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 unlikely that gravitational theory can be united with quantum theory. The fundamentals of gravitational theory have hardly changed since 1916, whereas quantum theory has advanced by leaps and bounds. It is likely that the problems with unification will lie within gravitational theory and not with quantum theory.

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.

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, could the two identical forms of mass be fundamentally related and have a common origin? For example, could the inertial motion of distant matter in the universe actually generate local gravitational forces?

Up until now everyone has considered it ridiculous to suggest that the inertial motion of matter could generate gravitational forces. However, this idea may not be so outrageous as it appears at first sight. It conforms with Mach’s proposals, which have never been fully implemented.

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, the gravitational constant. 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 value for G, are possible. Alternative motions of distant matter range from the radial motion associated with the expansion of all matter in the universe, to the rotational motion associated with the possible rotation of the universe as a whole.The author originally suggested that the rotation of a galaxy, relative to the most 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 this 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, recent observations of many rotating spiral galaxies, like our own Milky Way galaxy, indicate that the orbital velocities of stars are nearly constant right across the galactic disk. This motion is remarkable and is unlike the orbits of the planets round the sun, where a planet's velocity is inversely proportional to the square root of the orbit radius. The suggested explanation to account for a near constant orbit velocity for the stars in the galactic disk is even more strange. Not only does one need to assume the existence of hidden dark matter to make the usual law of gravitation work for the orbits, but this dark matter must lie outside the galactic disk and must have a mass that is at least ten times larger than the visible mass.

A still further complication is that our galaxy has only rotated through about 40 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 to base any search for the origin of gravity.

Current astronomical evidence appears to be clear. When related to the currently accepted Big Bang origin for our Universe, and a Hubble rate of expansion, all astronomical observations indicate that the free space value of the 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. However, 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 value of G in our Universe to the inertial motion of distant matter, then the most likely model is to suggest that the value of G is being directly created by the rotation of our whole Universe against the background of other distant universes. But it is unlikely that we should ever be able to detect this rotation directly.

Nevertheless, there may be further consequences if gravity is created in this way. It is still 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 in G was proposed in Paper 1 (see the end of this section). It was suggested that the rotation of an idealized fluid Earth might produce an abrupt 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 will occur within a band occupying about 1km on either side of the geoid (the mean surface level for a fluid Earth). Frank Stacey and his colleagues observed an increase in the value of G of nearly 1% from measurements taken in a deep Australian mineshaft.

Secondly, I analysed all twenty six pairs of the original laboratory test results for G undertaken at the National Bureau of Standards, Washington in 1930 and 1942 (L. M. Stephenson, Proc. Phys. Soc., 90, 601, 1967). These results showed that the predicted 0.4% increase in G when crossing the 2km band at the surface of the Earth may have been transformed into an annual variation in G, when measured near to the Earth’s mean surface level at Washington, with a maximum occurring at the vernal equinox and a minimum at the autumnal equinox. The analysis demonstrated that a proposed 0.2% annual variation in G at Washington reduced the spread of the means of the three sets of G results taken in 1930 by a factor of 3. For two more accurate sets of G results, taken in 1942, the spread of the means was reduced by a factor of 15. A statistical analysis shows that the probability of these improvements 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. 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 filaments is a regular variation of such nature as to be incredible.”

Within the past ten 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%. As is usual, in such circumstances, searches are being made to try to explain these anomalies in terms of experimental error. As an alternative approach, to check for possible actual variations in G, it would be worthwhile to study these results in more detail 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.

The 0.4% increase in G, which is predicted to occur when crossing the 2km band at the Earth’s boundary surface, will produce a much smaller increase in the acceleration due to gravity g. The related increase in g will be less than 1 part in 10⁶ when crossing this 2km band at the Earth’s surface into the interior. It is important to realize that this suggested change in g is much smaller than, and additional to, any normal change in g which occurs with any variation of height above and below the Earth’s surface level.

It was also shown in Paper 1 that the spin of an electron may create an extremely large value for G within the electron, and an internal gravitational force that is more than sufficient to stabilize the electron against the internal electrostatic repulsion forces. Any spinning atomic particle would be similarly stabilized gravitationally, in a way that Einstein believed should occur. This prediction attracted further comments from readers of my web site. In obtaining the stability result for the electron, given in Paper 1, classical values for both the radius of the electron and the spin velocity were used. The spin velocity is then much greater than c, the velocity of light. However, I defend the use of classical values for this case. First, “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). Secondly, Maxwell’s equations, and hence their required limitations on velocity, do not apply at the atomic level (see the next chapters and Papers 2 and 3 in the next section for further clarification).

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. Further gravitational links between all four fundamental forces have also been proposed (L. M. Stephenson, J. Phys. A, 2, 475, 1969).

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