## CHAPTER 13## Physical ConstantsBecause motion and its components, space and time, exist only in units, the derivatives of motion, dimensional variations of the basic relation between space and time, such as acceleration, force, etc., also exist only in natural units. A natural unit of force, for example, is a natural unit of time divided by a two-dimensional natural unit of space. It then follows that where a relation of the kind discussed in Chapter 12 is correctly stated, it is valid as a quantitative relation between units without any arbitrary “constant.” The expression F = ma, for example, tells us that one natural unit of force applied to one natural unit of mass will produce an acceleration of one natural unit. When all quantities are expressed in natural units there are no numerical constants in equations of this kind aside from what we may call structural factors: geometrical factors such as the number of effective dimensions, numerical factors such as the second and third powers of the quantities entering into the relations, and so on. There has been a great deal of speculation as to the
nature and origin of the “fundamental constants” of present-day physics.
An article in the Sept. 4, 1976 issue of A preliminary step that has to be taken before we can compare the mathematical results derived from the new theory with the numerical values obtained by measurement is to ascertain the conversion ratios by which the values in the natural system can be converted to the conventional system of units in which the measurements are reported. Inasmuch as the conventional units are arbitrary, there is no way in which the conversion factors can be calculated theoretically. It is necessary to utilize a measurement of some specific physical quantity for each independent conventional unit. Any physical quantity, which involves the item in question, and can be clearly identified, will theoretically serve the purpose, but for maximum accu1acy certain basic phenomena that are relatively simple, and have been carefully studied observationally, are clearly preferable. There is no question as to where we should obtain the value of the natural
unit of speed, or velocity. The speed of radiation, measured as the speed
of light in a vacuum, 2.99793 x 10 One of the problems in this connection is that of arriving at a decision as to which of the reported measured values should be used in the calculations. Ordinarily it would be assumed that the more recent results are the more accurate, but an examination of these recent values and the methods by which they have been obtained indicates that this is not necessarily true. Apparently the “consistent” values listed in the up-to-date tabulations involve some adjustments of the raw data to conform with current theoretical ideas as to the relations that should exist between the various individual values. For purposes of this present work the unadjusted data are preferable. The principal question at this point concerns the experimental
values of Avogadro's number, as only three conversion constants are required
for present purposes, and there are no significant differences in the
measurements of the quantities that will be used in calculating two of
these constants. The more recent values reported for Avogadro’s number
are somewhat lower than those reported earlier, but the correlation with
the gravitational constant, which will be discussed shortly, favors some
of the earlier results. The value adopted for use in evaluating the conversion
constant for mass, 6.02486 x 10 In any event, it should be understood that wherever the results obtained in this work are expressed in the arbitrary units of a conventional system, they are accurate only to the degree of accuracy of the experimental values of the quantities used in determining the conversion constants. Any future change in these values resulting from improvement of experimental techniques will involve a corresponding change in the values calculated from theoretical premises. However, this degree of uncertainty does not apply to any results that are stated in natural units, or in conventional terms such as units of atomic number that are equivalent to natural units. As in the first edition, the natural unit of time has been calculated from the Rydberg fundamental frequency. A question has arisen here because this frequency varies with the mass of the emitting atom. The original calculation was based on the value applicable to hydrogen, but this has been questioned, as the prevailing opinion regards the vague applicable to infinite mass as the fundamental magnitude. A definitive answer to this question will not be available until the theory of the variation in the frequency has been worked out, but in the meantime a review of the situation indicates that we should stay with the hydrogen value in the interim. From the theoretical viewpoint it would seem that the unit value would come from an atom of unit magnitude, rather than from an infinite number of atoms. Also, even though the difference is small, the value thus derived seems to be more consistent with the general pattern of measured magnitudes than the alternative. From the manner in which the Rydberg frequency appears in the mathematics
of radiation, particularly in such simple relations as the Balmer series
of spectral lines, it is evident that this frequency is another physical
manifestation of a natural unit, similar in this respect to the speed
of light. It is customarily expressed in cycles per second on the assumption
that it is a function of time only. From the explanation previously given,
it is apparent that the frequency of radiation is actually a velocity.
The cycle is an oscillating motion over a spatial or temporal path, and
it is possible to use the cycle as a unit only because that path is constant.
The true unit is one unit of space per unit of time (or the inverse of
this quantity). This is the equivalent of one half-cycle per unit of time
rather than one full cycle, as a full cycle involves one unit of space
in each direction. For present purposes the measured value of the Rydberg
frequency should therefore be expressed as 6.576115 x 10 By combing these two natural units as required, the natural
units of all of the quantities of the velocity group can be calculated.
Those of the inverse quantities, the energy group, can also be calculated
in the same centimeter-second terms, but this gives us expressions such
as 3.711381 x 10 With the addition of the value of the natural unit of inertial mass to the values previously derived for the natural units of space and time, we now have all of the information required for calculation of the natural units of the other primary quantities of the mechanical system. The mechanical units can be summarized as follows:
The values given in the first column of this tabulation
are those derived by applying the natural units of space and time to the
space-time expressions for each physical quantity. In the case of the
quantities of the speed or velocity type, these are also the values applicable
in the conventional systems of measurement. However, mass is regarded
as an independent fundamental variable in the conventional systems, and
a mass term is introduced into each of the quantities of the energy type.
Momentum, for example, is not treated as
As we saw in Chapter 10, the masses of the atoms of matter can be expressed
in terms of units of equivalent electric displacement. The minimum quantity
of displacement is one atomic weight unit. It is therefore evident that
this displacement unit is some kind of a natural unit of mass. In the
first edition it was identified as the natural unit of mass in general.
The continuing theoretical development has revealed, however, that this
atomic weight unit, the unit of inertial mass, is actually a composite
that includes not only a unit of what we will now call The concept of secondary mass was introduced in the first edition, without being developed very far. A considerably more detailed treatment is now available. The inward motion in space which gives rise to the primary mass does not take place from an initial level occupying a fixed location in a stationary frame of reference. Instead, the initial level itself is in motion in the region inside unit space. Since mass is an expression of the inward motion that is effective in the context of a stationary reference system, the primary mass is modified by the mass equivalent of the motion of the initial level. While the previous deductions with respect to the essential features of the secondary mass component have been confirmed in the subsequent studies, a few of the details take on a somewhat different appearance when viewed in the light of the more complete information now available. The recent findings indicate that although the primary mass is a function of the net total effective positive rotational displacement, the movement of the initial level that is responsible for the existence of the secondary mass depends on the magnitudes of the displacements in the different dimensions separately. The scalar directions of the motions inside unit distance
play an important part in determining these magnitudes. Outside unit distance,
the scalar direction of the rotational motion is inward because it must
oppose the outward motion of the natural reference system. However, as
we saw in Chapter10, the magnitude of that inward
motion depends to some extent on whether the displacement in the electric
dimension is positive or negative. Inside unit space there is still more
variability, as the motion in this region is in time, and there is no
fixed relation between direction in time and direction in space. (The
rotational motion of which the material atom or particle is constructed
is motion in space, but inside one spatial unit the translational motion
Because of this directional freedom in the time region, the secondary mass may be either positive or negative. Furthermore, the directions of the individual displacement units are independent of each other, and the net total secondary mass of a complex atom may be relatively small because of the presence of nearly equal numbers of positive and negative secondary mass components. This directional variability introduces a number of complications into the secondary mass pattern of the elements. The complete pattern has not yet been identified, but a substantial amount of information is now available with respect to the values applying to sub-atomic particles and the elements of low atomic number. The magnitudes of the natural units applicable to physical quantities are independent of the sector or region of the universe in which the phenomena to which they relate are located. As explained in Chapter12, however, only a fraction of any physical effect can be transmitted across a regional boundary, and the measured value beyond that boundary is substantially less than the original unit. This is the principal reason for the great disparity between the magnitudes of the primary and secondary mass. A unit of mass in the region inside unit distance is inherently just as large as a unit of mass in the region outside unit distance. But when both are measured in terms of their effect in the outside region, the inside, or secondary, mass is reduced by the interregional ratio. In this chapter we are dealing with some very small quantities, and for greater accuracy we will extend the previously calculated value of the inter-regional ratio to two more decimal places, making it 156.4444. The reciprocal of this ratio, 0.00639205, is the fraction of a time region unit that is effective outside unit distance. It is therefore the unit of secondary mass applicable to the basic two-dimensional rotation of the atom or particle. The unit of inertial mass is one such secondary unit plus one unit of primary mass, or a total of 1.00639205. An analysis of the secondary mass relations enables us to compute the mass of each of the sub-atomic particles, a magnitude that is of interest not only as one more item of information about the physical universe, but also because of the light that it throws on the structure of the individual particle. Here we must take into account not only the two-dimensional component of the secondary mass, the magnetic component, as we will call it, following our usual terminology, but also the other components that may be involved in the secondary mass. One of these is the component due to the electric rotation, if any. Inasmuch as this electric rotation, the rotation in the third dimension, is not an independent motion, but a reverse rotation of the pre-existing two-dimensional rotating system, or systems, it adds neither primary mass nor the magnetic unit that is the principal component of the secondary mass. It contributes only the mass equivalent of a unit of one-dimensional rotation. In this case, the 1/9 factor representing the possible positions of the basic photon applies directly against the basic 1/128 relation. We then have for the unit of electric mass: 1/9 x 1/128 = 0.00086806 This value applies specifically where the motion around the electric axis is a rotation of a two-dimensional displacement distributed over all three dimensions, as in a double rotating system. Where only one two-dimensional rotation is involved, the electric mass is 2/3 of the full unit, or 0.00057870. When two of the two-dimensional rotations (four dimensions in all) are consolidated to form a double rotating system (three dimensions), the two 0.00057870 mass units become one 0.00086806 unit. Another secondary mass component that may be present is the mass due
to an electric The relation between the time region and the third region,
in which the motion of the charge takes place, is similar to that between
the time region and the region outside unit space. The inter-regional
ratio is the same, except that because the electric charge is one-dimensional
the factor 1 + 1/9 has to be substituted for the factor 1+2/9 that appears
in the inter-regional ratio previously calculated. This makes the interregional
ratio applicable to the relation with the third region The charge applicable to electrons and positrons deviates from this normal value because these particles have effective rotations in only one dimension, leaving the other two dimensions open. In some way, the exact nature of which is not yet clear, the motion of the charge is able to take place in these two dimensions of the time region instead of in the normal manner. Since this is on the opposite side of the unit boundary, the direction of the effect is reversed, making the mass increment due to the charge negative, as well as reducing its magnitude by one third. The effective mass of a charge applied to an electron or positron is therefore -2/3 x 0.00004494= -0.00002996 We may now apply the calculated values of the several mass components, as given in the foregoing paragraphs, to a determination of the masses of the sub-atomic particles described in Chapter 11. For convenience, these values will be recapitulated as follows:
These are the masses of the various components on the Here again we face the same issue that was encountered early in this
chapter in connection with the selection of an empirical value c Avogadro’s
number as a basis for calculating the unit of mass: the question as to
whether we should regard the most recent determination as the most accurate.
It would appear that the arguments that led to the acceptance of the 1957
value of Avogadro’s number are also applicable to the particle masses,
particularly since the agreement between the calculated and observed masses
of the electron and proton is quit satisfactory on this basis. The empirical
values cited in the paragraphs that follow have therefore been taken from
the 1957 compilation by Cohen, Crowe and Du Mond. Since mass is three-dimensional, an independent one-dimensional or two-dimensional
rotation has no mass. Nevertheless, when such a rotation becomes a component
of a three-dimensional rotation, it contributes to the mass equivalent
of that rotation. This amount that a rotation which is massless when independent
will add to the mass of a particle or atom when it joins that combination
of motions constitutes what we will call In the case of the particles with no effective two-dimensional rotational displacement, the electron and the positron, the appropriate unit of electric mass, 0.00057870, is the entire mass of the particle, and even that mass is only potential, rather than actual, as long as the particle is in the basic uncharged condition. When a charge is added, the effect of the charge is distributed over all three dimensions by the chance process that governs the directions of the motion of the charge in the time region. Thus the charged particle has effective motion in all three dimensions, irrespective of the number of dimensions of rotation. This not only makes the mass of the charge itself an effective quantity, but, as indicated in Chapter 11, it also raises the potential mass of the rotation of the particles to the effective status. The net effective mass of the electron or the positron is then the rotational value 0.00057870 less the mass of the charge 0.00002996, or 0.00054874. The observed value is 0.00054877. The massless neutron, the M ½-½-0 combination, has no effective rotation
in the third dimension, but no rotation from the natural standpoint is
rotation at unit speed from the standpoint of a fixed reference system.
This rotational combination therefore has an initial unit of electric
rotation, with a potential mass of 0.00057870, in addition to the mass
of the two-dimensional basic rotation 1.00639205, making the total In this connection, it should be noted that the electron and positron also have rotation at unit speed (no rotation, in terms of the natural system) in the two inactive dimensions, but these rotations involve no mass, as they are independent, and are not rotating anything. The initial unit of rotation in the third dimension of the massless neutron, on the other hand, is a reverse rotation of the two-dimensional structure, and it therefore adds an electric mass unit. The neutrino, M ½-½-(1), has the same unit positive displacement in the magnetic dimensions as the massless neutron, but it has neither primary nor magnetic mass because these are functions of the net total displacement, and that quantity is zero for the neutrino. But since the electric mass is independent of the basic rotation, and has its own initial unit, the neutrino has the same potential mass as the uncharged electron or positron, 0.00057870. The potential mass of both the massless neutron and the neutrino is actualized when the rotations of these particles are joined to produce a three-dimensional rotation. The mass of the resulting particle is then 1.00754945. As indicated in Chapter 11, this particle is the proton. As it is observed, however, the proton is positively charged, and in this condition the foregoing figure is increased by the mass of a unit charge, 0.00004494. The resulting mass of the theoretical charged proton is 1.00759439. The mass of the observed proton has been measured as 1.007600. Consolidation of two protons results in the formation of a double rotating system. As stated earlier, this substitutes one three-dimensional electric unit of mass- for two of the two-dimensional units, reducing the combined mass by 0.00028935. The mass of the product, the deuterium atom (H²), is the sum of two (uncharged) proton masses less this amount, or 2.014810. The corresponding observed value is 2.014735. Inasmuch as the proton already has a three-dimensional status, addition of another neutrino alters only the electric mass. The material neutrino adds the normal two-dimensional electric unit, 0.00057870, making the total for the product, the mass one isotope of hydrogen, 1.00812815. The measured value is reported as 1.008142. The successive additions of neutrinos to the massless neutron that eventually produce the mass one isotope of hydrogen should be given special attention, as the considerations which will be discussed in Chapter 17 indicate that this addition process plays a very significant part in the overall cyclic mechanism of the universe. The following tabulation shows how the mass of the hydrogen isotope is built up step by step.
Neutrinos are plentiful in the local environment. The requirement for production of new matter in the form of hydrogen by the addition process is therefore a continuing supply of massless neutrons. In Chapter 15 we will find that there is in operation a gigantic process that furnishes just such a supply. Addition of a cosmic neutrino, the rotational displacements of which are on the opposite side of the unit boundary, to the proton, involves an additional initial electric unit, as both the rotation in time and the rotation in space must start from unity. Also the spatial effect of the cosmic neutrino rotation is three-dimensional, since the spatial direction of motion in time is indeterminate. The total addition of mass to the proton in the production of the compound neutron is then 0.00144676, and the resulting mass of the particle is 1.00899621. It has been measured as 1.008982. The following is a summary of the particle masses and the mass components from which these masses are built up. The empirical values from the 1957 compilation are given for comparison. As noted earlier, the correlation is quite satisfactory for the electron and the proton, as it is within the estimated range of experimental error. The divergence in the case of the heavier particles is not large, but it exceeds the estimated error. Whether the source of this discrepancy is in the theoretical development or in the experimental determinations remains to be ascertained.
In the first edition the relation between the natural unit of mass and the arbitrary unit in the cgs system was identified in terms of the gravitational constant. It has recently been pointed out by Todd Kelso and Steven Berline that the relation thus established cannot be converted to a different system of units such as the SI (mks) system. This made it evident that the interpretation of the gravitational phenomenon on which the previous determination was based was, in some way, erroneous An analysis of the situation was therefore carried out in order to locate the point of error. The invalidation of the interpretation of the gravitational equation
has no effect on any other feature of the theoretical results that have
been obtained from the Reciprocal System, as described in this volume
Its sole result has been to leave this system of theory without an, connection
between the gravitational equation and the theoretical structure. Once
the situation is viewed in this light, it is immediately apparent that
the lack of connection between the equation and physical theory is not
peculiar to the Reciprocal System. Conventional theory does not identify
the connection either. The physics textbooks find it necessary to admit
this fact in statements such as the following: “It should be noted that
Newton's law of universal gravitation is It follows from this that the error lies in some interpretation of that “observed relation” that has been common to A force can only be exerted against a resistance. Hence, when we attribute
a force to the motion of one mass we cannot also attribute a force to
the motion of the other. We must attribute a This is a situation in which the ability to reduce all physical quantities to space-time terms is very helpful. It will also be convenient to exam the dimensional situation independently before taking up the question of the numerical values. The gravitational equation, as expressed in current practice, is assigned dimensions as follows:
Reducing equation 13-1 to space-time terms in accordance with the relations
established in Chapter 12 (in which dynes, as g-cm/sec², are
In the light of the new understanding of the
Turning now to the numerical magnitudes, we note that while the One gram is 6.02486 x 10 A small additional correction is required because of the effect of the
secondary mass. Gravitation and inertia are inversely related relative
to the primary mass; that is, the primary mass is p/(p + s) units of gravitational
mass and also p/(p + s) units of inertial mass, where p and s are the
primary and secondary masses respectively. The product of a unit of gravitational
mass and a unit of inertial mass is therefore 1/(1 + s)² units of primary
mass. Where the result is expressed in terms of inertial mass, another
1 + s factor is introduced. The total effect of the secondary mass is
then the introduction of a factor of 1.019299. Applying this factor to
the value 9.95874 x 10 Replacing the 1/s² distance term by a t/s² force term has the effect
of introducing a time dimension, which must be expressed in natural units
to avoid creating a numerical unbalance. The numerical value of the natural
unit of time, 1.520655 x 10 Looking now at the question of conversion to a different system of units,
the issue that initiated the restudy of the situation, we find that a
change from cgs to mks units in the conventional form of the equation
(13-1) results in a change of 10 Those who are familiar with the first edition will have noticed that the values of the natural unit of inertial mass and related quantities, as given earlier in this chapter, are larger than the values given in the original publication. At the time of the original investigation it seemed clear that a factor of 1/3 entered into the mass situation in some way, and there appeared to be sufficient justification for applying this factor to the size of the basic unit. As brought out in the preceding paragraphs, we now find that the 1/3 factor is a result of the one-dimensional nature of the “as if” gravitational interaction. This factor has therefore been eliminated from the mass units. As a result, the natural unit of inertial mass, as defined in this edition, is three times the value given in the first edition (with a small adjustment to reflect the results of the continuing studies of the details of the phenomena involved). The use of these larger units has no effect on the physical relations involving inertial mass, as the expressions of these relations are balanced equations in which the mass terms are in equilibrium with terms representing quantities derived from mass. |