Physics Essays volume 16, number 2, 2003
A Dynamical Origin for the Gravitational Constant that Explains Gravitational and Inertial Mass Equality and Rejects Dark Matter and Dark Energy
Lawrence M. Stephenson
Abstract
There is no accepted theory that explains why the gravitational mass and the inertial mass of a given body are equal. Any new theory that predicted a dynamical origin for the gravitational constant should also predict that the gravitational mass of a body is equal to its inertial mass. Two dynamical theories are outlined. One of these theories does not require the assumptions of dark matter and dark energy to explain cosmological observations.
Key words: gravitational mass, gravitational constant, dark matter, dark energy.
1. INTRODUCTION
There is a fundamental difference, in principle, between the gravitational mass and the inertial mass of a given body. However, the most accurate experiments1 ever performed in physics have demonstrated that, for all of the materials tested, the gravitational mass of a body is found to be identical to its inertial mass to an accuracy of one part in 1011.
Because neither Newtonian theory nor general relativity (see below) is able to explain the physical reason for the observed equality of gravitational and inertial mass, it was suggested in 19862 that a fifth force could exist (the four basic forces being the electromagnetic force, the strong and weak nuclear forces, and the gravitational force). To check this idea a comparison was made of the gravitational and inertial masses of materials having very different nuclear constitutions. If a difference had been found, then it would indicate both the existence of a fifth force and an observable difference between the inertial mass and the gravitational mass for a few specific materials. But no difference was found. As a result of these latest experiments, it now seems certain that there is an undiscovered theoretical link between inertial and gravitational mass.
The lack of a theory to link gravitational and inertial mass is not the only problem. A secondary mystery concerning mass surrounds the value of the gravitational constant G, which controls the magnitude of the gravitational force of attraction between any two given bodies. All of the other constants of physics, such as the charge and mass of the electron, the radius of an atom, the velocity of light, and Planck's constant are interrelated. But G stands out as a totally isolated constant. Again, there is no accepted theory that explains either why G should stand alone or why it has the particular value it has.
The basis of general relativity lies in the experimental observation that a gravitational field imparts the same acceleration to all bodies. Hence a gravitational acceleration of a test particle cannot be distinguished from an inertial acceleration of the particle. It is then sometimes incorrectly argued that the introduction of “mass” is unnecessary, and that general relativity itself proves that the inertial mass of a body is equal to its gravitational mass. But what would be the case if a fifth force had been found to exist and, for some materials, the inertial mass of a body had been observed to be different from its gravitational mass? It is clear that general relativity cannot offer a theoretical explanation of why inertial mass and gravitational masses are identical. Also, general relativity adopts the Newtonian assumption that the value of G must be a universal constant. But there is nothing in general relativity that either establishes a value for G or proves that G is a universal constant3.
Mass is the fundamental quantity upon which the whole of physics is built, and yet we have no understanding of its true nature. What can be done to achieve a theoretical understanding of why the gravitational mass of a body is found to be identical to its inertial mass? When any two items are observed to be identical, we deduce that they must be linked to some form of common origin. One might then expect inertial mass and gravitational mass to have a common origin. Two separate theories, which are based on assuming such a common origin, will now be considered. Both of these theories also yield a particular value for G.
2. MACH’S IDEAS ON THE ORIGIN OF INERTIAL FORCES
Mach was troubled by the deeper problems associated with rotational motion. One of the many early arguments that showed the difficulties with Newtonian theory went as follows. Consider two fluid stars that are well separated in the sky. One of these stars is stationary and has a spherical shape. The second star is spinning about an axis that passes through the center of the first star, and has an oblate shape due to its rotation. Now let all of the rest of the matter in the universe be gradually removed. At the limit one is apparently left with one spherical star and one oblate star. One can deduce that the latter are spinning because of its oblate shape. But with all other matter removed there is no external frame of reference. To have one star spinning and oblate, and the other star stationary and spherical, is not acceptable. As they can only relate to each other, why is there a difference between them? One can go further and remove the stationary star. One now has a single star spinning in an otherwise empty universe, but what is it spinning with respect to?
On a Newtonian basis the above argument is plausible, but Mach considered that removing parts of the universe might have dramatic effects. Mach's answer to this problem was to postulate that our basic concepts of the mass of a body, the inertial acceleration of the body, and the gravitational attraction within the body must all depend on causes lying well outside the immediate system being considered. Mach suggested that the totality of matter making up the whole of the universe, in some way yet to be defined by experiment, gives rise to the concepts of mass and acceleration forces. Mach is suggesting that, if distant parts of the universe could be removed, then some of the basic laws of physics might change. For example, the value of the gravitational constant G might change. Such a hypothesis is close to the gravitational theory of Hoyle and Narlikar4, who suggested that G might be inversely proportional to the mean density of the universe. Hence, if half of the universe disappeared, then the value of G would double. However, it is unlikely that any experimental evidence will be forthcoming if the only suggested test is to remove a substantial part of the universe. Hoyle and Narlikar’s theory is limited to defining just the value of G, and it does not produce any link between inertial mass and gravitational mass. It is necessary to develop Mach’s ideas still further to find the origin of the equality of gravitational and inertial mass.
3. A DYNAMICAL ORIGIN FOR GRAVITATIONAL MASS
Hoyle and Narlikar’s theory involves relating G to the total static distribution of matter in the universe. But possible alternative theories exist that relate G to the dynamic distribution of matter in the universe. As it was the rotational motion of matter that caused many of the difficulties that Mach highlighted, there is a good chance that a dynamical theory for G might be more appropriate, especially one that involves rotational motion.
To achieve a dynamical theory for G one needs to postulate that it is the motion of distant matter, on a very large scale, relative to the cosmic background frame, that produces all of the terrestrial gravitational forces that we observe. If there is no motion of distant matter, then the value of G will go to zero and all gravitational effects will disappear. Hence the gravitational mass of any given body will be directly related to the motion of distant matter. As well as predicting a value for G, this type of dynamical theory indicates that the gravitational mass of a body should inherently be equal to its inertial mass. If the origin of the local gravitational mass of a body arises from the (inertial) motion of distant matter, then one would expect the gravitational mass of the body to be equal to its inertial mass because the apparently different types of mass have a common origin.
Any new dynamical theory for G must be compatible with our knowledge that locally moving matter does not produce a significant increase in locally observed gravitational forces. Fortunately, the detailed work of Hoyle and Narlikar supports a dynamical theory based on the motion of distant matter, because they showed that local matter has a negligible effect on G compared with distant matter.
The obvious way of transforming this idea of a dynamical origin for G into an analytical statement is to look initially at the dimensions of the gravitational constant. The dimensions of G may be expressed as
G ~ 1/ρT2, (1)
where ρ represents the dimensions of density and T the dimensions of time. It can immediately be seen that Hoyle and Narlikar's gravitational theory, in which G is inversely proportional to the mean density of the universe, is based on carrying over just a part of this dimensional relationship into a full theory. In a similar way, Dirac made the well-known suggestion that G may be slowly decreasing with time, with a time constant of the same order of magnitude as the age of the universe. Hence Dirac's suggestion is based on carrying over the other part of this dimensional relationship into a formal relationship.
If we are searching for a dynamical theory for G, based on the inertial motion of distant matter, then the above dimensional relationship can be changed slightly to become
G α v2/r2ρ, (2)
where v is now the velocity of distant matter at a distance r. Note that, as we require a theory relating G to both the density and the inertial motion of distant matter, we have changed from making a simple dimensional statement about G to making G proportional to all of the quantities involved.
Alternatively, for a possible rotational motion of distant matter, (1) may be restated as
G α ω2/ρ, (3)
where ω is the angular velocity of the matter relative to the cosmic background frame.
Let us consider (2) first. Suppose that G is related to the recessional velocity of distant matter in the universe associated with the expansion of the universe. For the usually accepted linear expansion of the universe, proposed by Hubble, v/r is a constant and hence G will be inversely proportional to the density of matter. One interpretation of (2) then reduces to a new dynamical theory of gravitation that is similar in its outcome to the static theory of Hoyle and Narlikar. However, this dynamical theory explains both the value of G and the equality of inertial and gravitational mass. If this particular dynamical theory happens to be correct, then, again, as with Hoyle and Narlikar's theory, it is unlikely that any direct experimental test could be performed to establish the theory.
But (3) is much more interesting. This equation allows G to be related to the rotational inertial motion of distant matter. If we ignore the concept of the whole universe rotating, then the next level down where rotation occurs is with galaxies or groups of galaxies. Hence one may propose a second dynamical theory of gravitation, based on the rotational motion of a galaxy, by making use of the known Newtonian stability condition for an idealized, gravitationally stable galaxy, which is
G = ω2/2πρk, (4)
where ω is the angular velocity of the galaxy, ρ is the mean density, and k is a boundary shape constant (≤1). In this second dynamical theory for G the value of G is no longer assumed to be a universal constant. It is proposed that the value of G is being specifically created by the rotational inertial motion of the matter within the galaxy. Even within our own galaxy, the Milky Way, the value of G need not be constant – the value will depend on the values of ω and ρ for the particular region. If G is created in this way, the predicted value of G will only apply within the boundary surface of the galaxy, and the value of G will be zero in any region of intergalactic space where no rotation of matter is taking place. This theory was originally proposed in 1965, and was elaborated on in 19695 and 19766. But at that time there were no observations that were considered to be unexplained by current theories. The situation has now changed as a result of two cosmological anomalies.
Cosmologists have found that the gravitational attraction of visible matter is insufficient to account for the orbital velocities of stars in individual galaxies, and of galaxies within clusters of galaxies. To explain this discrepancy vast amounts of dark matter are assumed to be present. More recently, Krauss7 has provided evidence that the gravity of visible matter is not consistent with an observed accelerating expansion of the universe; he proposes that an assumption of large amounts of dark energy, associated with a cosmological constant, is required to account for this second discrepancy. No direct observational evidence has been found to account for either dark matter or dark energy, and alternative theories cannot be discounted. In particular, many cosmologists are reluctant to accept a cosmological constant in general relativity.
The new interpretation of (4) eliminates the need for both dark matter and dark energy. The need for dark matter is removed because the new interpretation of (4) predicts that a galaxy is self-stabilizing in all regions. For a given value of angular velocity, (4) predicts that G will vary inversely with the value of the density of the region of the galaxy being considered. Hence in the low-density, outer regions of the galaxy there will be an increased gravitational attraction to correspond with the high angular velocities of stars in these regions. Artificial dark matter is not needed to explain galactic stability.
The need for dark energy is removed because this dynamical theory for G predicts that G is zero outside the boundary of any individual galaxy or any rotating galaxy cluster. Without gravitational attractions between individual galaxies, or galaxy clusters, one would expect a faster expansion of the universe than current theory predicts, where G is assumed to be a universal constant and gravitational attraction between galaxies will slow expansion.
This theory may also give rise to terrestrially observable effects. It is proposed that it is only for gravitationally-stable, rotating, fluid bodies, like the galaxy, that G will be created by the rotational motion of matter as given by (4). In a solid rotating body, such as a flywheel, additional centrifugal forces can be developed because ω is not limited by the fluid boundary-shape condition. Hence (4) will not apply to a flywheel. However, although the Earth is semi-rigid, its boundary shape is very near to that which would be adopted by a truly fluid Earth. If (4) applies to the Earth, then for a truly fluid Earth an additional increase in G of approximately 0.4% will occur when passing across the boundary surface into the interior. To observe the possible predicted increase in G below the Earth's surface it is necessary to perform two full G experiments. The first would be at a height of 1-2km above the Earth's mean surface level (at a location not in an anomalous g gravity high). The second experiment would be performed down a 1-2km deep mineshaft (at a location not in an anomalous g gravity low). Such experiments would establish whether there is an increase in G when crossing the mean boundary surface into the interior.
At the other extreme to predicting galactic stability, (4) predicts that the electron, or any other spinning atomic particle, is stabilized by gravitational forces5 and not by arbitrary short-range forces. If one substitute a value for ρ deduced from the classical radius e2/mc2 and a value for ω deduced from the angular momentum h/4π, then the internal value of G for an electron comes out to 1045 times the terrestrial value for G. Since the ratio of the electromagnetic coupling constant to the usual gravitational coupling constant is about 1043, the internal values of the gravitational forces within an electron are then predicted to be more than sufficient to stabilize the electron against the electrostatic repulsion forces. Einstein was convinced that atomic particles were stabilized by gravitational forces and he attempted to account for this stability by renormalizing the gravitational field at the boundary of the particle. As the nature of this renormalizing process was ill defined, arbitrary short-range forces are now assumed in order to explain electron stability. It seems likely that Einstein’s instincts concerning atomic particle stability were correct.
4. CONCLUSION
The most accurate experiments ever performed in physics indicate that the gravitational mass of a body is identical to its inertial mass. A theoretical link would be anticipated, but neither Newtonian theory nor general relativity is able to provide this link. If the gravitational mass of a body is to be directly related to its inertial mass, then the gravitational constant, which controls the gravitational mass of the body, must have a dynamical origin. One particular dynamical origin for the gravitational constant, based on the rotation of galaxies, explains two cosmological anomalies, in addition to linking gravitational and inertial mass.
Acknowledgement
I am very grateful to Professor C.W.Kilmister for some valuable comments, and to both referees for suggesting some clarifications.
Received 6 June 2003
References
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6. Idem, Found. Phys. 6, 143 (1976).
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Lawrence M. Stephenson
Department of Electronic and Electrical Engineering
University College
London WC1E 7JE England