Why gravitational and inertial mass are equal

Physicists have no idea why the gravitational mass of a given body is equal to its inertial mass. Until this basic problem of classical physics is sorted out, it is most unlikely that gravitational theory can be united with quantum theory. The fundamental interpretations of gravitational theory have hardly changed since 1916, whereas the interpretations of quantum theory have advanced by leaps and bounds over the years since 1916. It is therefore likely that the problems with the unification of the two theories will be solved by a better understanding of gravitational theory.

Gravitational and Inertial Mass

Because we are so familiar with the concept of the mass of a body, we tend to forget that there is a fundamental difference between the gravitational mass of a body and its inertial mass. Consider a simple pendulum:

When the pendulum is given a small push, to start it swinging, the pendulum bob would continue to revolve continuously forever, in a vertical circle, (assuming the right sort of pivot), if it were not for the effects of gravity and friction. The restoring force, which makes the pendulum swing to and fro, is the force of gravity acting on the bob. When one calculates the formula for the periodic time of swing of the pendulum, one finds that the periodic time depends not only on the length of the pendulum and the acceleration due to gravity, but also on the ratio of the gravitational mass of the bob to its inertial mass. However, whatever material is chosen for the bob, this ratio is always found to be unity. Hence, experiment shows that the gravitational mass of any given body is equal to its inertial mass.

Why should all of the different atomic particles within a body interact in just the same way with a gravitational field as they do when the surface of the body is given a push? There is no agreed explanation. We have had to accept that the gravitational mass of a given body is identical to its inertial mass.

But we can go further that just accepting this equality as a fact. When it is found that any two items are identical, we always deduce that the two items must be fundamentally related to one another and have a common origin.

The obvious question then arises, how may the two identical forms of mass be considered to have a common origin? If one accepts Einstein’s statement concerning Mach’s Principle, then one is left with only two options. Either every local inertial reaction force must be caused by an instantaneous gravitational interaction with distant matter (the currently accepted viewpoint of many physicists), or the value of the gravitational constant G must result from the motion of local matter relative to distant matter.

Up until now everyone has considered it ridiculous to suggest that the large-scale inertial motion of matter could generate gravitational forces. However, this idea may not be so outrageous as it appears at first sight.

In 1964 Hoyle tried to partially introduce Mach’s ideas by suggesting that the mean density of matter throughout the Universe might establish the value of G. Hoyle predicted that if the mean density of the Universe halved then the value of G would double. It seems logical to extend Hoyle’s idea to include the motion of distant matter, as well as its density, when predicting G.

A number of motions of distant matter, that would fully incorporate Mach's Principle and establish a specific free-space value for G, are possible. Alternative motions of distant matter range from the radial motion associated with the Hubble expansion of all matter in the Universe, to the rotational motion associated with the possible rotation of the Universe as a whole. In one of two possible models I have proposed in the past (see Appendix, Paper 1) it was suggested that the rotation of a galaxy, relative to the distant galaxies in the Universe, might generate the value for G that applied just within that galaxy. Such a model might have enabled one to reject both dark matter and dark energy at a stroke! However, comments from readers of my web site have shown that recent astronomical observations may contradict such a galactic model on two grounds.

First, it is now known that irregular galaxies do not rotate significantly, and yet they have a vigorous ongoing star formation that conforms, according to current theory, with the usual value of G. Secondly, our galaxy has only rotated through about 30 revolutions since it was formed. But, according to current theory, the spiral arms should not have had time to form within so few revolutions! Hence the galaxy is not a good starting point for us to commence any search for the origin of gravity.

Current astronomical evidence appears to be clear. When related to the currently accepted models for the formation of stars in our Universe, all astronomical observations indicate that the free-space value of G, within the whole of the observable Universe, has been virtually constant for the past 10¹⁰ years. This requirement puts severe restrictions on any inertial model for the origin of G. But all is not lost. We are becoming familiar with the fact that the Universe we observe may be a smaller entity than we have previously thought, and there may be many universes within a much wider horizon.

If we wish to relate the creation of the particular free-space value of G in our Universe to an inertial motion relative to distant matter, then the most likely model is one where the value of G is being directly created by the rotation of our whole Universe against the background of other distant universes. It is unlikely that we should ever be able to detect this rotation directly, and there are difficulties with this model. Nevertheless, in view of the problems with all of the alternative interpretations of Mach’s Principle, it is worth pursuing.

There may be further consequences if gravity is created in this way. It is possible that small perturbations in the observed local value of G might occur within the boundary surface of a near-fluid body, such as the Earth, which is rotating relative to the distant matter in just our Universe.

First, a possible observable variation of G was proposed in Paper 1 (see end of this section). It was suggested that the rotation of an idealized fluid Earth might produce an abrupt, discontinuous, increase of about 0.4% in the value of G when crossing the boundary surface into the interior. For the real, near-fluid, Earth it is likely that most of this 0.4% increase of G will occur within a shell that straddles the surface level of a hypothetical, idealized, fluid Earth. The thickness of this shell might vary from around a few meters in land areas close to sea level, to a few kilometers in moderately hilly areas.To be certain of observing this spatial change in G two separate measurements of G would have to be made at points about 1km above and below sea level, at carefully selected sites. Many recent measurements of G have differed by up to 0.6% from the expected mean value.

But there may be further consequences if the shell is only a few meters thick at some particular land areas. Such land areas might be flat land areas close to sea level in elevation. As the Earth rotates and orbits the sun, some of these locations, which are fixed on the Earth’s land surface, may pass back-and-forth across the hypothetical boundary surface of an idealized, fluid Earth. At these points a monthly and annual variation in the value of G might be anticipated.

I analysed all twenty six pairs of the original laboratory test results for G published by the National Bureau of Standards, Washington in 1930 and 1942 (L. M. Stephenson, Proc. Phys. Soc., 90, 601, 1967, and Found. Phys., 6, 143, 1976). The analysis demonstrated that a 0.2% sinusoidal annual variation in G at Washington, with a maximum occurring at the vernal equinox and a minimum at the autumnal equinox, reduced the spread of the mean values of the three sets of G results published in 1930 by a factor of 3. For two more accurate sets of G results, published in 1942, the spread of the mean values of the two widely separated sets was reduced by a factor of 12. A statistical analysis shows that the probability of these reductions in the spreads of the means arising by chance, if an arbitrary variable had been applied, is less than one in ten thousand. The probability of the existence of an annual variation of G at Washington is therefore significant.

It is interesting to note that, when publishing the 1942 G results, Heyl and Chrzanowski concluded with the remark: “...what is needed to account for the observed anomaly in the results with the two (suspension) filaments is a regular variation of such nature as to be incredible.” Heyl and Chrzanowski could not have seriously considered the possibility of a 0.2% annual variation of G, as they would have assumed that this variation would also require a totally unacceptable 0.2% annual variation in the acceleration due to gravity g. It is well known that pendulum clocks do not vary in their timekeeping by 0.2% over the course of a year!

But most of the proposed 0.4% increase of G is predicted to occur only when crossing a narrow shell, ranging in thickness from a few metres to a few kilometres at the Earth’s surface. This specific local increase of G will produce a much smaller increase in the value of g. The value of g depends on the mean value of G taken over the total path length between the centre of the Earth and a test particle. The related increase of g will be about 1 part in 10⁶ when crossing a 1km thick shell at the Earth’s surface. It is important to realize that this suggested change in g is much smaller than, and additional to, any normal change of g which occurs with variations of height both above and below the Earth’s mean surface level.

For the particular flat land area locations close to sea level, which might lie within a possible 10m thick shell, the related annual variation of g within the shell will be about 1 part in 10⁸. This annual variation of g might be observable by comparing the readings of two gravimeters. The accuracy of gravimeters is now of the order of 2μGal, which is equivalent to a change in g of 2 parts in 10⁹. If one gravimeter were to be located at the site in Washington where Heyl and Chranzanowski undertook their G experiments, then this gravimeter might show an annual variation in its reading of about 10μGal for the assumed shell thickness of 10m at this point. There would be a maximum at the vernal equinox and a minimum at the autumnal equinox. A second identical gravimeter, located in the same region, but at a height of at least 100m above or below the first gravimeter, would show negligible annual variation in its reading due to this specific effect in the 10m shell. The proposed annual variation of g would be additional to the normal annual variations of g, due to solar tides, of about 40μGal. However, the solar tide effect produces a minimum value for g at both equinoxes.

Within the past sixteen years many measurements of G have been made, in laboratories across the world, that show some unexpected variations of G of up to 0.6%, when errors of 0.05% were expected. It would be worthwhile to study these results to see if any annual variations of G were present for those laboratories that were close to sea level, and also to check whether these variations were dependent on latitude.

It was also shown in Paper 1, using equation (4), that the spin of an isolated electron is predicted to create a very large value for G within the electron that results in an internal gravitational attraction force which is 4,000 times greater than the internal electrostatic repulsion force (see page 38 for the calculation, where: 1/4α² ≈4000). The radius of an isolated electron must then be very much smaller than the classical radius r_e. This predicted stability of an electron when outside an atom, and the predicted near point-size, agrees with experiment but has never been explained previously. Any spinning atomic particle will be similarly stabilized gravitationally, in a way that Einstein believed should occur. For a proton, assuming initially a radius of r_e = 3x10^-13 cm, the gravitational force is predicted to be only about twice the strength of the electrostatic repulsion force. Hence, the radius of a proton should be slightly less than the assumed figure r_e, again agreeing with experiment.

In obtaining the gravitational stability result for the electron on page 38 classical values for r_e and ω_spin have been used. I defend the use of a classical value for this case. “The complementary views provided by both classical and quantum pictures are both essential to the understanding of nature” (a quote by Freeman J Dyson concerning the analysis of the uranium 236 nucleus). The limitations imposed by Maxwell’s equations do not apply to the peripheral spin velocity of an electron.

One is left with the important deduction that if the internal stabilizing force for all spinning atomic particles is a gravitational force then an initial link has been established between quantum theory and gravitational theory.

Specific gravitational origins for the two remaining fundamental forces, calculated using the same equation (4) in Paper 1, have also been proposed (L. M. Stephenson, J. Phys. A, 2, 475, 1969). By inserting the angular velocity of the electron in the first Bohr orbit, and the classical density of the electron, one obtains a gravitational force that equates with the magnitude of the weak force. By inserting this same angular velocity, and the density of the electron when spread out over the volume of the Bohr atom (which simulates the wave nature of the electron) one achieves a gravitational force that equates with the magnitude of the electromagnetic force within an atom. Hence, by combining the only two angular velocities of the electron in the Bohr atom with the only two densities of the electron in the atom, a single equation predicts three specific gravitational forces within the atom which correspond to the known magnitudes of the weak, electromagnetic and strong forces. It is inconceivable that these three correspondences arise by chance.

Details of the calculations from equation (4) are given below.

Conclusions

If one accepts Einstein’s statement concerning Mach’s Principle, then one is left with only two viable options. Either every local inertial reaction force must be caused by an instantaneous gravitational interaction with distant matter, or the value of G must result from the motion of local matter relative to distant matter. For the second option, the most likely relevant motion would be a rotational motion of our Universe relative to distant universes. Some observable, local variations in G might occur.

A common gravitational origin for each of the four fundamental forces in nature is predicted, based solely on equation (4) in Paper 1. The derivations are shown below. All of these forces appear to arise from the rotational inertial motion of matter. The usual gravitational force probably arises from the rotation of the Universe as a whole. Weak, electromagnetic and strong forces all seem to be additional, localized gravitational forces that arise directly from the spin and orbital angular velocities of electrons, and other atomic particles, within atoms.

Derivation of Gravitational Coupling Constants

Equation (4) from Paper 1 predicts the free-space value of G as:

                                           G = ω²/2πρk                       (4) 

where ω is the angular velocity of the Universe, ρ is the mean density, and k is a boundary shape constant (≤1). It is proposed that the free-space value of G is being generated by the rotational motion of the Universe.

          Other values of G may be generated within the boundary surfaces of smaller rotating bodies. For the values of ω which apply to an electron’s orbital and spin angular velocities in a Bohr atom this single equation predicts three further localized values of the gravitational constant at the atomic level, denoted by n = 2, n = 3 and n = 4.

The last three coupling constants correspond with the relative magnitudes of weak, electromagnetic and strong forces. They arise from a single equation when combining the three viable combinations of ω_orbit , ω_spin , ρ_e and ρ_e(r_e/a_o)³. Hence, a common gravitational origin for the weak, electromagnetic and strong forces within an atom is indicated.

For an electron within the two closely related regions, designated by n = 2 and n = 3, there is an indication that the coupling constant for the combined electromagnetic and weak forces (the electroweak force) may range in value from 10^-13 to 1.

For an isolated electron, outside an atom, the region designated by n = 4 is relevant. The value of the gravitational constant within the boundary surface of an electron is predicted to be G_s. As mentioned earlier, the gravitational attraction forces within an electron will then be 1/4α² (≈4000) times stronger than the electrostatic repulsion forces. Hence the theory requires an electron to be both inherently stable and have a radius which is very much less than the classical radius r_e. These results are confirmed by experiment.

The stability problem of the electron has been consistently ignored in main stream physics. Vague suggestions have been made that some unspecified quantum effects within an atom must be present that would explain the stability of both electrons and protons, and would ensure a point size for electrons. But what about the free electrons that are present in electron beams and in the conduction bands of metals? It is essential that the stability problem of the electron should be treated seriously. Einstein was convinced that gravitational forces were needed to account for the stability of electrons and protons. The above analysis suggests that his instincts were correct. Additionally, a single equation has been proposed that unites all four fundamental forces.

Paper 1:
A Dynamical Origin for the Gravitational Constant that Explains Gravitational and Inertial Mass Equality and Rejects Dark Matter and Dark Energy