# CHAPTER 13

# Physical Constants

Because 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 September 4, 1976 issue of *Science News,* for example, contends that we are confronted with a dilemma, inasmuch as there are only two ways of looking at these constants, neither of which is really acceptable. We must either, the article says, “swallow them ad hoc” without justification for “their necessity, their constancy, or their values,” or we must accept the Machian hypothesis that they are, in some unknown way, determined by the contents of the universe as a whole. The development of the Reciprocal System of theory has now resolved this dilemma in the same way that it handled a number of the long-standing problems considered in the earlier pages; that is, by exposing it as fictitious. When all quantities are expressed in the proper units—the natural units of which the universe of motion is constructed—the “fundamental constants” reduce to unity and vanish.

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, 2.99793×10^{10} cm/sec, is an accurately measured quantity that is definitely identified as the natural unit by the theoretical development. There are some uncertainties with respect to the other conversion factors, both as to the accuracy of the experimental values from which they have to be calculated, and as to whether all of the minor factors that enter into the theoretical situation have been fully taken into account. Some improvement has been made in both respects since the first edition was published, and the principal discrepancies that existed in the original findings have been eliminated, or at least greatly minimized. No significant changes were required in the values of the basic natural units, but some of the details of the manner in which these units enter into the determination of the “constants” and other physical magnitudes have been clarified in the course of extending the development of the theoretical structure.

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, 6.02486×10^{23} g-mol^{-1}, has therefore been taken from a 1957 tabulation by Cohen, Crowe and DuMond.^{59}

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 6.576115×10^{15} half-cycles per second. The natural unit of time is the reciprocal of this figure, or 1.520655×10^{-16} seconds. Multiplying the unit of time by the natural unit of speed, we obtain the value of the natural unit of space, 4.558816×10^{-6} centimeters.

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 3.711381×10^{-32} sec^{3}/cm^{3}, which is the natural unit of mass. This value has no practical use because the inverse relations between the quantities of the velocity group and those of the energy group have not hitherto been recognized. In setting up the conventional system of units it has been assumed that mass is another fundamental quantity for which an additional arbitrary unit is necessary. The ratio of the velocity-based unit of mass to this arbitrary unit, the gram, can be derived from any clearly defined physical relation involving mass that has been accurately measured in conventional units. As indicated earlier, the measurement selected for this purpose is that of Avogadro’s constant. This constant is the number of molecules per gram molecular weight, or in application to atoms, the number of atoms per gram atomic weight. The reported value is 6.02486×10^{23}. The reciprocal of this number, 1.65979×10^{-24}, in grams, is therefore the mass equivalent of unit atomic weight, the unit of *inertial mass, as* we will call it.

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:

Space-time Units | Conventional Units | ||||
---|---|---|---|---|---|

s | space | 4.558816×10^{-6} cm |
4.558816×10^{-6} cm |
||

t | time | 1.520655×10^{-16} sec |
1.520655×10^{-16} sec |
||

s/t | speed | 2.997930×10^{10} cm/sec |
2.997930×10^{10} cm/sec |
||

s/t^{2} |
acceleration | 1.971473×10^{26} cm/sec^{2} |
1.971473×10^{26} cm/sec^{2} |
||

t/s | energy | 3.335635×10^{-11} see/cm |
1.49175×10^{-3} ergs |
||

t/s^{2} |
force | 7.316889×10^{-6} sec/cm^{2} |
3.27223×10^{2} dynes |
||

t/s^{4} |
pressure | 3.520646×10^{5} sec/cm^{4} |
1.57449×10^{13} dynes/cm^{2} |
||

t^{2}/s^{2} |
momentum | 1.112646×10^{-21} sec^{2}/cm^{2} |
4.97593×10^{-14} g-cm/sec |
||

t^{3}/s^{3} |
inertial mass | 3.711381×10^{-32} sec^{3}/cm^{3} |
1.65979×10^{-24} g |

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 t^{2}/s^{2}, but as the product of mass and velocity, which, in space-time terms, is t^{3}/s^{3} × s/t. The use of an arbitrary unit of mass then introduces a numerical factor. Thus, in order to arrive at the values of the natural units in terms of the cgs system of measurement, each of the values given for the energy group in the first column of the tabulation must be divided by this factor: 2.236055×10^{-8}.

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 *primary mass,* the basic mass quantity, but also a unit of *secondary mass.*

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 Chapter 10, 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 *of* the atom is in time.)

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 Chapter 12, 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 × 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 *charge.* Like all other phenomena in a universe of motion, a charge is a motion, an additional motion of the atom or particle. We are not ready to discuss charges in detail at this stage of the presentation, so for the present we will merely note that on the basis of the restrictions on combinations of motions defined in Chapter 9, the charge, as a motion *of* the rotating particle or atom, must have a displacement opposite to that of the rotation in order to be stable. This means that the motion that constitutes the charge is on the far side of another regional boundary—another unit level—and it is subject *to two* successive inter-regional transmission factors.

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 128 × (1+1/9)= 142.2222. The mass of unit charge is the reciprocal of the product of the two inter-regional ratios, 156.4444 and 142.2222, and amounts to 0.00004494.

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 × 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:

p | primary mass | 1.00000000 |
---|---|---|

m | magnetic mass | 0.00639205 |

gravitational mass | 1.00639205 | |

E |
electric mass (3 dim.) | 0.00086806 |

e | electric mass (2 dim.) | 0.00057870 |

C |
mass of normal charge | 0.00004494 |

c |
mass of electron charge | -0.00002996 |

These are the masses of the various components on the *natural scale.* The measured values are reported in terms of a scale based on an arbiter assumed mass for some atom or isotope that is taken as a standard. For a number of years there were two such scales in common use, the * chemical scale,* based on the atomic weight of oxygen as 16, an * the physical scale,* which assigned the 16 value to the O^{16} isotope. More recently, a scale based on an atomic weight of 12 for the C^{12} isotope has found favor, and most of the values given in the current literature are expressed in terms of the C^{12} scale. In the light of the finding of this work the shift away from the O^{16} scale is unfortunate, as the theoretical development indicates that the O^{16} isotope has a mass c exactly 16 on the natural scale, and the physical scale (O^{16 }= 16) is therefore coincident with the natural scale. It will, of course, be necessary to use the natural scale for our purposes. The observed values quote for comparison with the theoretical masses will therefore be stated in terms of the equivalent O^{16} physical scale.

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.^{59}

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 * potential mass.*

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 *potential *mass of this particle 1.00697075.

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 (H^{2}), 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.

primary mass | 1.00000000 | |

magnetic mass | 0.00639205 | |

electric mass | 0.00057870 | |

M ½-½-0 | massless neutron | 1.00697075* |

M ½-½-(1) | neutrino | 0.00057870* |

M 1-1-(1) | proton | 1.00754945 |

M 2-2-(1) | neutrino | 0.00057870* |

M 1½-1½-(2) | hydrogen (H^{1}) |
1.008l28l5 |

* potential mass |

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.

Mass Composition |
Particle | Mass | |
---|---|---|---|

Calculated | Observed | ||

e - c | charged electron | 0.00054874 | 0.00054876 |

e - c | charged positron | 0.00054874 | 0.00054876 |

e | electron | 0.00057870* | massless |

e | positron | 0.00057870* | massless |

e | neutrino | 0.00057870* | massless |

p + m + e | massless neutron | 1.00697075* | massless |

p + m + 2e | proton | 1.00754945 | unobserved |

p + m + 2e + C | charged proton | 1.00759439 | 1.007593 |

p + m + 3e | hydrogen (H^{1}) |
1.008l28l5 | 1.008142 |

p + m + 3e + E | compound neutron | 1.00899621 | 1.008982 |

* potential mass |

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 *not a * defining equation like Newton’s second principle of mechanics and cannot be derived from defining equations. It represents an *observed relation” .* This is a theoretical discrepancy that conventional physics has not been able to resolve. But it is an isolated discrepancy, and it has been swept, under the rug by assigning fictitious dimensions to the gravitational constant.

It follows from this that the error lies in some interpretation of that “observed relation” that has been common to *both* conventional theory and the Reciprocal System. Evidently the developers of both systems of theory have misunderstood the true nature of the phenomenon. Here, again, recognition of the source of the difficulty points the way to the resolution of the problem. As brought out in the earlier chapters, one mass does not actually exert a force on another—each is pursuing its own course independently of all others—but the *results* of the inward motions of two masses are similar to those that would follow if the masses *did* attract each other. These results can therefore be represented in terms of an attractive force, on an “as if” basis. But in order to do this we must put the “as if” forces on the same footing as real forces.

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 *resistance* to the second mass. Thus, an “as if” force, a gravitational force, is exerted against an “as if,” inertial resistance. In the previous discussion we identified gravitation as three-dimensional motion, * s ^{3}/t^{3},* and inertia as three-dimensional resistance to motion,

*t*The product of the gravitational motion and the inertial resistance therefore does not have the dimensions of mass to the second power, as the conventional expression of the gravitational equation indicates; it is dimensionless.

^{3}/s^{3}.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:

(dynes cm^{2} g-^{2}) × g^{2} × cm-^{2} = dynes |
(13-1) |

Reducing equation 13-1 to space-time terms in accordance with the relations established in Chapter 12 (in which dynes, as g-cm/sec^{2}, are * t ^{3}/s^{3}*×

*s*×

*1/t*e have

^{2}= t/s^{2}), w(t/s^{2} × s^{2} × s^{6}/t^{6}) × t^{6}/s^{6} × 1/s^{2} = t/s^{2} |
(13-2) |

In the light of the new understanding of the *mm’* term as the dimensionless product of gravitational and inertial mass, it is now evident that the s^{6}/t^{6} dimensions belong with * mm’* rather than with the gravitational constant. When they are so applied, the resulting dimensions of *mm’ *cancel out, as the true theoretical dimensions do. We can therefore replace them with the correct dimensions. As pointed out in the first edition, there are also two other errors in the customary assignment of dimensions to this equation. The distance term is actually dimensionless. It is the ratio of 1/n^{2} to 1/1^{2} The dimensions that are mistakenly assigned to this term belong to a term whose existence has not been recognized because it has unit value, and therefore does not enter into the numerical calculation. In order to put the “as if” gravitational interaction on the same basis as a real interaction, we have to express it in terms of the action of a * force* on a resistance, not as the action of a *mass* on a resistance. And since the dimensions of the mass term cancel, so that the gravitational mass enters the equation only as a dimensionless number, the force of gravitation has to be expressed in actual force terms; that is, as t/s^{2}. The correct dimensional form of the equation is then

(s^{3}/t^{3} × t^{3}/s^{3}) × t/s^{2} = t/s^{2} |
(13-3) |

Turning now to the numerical magnitudes, we note that while the * dimensions* of the *mm’* term cancel out, the magnitudes do not. Every unit of mass is both a unit of s^{3}/t^{3} and a unit of t^{3}/s^{3}, each in its proper context. Since the units are independent, the effective magnitude of the “as if” action of *m* units of gravitation against * m’* units of inertial resistance is *mm’.* However, expressing both of the mass terms in conventional units introduces a numerical error, as only the inertial mass term is counterbalanced by a conventional mass magnitude on the other side of the equation. To compensate for this error a corresponding inverse factor must be introduced into the gravitational constant. There is no error if the gravitational mass is expressed in natural units, as the value 1 does not require any counterbalancing term. The relation between the natural and conventional units therefore determines the magnitude of the necessary correction factor.

One gram is 6.02486×10^{23} units of inertial mass (t^{3}/s^{3}). The reciprocal of this number is 1.65979×10^{-24}. But only one sixth of the total number of mass units is effective in the gravitational interaction because this “as if” interaction takes place in only one dimension, and in only one of the two directions in this dimension. The total number of s^{3}/t^{3} units corresponding to an effective mass of one gram is therefore 9.95874×10^{-24}. Expressing this mass as one unit overstates the numerical value, and a correction of this magnitude must therefore be included as a component of the gravitational constant.

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 9.95874×10^{-24}, we obtain 1.015093×10^{-23}.

Replacing the 1/s^{2} distance term by a t/s^{2} 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, 1.520655×10^{-16}, offsets in part the errors in the mass term. The net correction to be made is 1.015093×10^{-23} divided by the natural unit of time, and amounts to 6.67537×10^{-8}. This is the gravitational constant in the cgs system of units.

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^{-6} in the mass term, 10^{-4} the distance term, and 10^{-5} in the force term. A change of 10^{-3} in the gravitational constant is then required for a balance. In the theoretical equation (13-3) the net effect of a change in the system of units is confined to the relation of the natural and conventional units of mass. As can be seen from the explanation that has been given, the gravitational constant is proportional to the ratio of these units. Changing the conventional unit from grams to kilograms alters this ratio by 10^{-3}. The gravitational constant is then changed by the same amount. This agrees with the result obtained from equation 13-1.

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.