from Euclidean geometry to Riemannian geometry. That is, the space-time structure of the physical world is curved in the sense of Riemannian geometry. The frequency shift of light waves emitted by a light source in a gravitational field is then interpreted as the rate of the clock being changed in the world of curved space-time distorted by gravity.

In contrast, the new theory of relativity is not a metric theory of space- time as is Einstein's theory of relativity. The space and time coordinate system is presumed in the beginning. Then we have the given space and time to measure physical quantities such as the frequency of light waves and the speed of a light source. The light frequency shift is interpreted as

due to changes of the state of motion of the light source relative to the

observer, not due to the distortion of space-time coordinates as interprded

by Einstein. If two identical light sources have the same state of motion, that is, the same velocity, the same acceleration, and so on, with respect to a reference frame, then the frequencies of light waves emitted by these identical light sources measured with respect to the reference frame are the same. The frequency of light waves emitted by a moving light source differs from that of the identical light source at rest with respect to another reference frame, because the states of motion of these identical light sources with respect to the different reference frames are different, not because the measuring space and time units of the different frames of reference are different.

To explicitly highlight these conceptual differences between Einstein's theory of relativity and the new theory of relativity, consider the following two examples: for the first example consider a light source moving in a central gravitational force field due to an object of mass M. According to Einstein's general relativity, we have the well-known Schwara~hild metric for the isotropic gravitational field(eli:

d,2 = (1 2GM'~dx02 - (1 2GM~ -1 - ca r J - ca r J dr2

-- r2dO 2 - r2sin2Od~2, or (73)

d'2 = [ ( 1 - 2GM~ -~r J - ( 1 - 2GM'~-I c-~--r J (~ d-~'r0)2

_ r 2 ~ - r2s in20 dx o2. (74)

From Eq. (73), if we let dr = dO = d o = O, then we have

d't: = (1 - 2Gm/c2r)1/2dx~ (75)

Similarly, if we let drldx ~ = dO/dx ~ = d~pldx ~ : O, then from Eq (74) we also obtain Eq. (75). Consequently, the time dilation due to gravity is then interpreted according to Eq. (75)3 24) So far, experiments measuring the light frequency shift in gravitational fields have claimed to corroborate the time dilation as predicted by Einstein. (15),(25) These experiments are claimed to provide firm evidence for the revolutionary concept of curved space- time introduced by Einstein. However, the physical meaning of space-time coordinates in Einstein's general relativity is far from clear. Consequently, conflicting interpretations of the physical meaning of time and proper time

have been given, but they reach the same conclusion of time dilation by gravity which is confirmed by experiments as claimed. (24L(26) In addition, the same predication Eq. (75) can be obtained from the conditions dr =

dO = d(p = 0 and dr/dx ~ = dO/dx ~ = d(p/dx ~ = 0, which are not exactly the same. The condition d r/ dx ~ = dO/ dx ~ = d W dx ~ = 0 states precisely that the light source has zero velocity in the Schwarzsehild space- time coordinates. However, the condition dr -- dO = dq~ = 0 states that the light source is at rest (zero velocity, zero acceleration, and so on) with respect to the Schwarzschild space-time coordinates. It seems that general relativity does not distinguish the difference between the following two cases: the light source is at rest in a gravitational field, and the light source has zero velocity but nonzero acceleration due to gravity. Are the frequencies of light waves emitted by the light source in these two cases the same? If they are different, how do they differ according to Einstein's general relativity?

In contrast, according to the new theory of relativity, the frequency of light waves emitted by a light source measured with respect to a reference frame depends on the state of motion of that light source with respect to that reference frame. Therefore, the frequencies of light waves emitted by a light source of those two cases mentioned in the above are different, because their states of motion with respect to the frame are different. If the light source is at rest in a gravitational field, there must be an interaction acting on this light source in order to balance the gravitational interaction. The effect on the state of motion of the light source by the balancing interaction is considered the same as that by gravity. Hence the frequency of light waves emitted by a light source at rest in a gravitational field is the same as the light frequency measured with respect to the rest frame of that light source. If a light moves freely under the influence of a gravitational field in a reference frame, then the state of motion of that light source will change with respect to that reference frame. Therefore, the frequency of light waves emitted by that light source measured with respect to that reference frame will shift in accord with the state of motion of that light source (see Sees. 7.1 and 7.2). It should be noted that the shift of light frequency does depend on the change of the state of motion of the light source, but does not depend on what kind of forces causes its state of motion to change.

For the second example, consider two identical light sources A and B held at a reference frame X, which moves with a uniform acceleration with respect to an inertial frame X (see Fig. 4). According to Einstein's general relativity, these two light sources behave just as if they are at rest in a uniform gravitational field with respect to the reference frame )(. Then the frequency of light waves emitted by the light source A is not the same as that emitted by the light source B due to the space-time coordinates of the frame X being distorted by gravity as interpreted by Einstein.

539

Relativistic Kinematics III: A Relativistic Modification for Newton's Gravitational Force Law

•

X •

z f

CI >

A B J

,,-r ~~ , d "

X t

Figure 4. The schematic diagram of a reference frame X which moves with a constant acceleration with respect to an inertial frame X. Two light sources A and B are held at rest in the frame ;~.

However, with respect to the inertial frame X, the state of motion of the light source A is exactly the same as that of the light source B. Is the frequency of light waves emitted by the light source A the same as that emitted by the light source B, measured with respect to the inertial frame X? If it is not, then the intervening space itself must have had its own effect on the physics of these light sources.

Both the light sources A and B are at rest in the frame X. Since these light sources have the same state of motion with respect to the frame X, the frequency of light waves emitted by these light sources measured with respect to the frame X are the same, according to this new theory of relativity. In addition, with respect to the frame X, the states of motion of these light sources are still the same; their states of motion change continuously in an identical way. Hence with respect to the inertial frame X, the frequency of light waves emitted by the light source A must be the same as that emitted by the light source B. Although the light frequencies emitted by the light sources A and B shift continuously with respect to the frame X, their light frequencies shift in an identical manner.

8. CONCLUSIONS The new theory of relativity provides a unified scheme to uniquely modify

any given classical force law. The new relativistic theory of gravitation presented here contains Newton's theory of gravitation as a low-speed limit. Nevertheless, the new relativistic theory of gravitation differs from Einstein's general relativity of gravitation.

If one simply accepted the claimed precisions and experimental tests in general relativity, for example, experiments of radar echo delay, then Einstein's general relativity of gravitation should be refuted. (2~ Furthermore, reanalyses of experimental observations of the precession of the perihelia of inner planets also indicate that Einstein's general relativity of gravitation

may be incorrect. Ii~ Unless these problems are resolved, it is beyond dispute that in general the experiments in general relativity are not reliable as claimed. Critical reexamination of experimental tests in general relativity is indispensable.

In addition, on the theoretical aspect the physical meaning of the concept of curved space-time introduced in Einstein's general relativity is ambiguous. (2),(3),(27) Consequently, in the theory of general relativity one often finds complicated manipulations, such as the arbitrary transforma- tion of space and time coordinates, performed without having any clear corresponding meanings in the physical world. There exist contradictory in- terpretations of the physical meaning of time and proper time of curved space-time coordinates./20,~261 Furthermore, Einstein's general relativity of gravitation is even criticized on its non-uniqueness in making theoretical pre- dictions; Einstein's theory should not be considered a physical theory. ~2s/,<29) Whether the contradictory interpretations and the criticism are due to mis- comprehension of Einstein's general relativity is disputable. The dispute will not be settled without even more controversies, unless the physical meaning of Einstein's concept of curved space-time is clarified.

Acknowledgment This author gratefully acknowledges the critical comments of Dr. C.M.L. Leonard in the preparation of this paper, especially for his encouragement. This author sincerely thanks Or. T.E. Phipps, Jr., for his encouragement and endorsement.

Received 30 October 1989.

R~sum~ Ceci est le tro'wibne d'une s&ie d'articles sur une nouvelle thborie de relativit~ [Phys. Essays 4, 68 (1991); ibid. 194; "Relativistic I6"nematics IV,," submitted to Phys. Essays]. Ceci pr&ente en ddtail une application de la th~orie h la modification relativiste de la force gravitationnelle de Newton. La nouvelle loi ainsi obtenue n'est corrigde que par un simple facteur 1 - (v/c) 2 el se reduit donc cl la loi de Newton pour petites vitesses. Toutefois, cette nouvelle th&rie differe de la relativ#d g~n&ale de la gravitation d'Einstein. Nous commentons aussi des tests m'p~a'mentaux de la relativit~ g~n&ale el voulons souligner que ces tests ne sont pas aussi fiables qu'on le prbtend.

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Young-Sea Huang

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Young-Sea Huaag Department of Physics Soochow University Shih-Lin, Taipei Taiwan

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