The discovery of the mass-energy relation E = mc2 by Einstein was a very significant advance in physical theory and has already had some far-reaching practical applications. It is, of course, entirely in harmony with the principles upon which this work is based and has been incorporated into the new theoretical structure, but when we develop this relationship from the Fundamental Postulates instead of following Einstein’s derivation we arrive at a somewhat different concept of the physical meaning of the equation which affects its applicability to a considerable extent.
From these postulates we find that mass and energy differ only in dimensions; that is, energy is the reciprocal of one-dimensional velocity while mass is the reciprocal of three-dimensional velocity. This mass-energy relation does not mean that a quantity of energy always has a certain mass associated with it; on the contrary it indicates that reciprocal velocity exists either as mass or as absolute momentum, or as absolute energy, depending on the effective dimensions, not as all three or any two simultaneously. Mass is equivalent to energy only when and if it is transformed from the one condition to the other, and the mass-energy equation merely gives the mathematical equivalents in the event of such a conversion. In other words, an existing quantity of energy does not correspond to any existing mass but to the mass that would exist if the energy were actually converted into mass. For these reasons Einstein’s hypothesis of an increase in mass accompanying increased velocity cannot be accepted. The kinetic energy increment could increase the mass only if it were actually converted to mass by some appropriate process, and in that event it would cease to be kinetic energy; that is, the corresponding velocity would no longer exist. Actually this hypothesis of Einstein’s is inconsistent with his concept of the conversion of mass into energy, regardless of the point of view from which the question is approached. Mass cannot be a by-product of kinetic energy and also an entity that can be converted into kinetic energy; the two concepts are mutually exclusive.
This hypothesis was formulated as a means of accounting for the otherwise unexplained decrease in acceleration at very high velocities, but in the system now being developed from the Fundamental Postulates this phenomenon is found to be due to the vanishing of force as velocity approaches unity, rather than to any variation in mass. In the theoretical universe now being described the mass-energy relation is applicable solely to those situations in which mass disappears and energy appears, or vice versa. The familiar process of this kind is the interchange between mass and energy which takes place as a result of radioactivity or other similar atomic transformations. We will first examine the characteristics of this process, after which we will go on to a consideration of some additional conversion processes which have not hitherto been recognized, but which are necessary consequences of the principles developed in the preceding discussion and will play important parts in the cosmological theories that will be derived by a further extension of these same principles.
It is evident from the facts brought out in the examination of the atomic transformation phenomena that both rotational and vibrational primary mass are conserved in these reactions. In the radioactive disintegration Ra226 → Rn222 + He4, for example, the total primary mass of the original radium atom is 226. The primary mass of the residual radon atom, 222, and that of the ejected alpha particle, 4, likewise add up to 226. Any mass-energy conversion involved in these atomic transformation processes is therefore confined to the secondary mass.
The nature of the secondary mass has already been indicated in connection with the evaluation of the atomic mass unit. As stated in the Principle of Inversion, any motion of an atom within a space unit is equivalent to and in equilibrium with a similar motion of the space unit in which it is located. Normally the two motions are in opposite directions, since one is the inverse of the other. As we have seen, the normal effect of the induced rotational space motion is to produce a negative mass of 0.0049 units per natural mass unit, which reduces the size of the atomic mass unit accordingly. In hydrogen, where the force integral is less than unity, the space motion is in the same direction as the effective atomic rotation and the secondary mass is therefore positive. In this case, however, only one dimension of motion is effective and the mass of the hydrogen atom is 1.0025 on the natural scale or 1.0074 amu.
If four hydrogen atoms combine, the positive secondary mass of the hydrogen atoms is replaced by negative secondary mass in the product, a helium atom. Helium has 11 units of secondary mass (0.0037 amu) more than the amu standard and the total reduction in mass is
(4 × 1.0074) - 4.0037 = 0.0259 amu.
The equivalent of this mass appears as kinetic energy of the products. Such reactions therefore constitute a means of energy production.
Hydrogen is the only element with positive secondary mass but the negative mass can vary all the way from zero, which corresponds to a total of 1.0049 amu per unit of primary mass, to the full three-dimensional value, 0.0074, which reduces the net effective mass to 0.9975 per primary unit. The values applicable to the individual isotopes depend on the characteristics of the particular rotational system and since each primary unit is independent from this standpoint there is a considerable range of variation, especially among the unstable isotopes. As in the vibrational mass, however, there is a definite center of the zone of variation and the stable isotopes are generally found at or near this center. Like most physical relations of this kind the secondary mass curve has a negative initial level and since the secondary mass is itself negative with respect to the primary mass, the initial level is positive on the primary basis. In the lower atomic mass range this initial level decreases in linear relation with the magnetic rotational displacement and it can be represented as 12-a, where a is the effective number of magnetic units above the rotational datum, 1-0-0. Beginning with this positive initial level 12-a, each successive element in the range up to element 26 adds one secondary mass unit (-0.0025 amu) and the deviation from 1.00 m amu for an isotope of primary mass m and atomic number n in this range can be expressed as
mass deviation = 0.0025 [12 - (a + n) ] amu (138)
Table CVIII shows how the experimental mass deviation of the most abundant isotope of each of the elements 3 to 26 inclusive compares with the value calculated from equation 138. Hydrogen is excluded from the application of this equation as its secondary mass is positive. Helium is also excluded because the deviation thus computed exceeds the maximum possible deviation as indicated in the previous discussion. Beginning with element number three, lithium, the measured values are in general reasonably close to the calculated center of the zone of variation as far as element 26, iron. Beyond iron there is a greater degree of uncertainty, both in the experimental values and in the theoretical relation, but it appears that the deviation remains in the neighborhood of the iron value, with the usual plus or minus variations in the individual isotopes, for another range of approximately the same number of elements. After this it decreases and returns to positive values in the very heavy elements. The effect of this secondary mass pattern is to make both the growth process in the light elements and the decay process in the heavy elements exothermic.
Many investigators have devoted considerable effort to the study and analysis of atomic transformations of this type which might possibly serve as a source of the energy generated in the sun and other stars. The general conclusion has been that the most likely reactions are those in which hydrogen is converted into helium, either directly or through a cycle of intermediate reactions. Hydrogen is the most abundant element in the stars and this process, if it is actually in operation, constitutes an energy source of sufficient magnitude to account for the observed energy production.
The existence of a fusion process of this kind is entirely compatible with the basic principles of this work, but the hypothesis that it is operative under the conditions prevailing in the stellar interiors and is the primary source of the energy of the stars is not consistent with the principles that have been developed in the preceding pages. A serious objection is that reactions of this kind are reversible and there is no adequate reason why the reaction between helium and the hydrogen isotope H1 should proceed preferentially in the direction H → He. The situation with respect to the H2 and H3 isotopes is entirely different. These isotopes are unstable under terrestrial or similar conditions and are therefore subject to reactions which convert them into stable isotopes. Such reactions take place spontaneously but can be speeded up by application of additional kinetic energy and if H2 or H3 are present in the stars in substantial quantities a process of conversion to He4 could be an important energy source. Available evidence indicates, however, that most of the hydrogen in the stars is in the H1 state, as would be expected from the probable level of magnetic ionization, and H1 is just as stable as helium.
At a very high temperature the chances of an atomic break-up and rearrangement are improved but this does not necessarily increase the proportion of helium in the final product; on the contrary, we have seen that a greater kinetic energy results in more fragmentation and it therefore favors the smaller unit rather than the larger. Furthermore, an increase in the amount of space displacement (thermal motion) is not conducive to building up time displacement (mass). The two principal processes which have been postulated as stellar energy sources begin with the reactions H1 + H1 → H2 and C12 + H1 → N13 respectively. These reactions involve combination of stable isotopes to form unstable isotopes and combination of smaller units to form larger units. In both of these respects the direction of the proposed reactions is in direct opposition to the normal probabilities under the prevailing conditions.
A second objection to the hypothetical fusion reaction is that it is a “dead end” process, and as such is open to criticism from both the theoretical and the observational standpoints. The Fundamental Postulates definitely require all basic physical processes to be cyclic and any one-way process such as the conversion of stellar hydrogen to helium violates this general principle. Also if this hypothesis were valid there should be some evidence of the existence of helium-rich structures, representing the later stages of the hypothetical stellar evolution. No such evidence is available. It is true that there are some peculiar structures, the white dwarf stars in particular, for which no satisfactory explanation has heretofore been found and which have therefore been postulated as the victims of hydrogen exhaustion and “collapse.” It should be understood, however, that this is pure speculation and there is no actual evidence that these white dwarf stars are rich in helium. There is, in fact, some collateral evidence that will be discussed in a later section which indicates that the white dwarfs contain much less helium than the average star, rather than more. At that time it will also be shown that the white dwarfs are not abnormal and that they are in the direct line of stellar evolution.
When the fusion process is thus ruled out as the source of stellar energy the question then arises as to what alternative energy generation process is operative under the existing conditions. Since the most distinctive physical condition within the stars is the very high temperature, this question reduces to the problem of determining what happens to matter under extreme temperature conditions. The answer to this problem is evident from the nature of the atoms of matter: if the temperature continues to rise the total space displacement, thermal energy and its equivalent, must eventually reach a destructive thermal limit.
There have been many instances in the preceding pages in which a limiting magnitude has been established for the particular quantity under consideration. The electric ionization of atoms, for instance, is limited to the equivalent of the net rotational displacement; that is, the element magnesium, which has 12 net effective electric rotational displacement units (equivalent basis) can take 12 units of electric vibrational displacement (ionization) but no more. Similarly we found that the maximum rotational base of the thermal vibration in the solid state is the primary magnetic rotation of the atom. Most of the limits thus far encountered have been of this type, which we may designate as the non-destructive limit. When such a limit is reached, further increase of this particular quantity is prevented, but there is no other effect.
We are now dealing with some physical phenomena which are subject to a different kind of limit: a destructive limit. The essential difference between the two stems from the fact that the phenomena to which the non-destructive limits apply are subsidiary properties, not the basic motions which are the essence of the unit under consideration. The electric rotation, for instance, is purely a supplement to the basic magnetic rotational motion of the atom, and reaching the electric ionization limit does not in any way imperil the existence of the atom itself. On the other hand, if the variable motion is in direct opposition to the basic motion of the system, the attainment of equality between the two motions has a deeper significance. When an oppositely directed velocity - a is superimposed on a rotational velocity + a, the net total is zero and there is no longer any rotational velocity at all. If this is the primary rotation of an atom or sub-material particle, or any full unit of that primary rotation, the existence of the rotating displacement unit automatically terminates and the displacement reverts to the linear basis (radiation).
A simple example is provided by the combination of a positron and an electron. The positron can combine in a normal manner with any other kind of a material or sub-material particle because the addition product still has an effective rotational displacement and therefore exists as a particle. When it combines with the electron, however, the resulting net rotational displacement is zero and the addition product is not a rotating system but a pair of oppositely directed photons. Since each particle entering into this reaction is only a single displacement unit, the destructive limit is reached by combination with a single unit of the opposite kind and the result is the complete destruction of both particles. In the more general case of the atomic rotational combinations, the basic rotation consists of several magnetic displacement units. Each of these is an independent entity and when the opposing displacement reaches equality with one of the basic units this unit is destroyed and the element is transformed into an element of a lower magnetic group, the neutralized displacement unit being converted into linear motion (energy).
The magnetic rotation is a two-dimensional motion with a time displacement, t2/s2. As indicated in the preceding discussion, any such n-dimensional motion is the equivalent of a specific amount of one-dimensional motion, t/s or energy. The thermal motion of the atom is an equivalent space displacement, the direct opposite of the displacement of the magnetic rotation. At the higher temperatures electric ionization also occurs, and since this ionization involves the addition of more space displacement, the total space displacement in opposition to the time displacement of the magnetic rotation is the sum of the thermal displacement and the electric ionization displacement. The thermal energy in fully dissociated gases is independent of the atomic mass but the maximum ionization level increases with atomic number, hence at extreme temperatures where all substances are completely ionized the heavier the atom the greater the total space displacement. This means that a heavier atom reaches the limiting value of the space displacement at a lower temperature. When the temperature of a star reaches the level which represents the destructive thermal limit for the heaviest atom present, one unit of the magnetic rotation of this atom is neutralized and the corresponding rotational displacement (mass) is converted into linear displacement (energy). As the rise in temperature continues one after another of the elements meets the same fate in the order of decreasing atomic number.
Here we have not only a source of practically unlimited energy but also just the sort of process which we found is necessary to account for the scarcity of heavy elements. This process of destruction of primary mass is, of course, purely theoretical. There is no direct experimental or observational evidence that such a neutralization of mass actually does take place, except to the extent that the observed neutralization of the electron and positron rotations can be extrapolated to apply to the atomic situation. It should be remembered, however, that all of the material in this presentation is theoretical; the specific objective of the work is to develop a theoretical universe from the two Fundamental Postulates, and all of the phenomena and relations previously described are theoretical deductions. The only difference is that it has usually been possible heretofore to verify each successive step by comparison with observation or measurement. We cannot verify the validity of this particular step by any direct method and we will have to develop the theory further before making the usual comparisons, but since the whole theoretical structure is a fully integrated unit a satisfactory correlation at a higher level should confirm the validity of the intervening steps.
When we turn to the second of the two destructive limits which should be considered at this time, the magnetic ionization limit, the gaps in the correlation between theory and observation are still greater. Here again, however, the theory as outlined consists of a series of straightforward deductions from principles whose validity has been established in the preceding pages, and wherever comparison with the results of observation or measurement can be made the correlation is satisfactory. The gaps are there only because we have no experimental knowledge at all in certain areas. In discussing the nature of the limiting value of the thermal energy we are dealing with a limit of which we have no direct observational evidence. In the case of the limiting magnetic ionization level it is not only the existence of the limit that cannot be verified experimentally at present; we have no actual evidence of the ionization level itself, except to the extent that the existence of isotopes can be accepted as confirmation. It is clear, however, that both magnetic ionization and a limit thereto are required by the Fundamental Postulates which define the theoretical universe that is being developed herein, and since (1) there are no observations that contradict these findings or the consequences thereof, and (2) the extension of these concepts in the preceding and subsequent pages leads to many conclusions which are fully confirmed by observation, the verification would appear to be as satisfactory as can be expected in the present state of experimental knowledge.
We have found that the accumulation of charged neutrinos within a material aggregate leads to the magnetic ionization of the atoms of matter, and we have further found that the increase in the neutrino population is cumulative, so that the magnetic ionization level increases with the age of the aggregate and ultimately reaches the destructive limit. In general, the magnetic ionization limit is the same kind of a phenomenon as the thermal energy limit. The points brought out in the discussion of the latter are therefore equally applicable to the magnetic limit and will not need to be repeated. There is one significant difference which should be pointed out. The magnetic ionization of the atoms is in time and its direction therefore coincides with that of the atomic rotation rather than opposing it. For this reason there is no level at which the displacements add up to zero in the manner of the space and time displacements at the thermal energy limit. As we have previously seen, however, there is an upper limit to the rotational displacement, which is in effect another physical zero point, and increasing magnetic ionization approaches this upper zero point rather than the mathematical zero. Attainment of this upper limit destroys the atomic rotation and terminates the existence of the particular element just as effectively as reaching the lower zero point, but there are some important differences in the details of the two processes which we shall consider in connection with some of the matters that will be examined later.