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. It is better to concentrate on terrestrial measurements, related to smaller scale rotations, that might disclose the orgin of G.

A paper will now be attached (that includes the invalid specific model for G mentioned above) because the paper shows why the gravitational mass of a given body should be identical to its inertial mass. One can then confirm that it is the law of gravitation which may change, for certain types of relative motion between local masses and distant masses, and not the laws of motion. The paper also includes details of increases of G that may occur within smaller spinning bodies, such as the Earth and all spinning atomic particles. These increases are not in conflict with astronomical observations which require a constant free space value for G.

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

Current astronomical evidence appears to be clear. When related to a Big Bang origin for our universe, and a Hubble rate of expansion, 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.

For the interpretation of Mach’s Principle we are investigating, the value of G applying within a given region is created by the motion of the region relative to distant matter. Hence, the only option is to suggest that our whole universe is moving relative to a background of even more distant matter. This motion would then create the free space value of G within our universe. It is unlikely that we would ever be able to detect such a motion.

Nevertheless, there may be further consequences if gravity has inertial origins. It is still possible that small perturbations in the observed local value of G might be expected when a local body is moving relative to the most distant matter in our universe.

First, a possible observable variation of G was proposed in Paper 1. It was suggested that the rotation of an idealised 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 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.  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, the author (L.M.Stephenson, Proc. Phys. Soc., 90, 601, 1967) analysed over fifty original individual laboratory test results for G made at the National Bureau of Standards, Washington. 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 results demonstrated a 0.2% annual variation in G at Washington.

Within the past ten years many measurements of G have been made, in laboratories across the world, that show some unexpected variations of up to 0.6%. As a first approach, it would be worthwhile to study these recent individual experimental readings to see if any annual variations of G were present, and also to check whether these annual 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 most important to realise that this suggested change in g is additional to any normal change in g which occurs with any  variation of height above and below the Earth's surface.

At the time these predictions were first made, in 1967, it was not anticipated that the suggested changes in G due to the rotation of the Earth would produce any significant related variations in g at locations  outside the narrow 2km band on either side of the geoid. However, it is conceivable that an extremely small increase in the local value of g, arising from the proposed increase in G , could extend much further out  from the surface of the Earth. There have been many observations of flyby anomalies in the trajectories of spacecraft coming close to the Earth. These anomalies appear to be directly related to the Earth's rotation and  depend on the difference in latitude between the incoming and outgoing trajectories of the spacecraft. It is possible that the flyby anomalies might  be explained by the specific proposed increase in g, which is linked to the G increase and which, in turn, arises from the Earth's rotation. The anomalies in the trajectory of the Pioneer spacecraft may also arise from a similar small local increase in g.

It was also shown in Paper 1 that the spin of an electron may create an extremely large value for G within an electron, and an internal gravitational force that is more than sufficient to stabilize the electron against the internal electrostatic repulsion force. Any spinning atomic particle would be similarly stabilized gravitationally, in a way that Einstein believed should occur. This prediction attracted further comments from viewers of the web site. In obtaining the gravitational stability result for an 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. I should like to make two points in defence of using classical values. First, "the complementary views provided by 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 section on advanced potentials and the next paper 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.

I will add here a few unresearched ideas on dark matter. The existence of invisible, dark matter has been assumed to account for the stability of spiral galaxies. The assumption of dark matter may not be necessary when a fuller classical analysis is applied to galactic stability.

Observations of spiral galaxies indicate that the orbital velocities of stars are approximately constant right across the disc of the galaxy. Conventional analysis initially assumes that each individual star is an isolated body, attracted only by the central hub of the galaxy. The predicted orbital velocity for a star is then inversely proportional to the square root of the radius. This relationship is the same as for a planet orbiting a massive star.

To explain the observed near constant velocity of stars across the disc of the galaxy, it is then assumed that invisible, dark matter must exist in a halo surrounding all spiral galaxies. The required mass of this dark matter has to be about 10 times the visible mass of the galaxy.

Little attention has been given to the fact that individual stars should not be considered in isolation when analysing the overall gravitational stability of a galaxy. A rotating galaxy approximates to being a homogeneous, gravitationally stable, fluid body. The simplest form of stability condition for such a body is given by:

                                            ω² = 2πGρk       (1)

where: ω is the angular velocity, G is the gravitational constant, ρ is the mean density and k is a shape constant (≤1).

For an idealised galaxy, having a uniform density and a constant angular velocity for all parts, one would then deduce that the velocity v of any star should be proportional to the radius r.

It might be reasonable to assume that the correct theoretical model for a star in the disc of a spiral galaxy might lie somewhere between considering the star as an isolated body (v α 1/r^1/2) and considering the star to be part of a uniform medium (v α r). It would then not be surprising if the predicted velocity of a star was approximately constant across the galactic disc. On this basis there would be no need to assume large quantities of dark matter.

Furthermore, equation (1) is still applicable when different shells of the fluid body rotate at different angular velocities, provided the mean density of the region varies accordingly. One may then achieve a constant velocity for stars across an idealised galactic disc, based on just this model, if ρ α 1/r².

For a cluster of galaxies one may also need to consider a quasi-uniform, fluid model for the cluster as a whole when analysing the gravitational stability of the cluster.