5.3.3 Deriving the Mass-Energy Relation
In addition to explaining the constancy and invariance of the speed of light and merging it with the second and third principles of Relativity (see sections 3 and 2.2), the most significant implication of the Duality of Time Postulate is that it appears to be the only way to explain the equivalence and transmutability between mass and energy (
).
Einstein gave various heuristic arguments for this relation without ever being able to prove it in any theoretical way, neither did anyone else. Based on Doppler effect and Maxwell’s theory of radiation, the reasoning that he gave in his 1905 original derivation was questioned by Planck and shown to be faulty. In 1907, Einstein acknowledged the controversy over his derivation, and in the following forty years, he produced more than half dozen proofs that all suffer from unjustified assumptions or approximations. He never succeeded in producing a valid general proof.
In 1955 he wrote in a letter to Carl Seelig: “I had already previously found that Maxwell’s theory did not account for the micro-structure of radiation and could therefore have no general validity.” Capria (2005). In 1990, Rohrlich tried to apply the relativistic Doppler shift, but he also had to introduce various approximations in order to reach the final equation Rohrlich (1990). Various other attempts have been made in this regard but until now there is no exact derivation of this famous formula.
It can be readily seen from Figure 5.4 that the transmutability between mass and energy can only occur in the inner levels of time, because it must involve motion at the speed of light that appears on the normal level of time as instantaneous, hence the same Relativity laws become inapplicable since they prohibit massive particles from moving at the speed of light, in which case the mass
would be infinite. In the inner levels of time, however, this would be the normal behavior because the geometrical points are still massless, and their continuous coupling and decoupling is what generates mass and energy on the inner and outer levels of time, respectively.
As we introduced in section 2.2 above, the normal limited velocities of massive physical particles and objects are a result of the spatial and temporal superposition of the various dual-state velocities of their individual points. This superposition of instantaneous velocities occurs in the inner levels of time where individually each point is massless and it is either at rest or moving at the speed of light, but collectively they have some non-zero inertial mass
, energy
, and limited total apparent velocity
, which can be calculated from equation 5.3.
Based on this metaphysical behavior in the inner levels of time, we will provide in the following three simple methods to derive the mass-energy equivalence relation directly from the classical equation of mechanical work
. The first two methods that lead to equations 5.9 and 5.10 are equivalent, and they respectively involve integration (or rather: summation) in the inner time when velocity changes abruptly from zero to
, or when the mass changes also abruptly from zero to
(or vice versa). This is obviously not allowed on the normal level of time when dealing with physical objects that have mass.
The third method is obtained from the geometry of Figure 5.5 which gives the relativistic (effective) mass equation
and the relativistic energy equation 5.15, as will be explained further in section 3.2 and section 3.3.8.
In normal classical mechanics, the kinetic energy is the work done in accelerating a particle during the infinitesimal time interval
, and it is given by the dot product of force
and displacement
:
(5.5)
Thus:
(5.6)
Now if we assume mass is constant, so that:
(we will discuss relativistic mass further below), we will get:
(5.7)
The science of Time is a noble science, that reveals the secret of Eternity. Only the Elites of Sages may ever come to know this secret. It is called the First Age, or the Age of ages, from which time is emerging.
