The Gravitational Potential for a Moving Observer, the Perihelion Shift of Mercury, and Photon Deflection

Introduction

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 [3], and to support all experimentally verified Doppler shift results.

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. In what follows, we will show that there is also a velocity dependent term, derivable from the RCM model of light. 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.

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 which leaves the source at a velocity of c+v. 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:

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 [4]. 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:

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 with respect to the source is c, and the shifted frequency is given by replacing v in equation (1) with the expression of (2):

As a check on the validity of the form of equation (3), we can derive the standard formula for the gravitational red-shift as follows, where gh<<c², g = GM/R², and (1/R₁-1/R₂) is approximately h/R²:

The Gravitational Red-Shift for a Moving Observer

If we now consider an observer moving away from the source with a velocity v, the observer is sensitive to light leaving the source at a velocity of c’ = c+v. 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 what follows, we will use g’ for the acceleration due to gravity as experienced by moving observers, and verify whether, for each type of motion (radial and transverse), g’ = g. Since the proper values for the velocity of light with respect to the source, c’, and the effective value for h in equation (3) yields:

Thus, the total red-shift in received energy is equal to the expected motion-induced Doppler shift (given by the first term in the last expression) times the expected gravitational red-shift (given by the second term). 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 the figure, 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:

The reason we are interested in light which 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 which appears in the reference frame of the test aprticle 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 is given by:

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 term, times the gravitational red-shift, given by the second term. 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.

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 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 backing out motion induced Doppler effects) should be the same for both observers. In order for this to be the case, equations (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:

If we now replace this radial motion v with , and append the dynamic adjustment of equation (6) to Newton's static potential, we arrive at the dynamic gravitational potential for a moving observer:

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, so we will present an example 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] for example). 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 ([1] and [7] for example).

Now, suppose we have three, identical, synchronous clocks, far removed from a gravitational field. We hold one of these clocks in space, stationary with respect to a distant, gravitating body. We lower another of the clocks into the field and leave it there, at rest with respect to the stationary space clock, though far removed from it and deep within the field. The clock in the field will tick more slowly due to its presence in the field, according to the strength of the field at that point. Next we accelerate the third clock outside the field to some known initial velocity with respect to the stationary clocks outside and within the field. Once this velocity is reached, we allow this clock to pass very near the stationary clock outside the field and note that this clock is running slow due to its velocity as measured by that clock, and we measure the rate of slowing. We allow the moving clock to continue on with the same velocity until it passes the stationary clock inside the field. We note again that this clock is running slow due to its velocity with respect to that clock, even though both clocks have also slowed due to their presence in the gravitational field, and we measure the rate of slowing. We compare the rate of the moving clock as determined by each of the stationary clocks.

Since the moving clock was at the same point in or out of the gravitational field as were the stationary clocks at the time each measured the moving clock’s rate, each of these clocks 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. However, if the gravitational red-shift for the observer with an initial velocity is different than that for the stationary observer, then the gravitational effect on the moving clock will also not be to the same degree as that on the stationary clock. This contradicts the original assumption and analysis which 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. Therefore, 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, and we see that 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:

Substituting equation (11) into (10) yields (ignoring terms of higher order than 1/c²) a useful form of the dynamic gravitational potential for a moving observer:

We can now write directly the force due to the potential of equation (11), where Gm₁m₂ has been replaced by k [2]:

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

This is, of course, the exact form derived under general relativity.

The Perihelion Shift of Mercury

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

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

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.

The Deflection of a Solar Grazing Photon

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). Thus, we can assume a homogenous solution of the form:

and obtain:

In equation (19), d is the tangent of the angle of deflection on one side of the sun. 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 =

Equation (21) is the exact value obtained in general relativity theory and results in 1.75" of deflection as confirmed by observation.

Summary

Acceptance of Einstein's second postulate, combined with the principle of equivalence, requires the introduction of space-time curvature in the presence of massive objects. In this paper it has been demonstrated that the RCM model of light results in a modification to Newton's static gravitational potential. 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, all 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.

References

[1] Renshaw, Curtis, E., 1996, "The Time Delay of a Solar Grazing Photon," IEEE: Aerospace and Electronic Systems Magazine, August.

[2] Marion, J. B., Classical Dynamics of Particles and Systems, Harcourt and Brace Publishers, New York, 1970, pp. 266-270

[3] Renshaw, Curtis E., 1996, "The Radiation Continuum Model of Light and the Galilean Invariance of Maxwell's Equations," Galilean Electrodynamics, Volume 7, 1

[4] Pound, R. V., Rebka, G. A., Jr. and Snider, J. L., 1965, "Effect of gravity on gamma radiation," Phys. Review, vol 140, B788-803

Questions? Comments?crenshaw@teleinc.com