Abstract. Utilizing the principle of equivalence and the radiation continuum model of EM radiation, it is demonstrated that Newton's gravitational potential applies only for static or slowly moving objects. The addition of a velocity dependent term, derivable from the principle of equivalence, produces the full dynamic gravitational potential. This dynamic potential is applied to the problem of Mercury's orbit and photon deflection, fully accounting for both, and providing a result identical in form and value to that obtained utilizing the curved space-time of GRT.

dépendant de la vitesse, produit le potentiel dynamique gravitationel. Ce potentiel dynamique justifie le problème de l'orbite de Mercure et la déviation des photons et fournit un résultat ayant une forme et une valeur identique à celui utilisant la courbure de l'espace-temps de la rélativité

générale.

It has been shown in a previous paper that under the radiation continuum model, RCM, of light, EM radiation may be treated as emanating from a source at all velocities from 0 to some upper value C, which is greater than c, and may be infinite. In this model, a photon may be viewed as an expanding spherical volume, with the leading edge of that volume expanding at a velocity of C. This is in contrast to the special relativistic model of a photon expanding in a spherical shell at a velocity of c, but with that value c dependent on the velocity of the observer, and not measured with respect to the source. As a result of modeling the photon as an expanding spherical volume, there can be found a component of this extended photon with any velocity one chooses between 0 and C. With respect to the source, all velocity components of the emitted continuum photon are at the same frequency, thus the wavelength increases proportionally with velocity. Any observer will be susceptible to that component with a velocity of c relative to the observer. For example, an observer moving away from the source with a velocity of v would be susceptible to that component which leaves the source with a velocity of c + v with respect to the source, and thus has a velocity of c with respect to the observer. This model has been shown to allow a Galilean invariance of Maxwell's equations, and to support every experiment used to "confirm" special relativity.^{(1, 2, 3, 4)}

Since the discovery of Mercury’s anomalous perihelion advance by LeVerrier in 1859, there have been extraordinary attempts to explain it. Many of these took the form of unseen planets or missing mass, but most involved modifications to the gravitational potential. An excellent treatise on most of these attempts is made by Roseveare.⁽⁵⁾ There have been no less than four very strong attempts to produce a velocity dependent form of the gravitational potential, most notably developed by Gerber, Weber, Maxwell and Ritz. Interestingly, with the exception of Einstein’s theory of general relativity, every such attempt failed either in accommodating Mercury’s anomaly, or photon deflection, or both. Many researchers theorized that the gravitational force likely propagated between bodies at a speed of c, and that this non-instantaneous action must be taken into account. Paul Gerber came very close to deriving an appropriate potential calculating the motion of the bodies during the time required for the force to propagate between them, but failed to fully justify his approach. He died shortly after its publication, and his method was later found to produce a value for photon deflection one and one-half times greater than the accepted value.

Newton's gravitational potential was developed by studying stationary or slowly moving objects (v<<c). As such, we can consider this to be the "static" gravitational potential. Since Newton assumed that gravitation involved instantaneous action-at-a-distance, he had no reason to concern himself with bodies in motion, as such motion would not affect instantaneous forces. In what follows, we will show that there is a velocity dependent term, derivable from the RCM model of light. This effect is shown to arise in much the same manner as Doppler effects in EM radiation or light. The modification takes into account that gravitational influences must have an effective velocity of c in the test-particle’s frame of reference just as observed light must have a velocity of c in the observer’s frame of reference. Thus, while Newton’s potential forms a starting point for discussion, the concept of instantaneous action-at-a-distance is abandoned. Once the full "dynamic" gravitational potential is developed, we will apply it to an analysis of Mercury's anomalous perihelion advance and the deflection of a solar grazing photon, and show that the form and results of the solutions are identical to those derived utilizing GRT.

2. The Gravitational Red-Shift

For an observer moving directly away from the source, with increasing distance represented as a positive velocity, the observer will be sensitive to that component of light that leaves the source at a velocity of c+v, as measured with respect to the source. In what follows, we will refer to the velocity of light measured with respect to the source as c’, while the velocity of light observed is always c. The red-shifted frequency due to such motion is equal to the ratio of c, the observed velocity, to the initial velocity component of light emitted with respect to the source, c’, times the source frequency, or:

(1)

Although it can be demonstrated that any given component of light increases its velocity as it leaves a gravitational field⁽¹⁾, the light also loses energy, as measured by a receiver (atom or clock, for example) located outside the field, accounting for the gravitational red-shift experienced by the observer. The Pound-Rebka-Snider experiment demonstrated that we can counter the effects of the gravitational red-shift by moving the observer toward the source at a certain velocity⁽⁶⁾. The velocity required is such that the increase in energy due to the motion induced Doppler blue shift exactly counters the loss in measured energy due to the gravitational red-shift. If the height of the Pound-Rebka test apparatus is h, then the velocity required will be given by:

(2)

where g is the acceleration due to gravity near the surface of the earth.

From this relation, we can express the formula for the gravitational red-shift in the form of equation (1). If we consider an observer stationary in the gravitational field, the velocity component of light with respect to the source is c, and the shifted frequency is given by replacing v in equation (1) with the expression of (2):

(3)

As a check on the validity of the form of equation (3), we can derive the standard formula for the gravitational red-shift for a uniform gravitational field. We assume that we are on a non-remarkable star or planet, thus gh<<c² and g = GM/R², and for simplicity of derivation, we let the distance traveled by the light, h, be small compared to our distance from the center of mass, thus (1/R₁-1/R₂) is approximately h/R². The derivation then becomes:

(4)

In what follows, we will use g_r’ and g_t’ for the acceleration due to gravity as experienced by radially and transverse moving observers, respectively. We will verify whether, for each type of motion (radial and transverse), g’ = g. If we consider an observer moving radially away from the source with a velocity v, the observer is sensitive to light leaving the source at a velocity of c’ = c+v (Fig. 1). For the moving observer carrying a receiver with him, the height, h, traversed by the light, depicted in the figure, is derived by the following expressions in (5). In (5), the height of the apparatus plus the radial distance moved must equal the velocity of light with respect to its source, c’, times the time required to reach the receiver:

(5)

In equation (3), the denominator contains the velocity of light measured with respect to the source, c’, while the numerator contains the velocity as observed, always c. Using the proper value for the velocity of light with respect to the source, c’, and the effective value for h in equation (3) yields:

(6)

Thus, the total red-shift in received energy is equal to the expected motion-induced Doppler shift (given by the first factor in the last expression) times the expected gravitational red-shift (given by the second factor), which implies g_r’ = g. From this we can deduce that radial motion has no effect on the measured gravitational red shift or on the effective gravitational potential for such motion.

Next we consider an observer carrying a receiver and moving transversely to the gravitational field. As can be seen in figure 2, the effective height, h, does not change. Since we are interested in the component of light that strikes the receiver with apparent perpendicular incidence, the aberrated velocity of light with respect to the source, c’, is given by:

(7)

The reason we are interested in light that appears to be perpendicularly incident to the receiver is that we are going to develop the equivalent gravitational relation, and we are interested in that force that appears in the reference frame of the test particle to be along the line joining the gravitational source and the test particle. In other words, all gravitational force is directed along this line. The gravitational force vector always appears radial, with no transverse components. Thus, the total equation for the change in received energy for light observed at an aberrated angle of ninety degrees to the direction of motion is given by:

(8)

In this case, the change in energy is equal to the product of the transverse Doppler shift for apparent perpendicular incidence, given by the first factor, times the gravitational red-shift, given by the second factor. However, if the above equation were correct, unlike the case of radial motion, the gravitational red-shift in energy observed in the moving system would be different from that measured at the same point in the field by a stationary observer, due to the extra factor (1+v²/c²) in the denominator. We must consider the implications of this factor closely.

Imagine two observers in free-fall in empty space, far removed from any gravitational fields. Allow one observer to have an initial velocity, v, with respect to the other observer. Further allow the moving observer to pass very near the stationary observer, at the same instant that each observes the light from a distant source. If each observer records the frequency of light received from a distant source, then, after accounting for the motion induced Doppler shift experienced by the moving observer, they will each obtain the same frequency for that light. Now, the principle of equivalence tells us that the same experiment performed by these same two observers in free-fall in a strong gravitational field should obtain the same results as they did in free-fall far removed from any gravitational fields. Thus, the observed shift in frequency due to the gravitational field alone (after accounting for motion induced Doppler effects) should be the same for both observers. In order for this to be the case, equation (3) and the right hand side of equation (8) must produce the same result--each observer will measure the same value for the gravitational red-shift. From this, the following relation becomes immediately apparent:

(9)

If we now replace this transverse motion v with , and append the dynamic adjustment of equation (9) to Newton's static potential, we arrive at the dynamic gravitational potential for a moving observer:

(10)

One might argue that the principle of equivalence doesn't actually go as far as to imply that a free fall observer with an initial velocity should see the same gravitational red shift at a particular point in space as does a stationary observer. Therefore, we will present a gedanken experiment that illustrates that if such is not the case, a contradiction will occur. It has been demonstrated experimentally that clocks in a gravitational field slow down according to a relation identical in form (though inverted) to that for the gravitational red shift⁽⁷⁾. In other words, the amount of slowing depends only on the strength of, and thus the position in, any particular gravitational field. It has also been demonstrated that a clock placed in motion slows down according to a well-defined formula dependent only on the velocity of the clock relative to its rest frame.

Now, suppose we have two, identical, synchronous clocks far removed from a gravitational field. We place one clock on the outside of a large turntable, while the second clock sits stationary at the center. With the turntable at rest, we measure the rate of the clock on the rotor edge by passing a signal from that clock to the clock at the center, and we synchronize the two clocks. We now spin up the turntable so that the clock on the edge of the rotor attains a large, fixed speed of rw with respect to the stationary clock, as depicted in Figure 3. We now find that, due to its attained velocity, the clock on the rotor’s edge is running slowly as measured by the clock at the rotor’s center, as has been confirmed by experiment using the Mossebauer effect. We measure and record the rate of slowing, and find it to be the expected value for a fixed speed of rw.

Next we lower the still rotating turntable and both clocks into the gravitational field. We note that the rotor edge clock is still running slow due to its velocity with respect to the center clock, even though both clocks have also slowed due to their presence in the gravitational field, and we again measure and record the rate of slowing. We compare the rate of the rotor edge clock as determined by the center clock in each case.

The moving clock was at the same point in or out of the gravitational field as was the stationary clock at the time of each rate measurement. Thus each such measurement should have determined the same rate of motion induced slowing for that clock, since the gravitational slowing would be the same for the moving and the stationary clock. Suppose instead the gravitational red-shift for the observer with an initial velocity is different than that for the stationary observer. In this case the gravitational effect on the moving clock will also not be to the same degree as that on the stationary clock, since the two effects derive from the same equation. Thus the rate of slowing of the rotor edge clock as measured by the center clock while both are in the field will be different than the rate measured when both clocks were far removed from the field. This contradicts the original assumption and analysis that showed that the slowing of a clock due to a gravitational field is due only to the strength of that field and the distance from the source of that field, not the velocity through that field as well, and that the slowing due to velocity is due only to the magnitude of the velocity, and not the presence of a gravitational field.

From the above, we see that the gravitational red shift as measured by a moving observer at a certain point in the field must be the same as that measured by a stationary observer. Consequently, Newton's static gravitational potential must be modified to reflect the effects of motion through a gravitational field as is done in equation (10).

Since we wish to consider the case of planetary orbits and solar grazing photons, we obtain the following relations regarding angular momentum, l:

(11)

Substituting equation (11) into (10) yields a useful form of the dynamic gravitational potential for a moving observer:

(12)

We can now write directly the force due to the potential of equation (12), where Gm₁m₂ has been replaced by k: ⁽⁷⁾

(13)

Utilizing a change of variables to u=1/r, and applying the LaGrangian treatment of motion to equation (13) yields:

(14)

This is, of course, the exact form derived under general relativity. The importance of equation (14) lies in the fact that it was derived without resort to space-time curvature proposed by general relativity.

Equation (14) has the form of an elliptical orbit plus a small perturbation term. If we use a solution of the form:

, (15)

Equation (15) shows that with each orbit of the planet, the perihelion will advance by an amount given by:

(16)

Equation (16) is of course the exact equation derived in relativity theory, and results in an anomalous perihelion advance for Mercury of 43" per century, as confirmed by observation.

Beginning with the equation of motion derived from the dynamic gravitational potential, and considering a photon approaching from , we see that , and we can neglect the first term on the right-hand side of equation (14)⁽⁵⁾. Thus, we can assume a homogenous solution of the form:

(17)

and obtain:

(18)

In equation (18), d is the tangent of the angle of deflection on one side of the sun as in figure 4. Since the value of d is very small, the tangent of the angle is almost identical to the angle itself, expressed in radians. The total deflection of the photon then, in radians, is 2d, or:

Total deflection = (19)

Equation (19) is the exact value obtained in general relativity theory and results in 1.75" of deflection as confirmed by observation.

Acceptance of Einstein's second postulate requires the introduction of length contraction and time dilation due to motion, culminating in the special theory, along with the concept of Lorentz invariance, as initially applied to Maxwell’s equations. That theory, combined with the principle of equivalence, requires the introduction of space-time curvature in the presence of massive objects, carrying with it the idea that all fields must be Lorentz invariant. The RCM theory has been shown to eliminate the need for length contraction, time dilation and Lorentz invariance as specified by Einstein (the Galilean transform, after all, is Lorentz invariant if the upper limit on attainable velocities is assumed infinite rather than capped at c). In this paper it has been demonstrated that the RCM model of light combined with the principle of equivalence results in a modification to Newton's static gravitational potential. Instantaneous action-at-a-distance is supplanted by effects propagating at a speed of c with respect to the observer. Applying this modified potential to the case of Mercury's anomalous perihelion advance accounts for the entire 43" per century, and provides a form equivalent to that of GRT. Thus this model will work equally well for any system to which it is applied, without resorting to the space-time curvature of GRT. The same potential, applied to the problem of a solar grazing photon, gives a solution identical in form and content to that of GRT as well. Such results lead to the conclusion that these effects are not attributable to massive objects curving space-time. In fact, the results are derived in the same manner as, and appear to be analogous to, the Doppler type effects experienced by moving observers of EM radiation or light.

7. J. B. Marion, Classical Dynamics of Particles and Systems, (Harcourt and Brace Publishers, 1970), p. 266-270.