As Galileo Galilei (1564-1642) showed by experiments, all bodies, independent of their masses, fall with the same acceleration in vacuum. This experimental reality contradicted Aristoteles's postulate, that a heavy body falls faster than a lighter body. In the Newtonian mechanics, it follows from Newton's 2nd law and his gravitational law, that bodies with different masses fall with the same acceleration near a gravitating mass. However, the condition for this is, that the the inertial mass of the body is equal to its gravitational mass. Since Newton discovered his gravitational law, there have been many reflections why these two forms for mass, although they are defined in different ways, should have the same value. The identity between inertial mass and gravitational mass is one of the basic conditions for the validity of Einstein's General Theory of Relativity from 1915, where he calls it 'the equivalence principle'. The inertial mass of a particle is in Newton's 2nd law defined as the ratio between the resulting force exerted on the particle and the resulting acceleration. The gravitational mass of a particle is defined by Newton's gravitational law. As everything 'existing', according to my Holistic Quantum Cosmology, is assumed to consist of unitons (see above), in smaller or greater concentrations, I shall here give a uniton based definition of mass. Let us define: A system's uniton mass m is proportional with the number, N_u, of unitons in the system, viz. the following is valid:

(1)

where m_u is a proportionality parameter, which we may call the mass proportionality parameter, and it is agreed that it shall have the unit kilogram. m_u is equal to the mass of a uniton.

Uniton determined gravitational forces on two different bodies.

In the following I shall show that bodies with different masses have the same acceleration when falling. I use my uniton mechanical gravitational theory. The basis for this theory is my discovery (see the uniton section) of the existence of a universal 'quantum medium' - the cosmic uniton field - consisting of the actually smallest 'matter quanta', called unitons. These unitons are present everywhere in the Universe. Unitons have an extremely small extension and move with the velocity of light (possibly faster). All 'matter' is assumed to consist of unitons. Thus an electron is assumed to be a dynamic subsystem, consisting of about 10³⁷ unitons.

That the electron is a uniton dynamic subsystem means that it, in a condition of balance, emits and absorbs the same number of unitons per unit of time. The repulsive force between two electrons can well be caused by the forces exerted by the unitons, arising by the emission and absorption processes (see above).

Let us consider two bodies, 1 and 2, with the different masses m₁ and m₂. Let first one body fall in a homogenouos and constant gravitational field, then the other body. In the uniton gravitational theory this means that the density of unitons (viz. the number of unitons per unit of volume) outside the bodies and between a body and the body it is falling towards, are not equal, but constant, see the figure. The reason for the difference in uniton densities is that the bodies are 'shading' the cosmic uniton field.

The masses, m₁ and m₂, of the bodies in question, can according to the definition in equation (1) be written as:

(2)

(3)

where N_u,1 is the number of unitons in body 1, and N_u,2 is the number of unitons in body 2. Let us consider the uniton forces exerted on the bodies. On body 1, an external total force F_1,ex is exerted, which must be proportional to both the external uniton density - the number of unitons per unit of volume - n_u,ex (corresponding to the cosmic uniton density), and the number of unitons, N_u,1, in body 1. F_1,ex must thus be given by:

(4)

where k_ex is a proportionality constant, characterizing the external uniton field. For the total internal forces (uniton forces exerted on the lower side of the body), F_1,in we similarly have:

(5)

where n_u,in is the internal uniton density, viz. the uniton density between the bodies being pushed together. k_in is a proportionality constant, characterizing the uniton field between the two bodies. The resulting force, F_1,res, exerted on body 1, is determined by:

(6)

For body 2, we quite similarly get the resulting force:

(7)

where N_u,2 is the number of unitons in body 2.

The condition for the linear expressions of the forces is that the density of unitons in the bodies is very small. This will also be the case, as the extension of a uniton is very small, equal to the elementary length. Most of the volume of a body of matter is thus vacuum.

The accelerations of the two bodies, a₁ and a₂, we can find by means of Newton's 2nd law:

(8)

(9)

where also the inertial mass is determined according to equation (1). As N_u,1 and N_u,2 vanish by division in equations (8) and (9) we see, that the two bodies move with the same acceleration, which is according to experience.

The average resulting force exerted on one uniton of the body, F_u is given by:

(10)

If the question is about a free fall near the surface of the Earth, where the acceleration of gravity of a body is 9.82 m/s², the value of F_u is about 2·10^-67 Newton.

Louis Nielsen, 20th October 1997.
louis44nielsen@gmail.com

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