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Key Words: Gravity, inertia, Mach's Principle, relativity, velocity of light, clock rate, mass, conservation of energy and momentum.

1. INTRODUCTION

Newton advanced three laws of motion and a separate law of universal gravitation. With these he explained and predicted the orbits of all the planets and their moons⁽¹⁾. However, Newton refused to speculate regarding the cause of these laws. Einstein extended Newton's laws by equations that predicted the effects of motion (SR) and the gravitational field (GR) on the rate at which time passes, as measured by clocks^(2,3). Einstein also extended the relativity principle, assumed by Newton, to include measurements of the velocity of light, which he predicted would give the same result in all directions and in all inertial frames. The predictions of both SR and GR have been confirmed by a large body of observations⁽⁴⁾. But, like Newton, Einstein did not speculate regarding the cause of the effects that he predicted. Nevertheless for the purpose of predicting observations, Einstein showed it was not necessary to assume, as Newton had done, that absolute space and time existed.

The equivalence of gravitational and inertial masses was noted by Galileo and assumed by Newton. An exact equivalence has been demonstrated in many experiments⁽⁴⁾ and was made the basis of GR. Gravity and inertia may be considered "forces"⁽⁵⁾ that involve both an exchange of energy and a change in the rate of atomic clocks. However, there is one important difference between the two which played an important part in the development of our proposal. With gravity the loss of energy from matter particles is associated with a slowing of atomic clocks, whereas with inertial motion, the clock slowing occurs with a gain of energy. But in both cases there is a strict correlation between energy loss or gain and clock slowing.

The above correlations suggest that gravity and inertia may be causally related. Since inertia appears to be independent of the presence of a gravitational gradient, it would seem unlikely that gravity causes inertia or vice versa. Thus, we are left with the third possible type of causal relationship -gravity and inertia have a common causative factor. The purpose of this paper is to propose such a common cause. To this end we postulate that the energy exchange common to both forces is derived from a change in the energy content of matter particles, proportional to a change in a length dimension of the particles. This in turn is determined by the velocity of light and the motion of particles relative to a preferred and isotropic light speed frame.

2. AN EXPANDED MACH'S PRINCIPLE

As evidence for the existence of an absolute space Newton cited the behavior of a bucket of water, in which the level of water at the edge rises when the bucket is rotated. But Newton's postulation of absolute space was criticized by Bishop Berkeley⁽⁶⁾ on the philosophical grounds that inertial and centrifugal forces are observed to occur with acceleration of motion relative to the stars. This suggestion was adopted by Ernst Mach⁽⁷⁾ in the development of his philosophy of Logical Positivism. Neither Berkeley nor Mach proposed any mechanism whereby distant stars could exert a force on moving objects. This was consistent with Mach's philosophy that science should deal only with observables, but the absence of a mechanism led to considerable skepticism in the scientific community. The choice was between an unobserved and an unimagined mechanism. Nevertheless, Mach's proposal has generally become known as Mach's Principle.

Mach's Principle implies that inertial energy is somehow related to motion relative to the rest of the universe. Einstein attempted to incorporate Mach's Principle into GR and he was partially successful with respect to rotational motion. He predicted that the rotation of the earth would have a dragging effect on the frame of reference produced by the stars. A test of this aspect of Mach's Principle is planned with the Stanford gravity probe⁽⁸⁾ which involves a highly accurate and stable gyroscope capable of detecting the tiny dragging effect of the earth's rotation. If this dragging effect can be observed, it will demonstrate that the framework of space is not absolute as postulated by Newton, but is created by an effect of local and distant matter.

Although neither Mach nor Einstein proposed a mechanism whereby the distant stars influence motion here, we have proposed that the integral of all the gravitational potentials from the entire universe not only has an observable effect on the velocity of light, but also creates a preferred frame of reference within which such velocity is isotropic. This extension of Mach's Principle has an advantage over Newton's absolute space in that it explains the identification of rotation with motion relative to the stars. And the postulate that the velocity of light is controlled by the gravitational potential is supported by experiments that measured the time delay of light passing near to the sun⁽⁹⁾. In the next three sections we will develop the rationale for the postulate that a preferred inertial frame exists in which the velocity of light is isotropic and explain why this frame cannot be detected in laboratory experiments.

3. RELATIVITY AND THE CONSERVATION OF LINEAR MOMENTUM.

The same logical situation exists today with respect to anisotropy in the velocity of light and changes in clock rate and measuring sticks. Rotation relative to the stars can be detected with the Sagnac effect⁽¹²⁾, but a preferred isotropic light frame cannot be detected in laboratory experiments.

4. RELATIVITY AND THE VELOCITY OF LIGHT

The velocity of light is clearly not affected by the velocity of its source,as demonstrated by observations of binary stars⁽¹³⁾. Within the narrow limits of our experience and technical ability, the measured value of the velocity of light is always the same. If, like Einstein, we assume that this will always be true, then the observation of the Shapiro time delay⁽⁹⁾ indicates that our clocks and measuring sticks are altered by a change in gravitational potential to exactly compensate for the change in the velocity of light. Because the observed fractional change in velocity of light is twice the observed fractional change in clock rate, there must be a symmetrically equivalent change in measuring stick length to explain the total compensatory effect on the measured velocity of light.

If an isotropic light speed frame exists, why is it that linear motion with respect to that frame cannot be detected in laboratory experiments? Our answer is the same as that given by Fitzgerald, Lorentz and Poincaré⁽¹⁴⁾, namely that changes in the instruments that measure time and distance exactly cancel the expected effects. In fact, Einstein⁽²⁾ calculated the expected effects of motion on clocks and measuring sticks by first assuming that the measured velocity of light would be the same in every inertial frame, and then he derived the changes in clock rate and distance that would have to exist. So it is not surprising that when these changes are observed, they are exactly those that would hide the isotropic light speed frame if such a frame did exist.

The constancy of the velocity of light is based on two-way measurements. This is necessary because the changes in clock rate caused by motion make it impossible to synchronize clocks in a way that would distinguish between absolute and relative motion. In fact, if one assumes any particular velocity of the earth relative to a putative isotropic light speed frame, the calculated change in clock rate induced by moving one of two synchronized clocks almost exactly cancels the expected difference in the two one-way velocities. A small residual remains that is proportional to the square of the velocity and independent of direction⁽¹⁵⁾. This is true regardless of the inertial frame from which one calculates velocity. For this reason an experiment in which atomic clocks were transported slowly over a 20 km course failed to show a significant variation in the velocity of light in the two directions⁽¹⁶⁾. The global positioning system (GPS) used to pinpoint accurately positions on the earth uses atomic clocks that are synchronized and corrected for their motion relative to the non-rotating earth. But any other inertial frame would work just as well⁽¹⁷⁾.

5. DOES THE ISOTROPIC LIGHT SPEED FRAME EXIST?

Another set of phenomena that are easier to explain with the assumption of a preferred frame are those dealing with rotation with respect to the stars. These include Newton's bucket, the Foucault pendulum, the Sagnac effect, and the Michelson-Gale experiment⁽¹⁹⁾. Why should the non-rotation of material objects relative to the stars be the position of least energy? This is easy to explain if one assumes that the preferred frame is created by the stars.

Furthermore, the aberration of star light, first observed by Bradley⁽²⁰⁾, depends on the revolution of the earth around the sun, and is independent of the motion of the source star. When the source star is a member of a binary system that shows semiannual changes in its light spectrum (Doppler effect), there is no alteration in the amount of aberration⁽²¹⁾. While there are other explanations of this phenomenon, it follows from our model as an effect of the motion of the earth relative to the reference frame of the universe.

Finally, we take up the Hafele-Keating experiment⁽²²⁾, the slowing of clocks by to and fro motion⁽²³⁾, and the so-called twin paradox. The asymmetry of the travelling and resting clocks is usually explained by the asymmetry of their experience. The travelling clock undergoes acceleration, whereas the resting clock does not. But the clock slowing in the Hafele-Keating experiment was clearly proportional to the velocity and not to the acceleration. SR does not mention an effect of acceleration on clocks.

If a preferred isotropic light speed frame is assumed then all of these experiments are easily explained⁽¹⁵⁾. Thus we conclude that the universe is so constructed that it is impossible to detect the isotropic light speed frame that we believe exists. To paraphrase Copernicus, who was faced with a similar problem, we believe that the appearance is relativity, but the reality is an isotropic light speed frame⁽²⁴⁾.

6. ENERGY AND CLOCKS

The second basic postulate of our model deals with the energy exchanged in gravitational and inertial interactions of matter. Our model postulates that this energy is derived from and stored in the fundamental particles that make up matter. However, the model does not speculate a mechanism of energy transfer.

This energy postulate explains in the most direct way possible the exact equivalence of gravitational and inertial mass. It posits that the two masses are identical, being derived in both cases from the total energy content of matter particles. As long as the energy exchanged in both cases is proportional to this energy content, no difference between gravitational and inertial mass will ever be observed.

All matter is made of energy. The fundamental particles that make up matter can be created from energy, and energy can be retrieved from these particles by the recombination of the matter and anti-matter created. As far as we know, matter particles do not contain anything except energy.

All forms of energy (heat, kinetic, atomic, chemical, electromagnetic and gravitational) are assumed to be equivalent. Only the total energy is conserved. Electromagnetic and electric charges are thought to be produced by an exchange of photons. The strong and weak nuclear forces are also thought to be produced by an exchange of energy containing particles. Free energy always travels at the fixed velocity, c, and the acceleration of matter particles meets increasing resistance, which approaches infinity, as their velocity approaches that of light.

Although this is probably an over simplification, it is useful to consider that the energy within all matter particles is trapped in a standing wave⁽²⁵⁾. This means that the group velocity of the standing wave within particles is c and responds to the same forces that control the velocity of free energy. This explains why matter cannot travel at a velocity greater than c. The particle wave also acts as a clock. Although fundamental particles are subject to the uncertainty principle, the half-lives of unstable particles are reproducible and subject to the same effects of motion and gravity as any other clock.

An additional assumption regarding energy is that the amount of energy required to produce a particular stable particle (e.g. an electron) is precisely determined so that one electron is exactly like every other. Our model assumes that this precise amount of energy varies according to the velocity of light and the motion of the particle relative to the isotropic light speed frame. Momentum appears to be an intrinsic property of energy and for that reason it is also a property of matter.

Although GR was based on the assumption of an equivalence of gravitational and inertial masses, the theory does not give a satisfactory explanation of inertia⁽²⁶⁾. For this reason, inertial forces have been referred to as fictitious. But the energy associated with inertia is real and can be converted into electric power (e.g. hydroelectric). It has also been demonstrated that when particles are accelerated near the speed of light, they increase in mass⁽²⁷⁾ by an amount proportional to the energy added according to the formula:

Equation (1) reduces at low velocities to

m = m₀ + m₀v²/2c². (2)

Although the mass exchanged at low velocities is equal to mv²/2c², it is impossible to measure the small change in mass or determine whether the velocity relative to the universe has increased or decreased. However, it is possible to detect a change in clock rate (T) at low to and fro velocities, and this has been found to follow the same relation at both low and high velocities^(22,23), namely

Thus, it can be observed that at high velocities the mass of a particle is inversely proportional to its clock rate, as they are measured from the laboratory frame. Since the change in clock rate can also be observed at low velocities, it is reasonable to assume that the same relationship between mass and clock rate occurs at low velocities.

If we assume that the clock rate exhibited by a matter particle is a reflection of the vibration of an internal standing wave, then the mass of the same matter particle at different velocities is inversely proportional to the frequency of this standing wave. This means that the mass of the particle must be directly proportional to some length dimension of the particle, which we define as

L = kc/T (4)

where L is the length dimension, k is a constant, c is the velocity of light, and T is the clock rate.

We should emphasize the difference between the length dimension (L) and the deBroglie wavelength⁽²⁸⁾. The latter decreases as the particle's velocity increases. The deBroglie wavelength, reflecting the uncertainty of a particle's position, is obtained by diffraction, whereas the length dimension is obtained by dividing the velocity of light by the particle's clock rate. The distinction between the two is based on the observations that as a particle increases in velocity and its mass increases, there are reductions in both the frequency of its internal vibrations as determined by clock rate and its deBroglie wavelength as determined by diffraction.

If photons do not lose energy as they climb out of the gravitational hole, then the gravitational field does not exchange energy with photons. Since matter particles are constituted of energy, just like photons, there is no reason to think that the gravitational field exchanges energy with matter particles as they fall or climb in a gravitational gradient.

The fact that the fractional change in clock rate is one-half the fractional change in velocity of light has another interesting consequence. This makes the length dimension (L), as determined by (4), vary directly with the clock rate (T), instead of varying inversely with T, as with inertial motion. Thus, when a matter particle falls through a gradient of gravitational potential, its clock rate (T), the length dimension (L) of its internal vibration and its energy content at rest all decrease by the same fraction. Thus, we conclude that the energy content of a matter particle at different gravitational potentials is proportional to the length dimension of its internal vibration. Since the energy content of a moving particle is also proportional to the length dimension of its internal vibration, we conclude that a change in particle energy caused by the change in the length dimension of its internal vibration is the common cause of gravity and inertia.

9. DISCUSSION

In a random universe, correlations are an indication of causal relationships.

In most instances, the correlation is not exact, which is interpreted to mean that other causal factors or an element of chance enters into the relationship, e.g. the correlation between smoking and lung cancer. But in the case of gravity and inertia, there are three exact correlations, which we have interpreted to indicate strong, unifactorial relationships. These are:

(1) The exact correlation between the kinetic energy gained from a gravitational fall from one altitude to another and the difference in clock rate between these same two altitudes.

(2) The exact correlation between the energy required to accelerate a particle, the change in clock rate of the rapidly moving particle, and the increase in mass of the particle.

(3) The exact correlation between the gravitational and inertial masses of different matter particles, which formed the basis of GR. We interpret this correlation as evidence that gravity and inertia have a common cause, which strengthens the independent evidence that both gravitational and inertial energies are derived from changes in particle mass.

Our model explains all three correlations by postulating that the energy content of the same matter particle under different conditions of motion and gravitational potential is proportional to a length dimension of its internal vibrations. The necessity of making this proportionality related to the length dimension rather than to the clock rate derives from the fact that with gravity the energy content of the particle is proportional to the clock rate, whereas with inertia, the energy content is inversely proportional to the clock rate. Because of the difference in the velocity of light at two different gravitational potentials, the mass of the particle in both situations is directly proportional to its length dimension.

A major theoretical advantage to the model is that all the known effects of the gravitational field can be explained as secondary to the observed change in the velocity of light. The bending of light around the sun follows Fermat's Principle⁽³¹⁾ of the least time path⁽³²⁾ and is not due to a gravitational attraction of photons by the sun. In fact photons slow down as they near the sun as demonstrated by the Shapiro time delay. There is no evidence that a photon loses energy as it leaves the sun, since the gravitational red shift is completely explained by the observed slowing of clocks with gravitational potential. This correlates exactly with the reduction in the velocity of light. Our model adds to this by showing that the change in velocity of light can explain both gravity and inertia. The exact equivalence between gravitational and inertial masses is explained by the postulation that the energy exchanges common to both are proportional to changes in the energy content of matter particles. Because all the gravitational energy is incorporated into the particle, the model explains the Strong Equivalence Principle as well as the Weak Equivalence Principle.

A conceptual difficulty with the model is the sharp distinction it requires between appearance and reality. This is particularly true with different inertial frames, which have different energy levels in reality according to the model, but the appearance is that they are all the same. This is quite different from the situation with different energy levels in a gravitational gradient. Matter falls spontaneously from higher to lower levels of energy and converts "gravitational energy" to inertial energy. But when there are differences in inertial energy, nothing happens unless energy is exchanged in such a way that the direction of energy exchange is impossible to determine.

Another way of stating this objection to our model is that it requires the postulation of an isotropic light frame, which cannot be detected by laboratory experiments. There are two reasons that this objection might seem valid. The first is the positivist argument that physics should consider only observables and their mathematical relationships⁽³³⁾. The second is the demonstration by Einstein that an assumption of a preferred inertial frame is not necessary for any predictive purpose.

Our answer to these arguments is that we do not accept the positivist position. Many of the most important scientific concepts were proposed before they were directly observable, e.g. the gene, the virus, the atom and the quark. The question of what is necessary is a matter of opinion and purpose. If one's goal is to understand the mechanism of inertia and its relationship to gravity, then we feel the postulation of an isotropic light frame is appropriate. Einstein was not directly concerned with this relationship, and he avoided the necessity of assuming a preferred inertial frame by using a mathematical transformation (Lorentz). It is not our intention to dispute the accuracy of SR and GR in predicting observations. What we have tried to do is suggest a possible mechanistic approach to explain these same observations.

The history of science is replete with examples that demonstrate that an increase in the understanding of reality leads to further experiments and an increase in knowledge. Thus, while it is conceivable that our answers to the questions raised here are wrong, we cannot imagine that the questions themselves are unimportant.

10. TESTS OF THE MODEL

The model presented here was designed to explain the observations predicted by GR and SR that are available with current technology. For this reason it may be difficult to propose new tests at this time. However, there are two areas where possibilities exist.

The concept that the energies of gravity and inertia reside entirely within the matter particles involves a prediction that the mass of the earth would be independent of its distance from the sun. However, if the kinetic energy of gravitational fall is derived from the field as assumed in GR, then the earth should have a greater mass when it is nearer to the sun. This change in the mass of the earth should be detectable by modern laser ranging techniques that can measure the distance to the moon down to a few centimeters.

Another possible test of the model involves the intense gravitational potential in the vicinity of a putative black hole. If the velocity of light falls linearly, as assumed in GR, and the kinetic energy of a gravitational fall is derived from the particle. as we have postulated, then, when the gravitational potential reaches the Schwarzchild limit, matter would cease to exist. All of its energy would have been converted to kinetic energy and radiated. This seems unlikely to us, and it is still possible and consistent with observations that the velocity of light and the energy content of matter are inversely proportional to the gravitational potential. At high gravitational potentials the velocity of light would become very low but never reach zero, thus avoiding the disaster of a singularity⁽³⁴⁾. The possibility of observing such behavior around putative black holes cannot be excluded.

A third area, which does not seem promising at this time, is the relationship between the velocity of a particle and its size as determined by collision cross section. Although the model suggests that a particle's size might increase with velocity, a determination of collision cross section depends on many factors. For example, a proton has a smaller cross section than an electron, a fact that is explained by the decreased uncertainty of its position.

The authors are grateful for stimulating discussions with Ron Hatch and for the numerous criticisms and requests for clarification from three reviewers.

La gravité et l'inertie sont unies par l'emploi de deux postulats non verifés et des observations qui ont étés prédites par la théorie de la relativité speciale (SR) et générale (GR). Le premier postulat est une forme étendue du principe de Mach qui déclare que le potentiel de gravitation de toute la matière de l'univers engendre un cadre référentiel inertiel duquel l'effet premier est le contrôle de la vitesse istropique de la lumière. Le deuxieme postulat est que l'énergie des particules fondamentales est allié à une dimension de longue qu'est determiné par la vitesse de la lumière et la motion relative des particules et le cadre référentiel. Avec ce model, c'est possible d'expliquer la corrélation précise des masses gravitationaux et inertiaux et un différence de gravité et d'inertie. Avec la gravité le ralentissement des chronomètres est en proportion à un diminution de l'energie des particules fondamentales, tandis que avec l'inertie le ralentissement des chronomètres est en proportion à un augmentation de l'energie des particules fondamentales. Le modele exige aussi les supersitions que l'energie et du moment sont gardés.

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David W. Talmage and Richard J. Sanderson

The Webb Waring Institute for Medical Research

University of Colorado Health Science Center

Denver, CO 80262, U.S.A.