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3 The Nature of Matter Energy in the Relativistic Space-Time Domain D0. 3.1 Einstein's Relativistic Energy/Momentum Relationship. Via a simple re-arrangement of (2.1) using (2.23), this relationship can be derived easily as follows. Here E is assumed to possess both spatial and temporal components as advanced earlier. Accordingly, its terms have been so designated in the following derivation.
E is spatial - temporal total energy. fv is spatial - temporal matter wave frequency. lv is spatial - temporal matter wave wavelength. vv is spatial - temporal matter wave wave number.and thus from (2.2) and (3.1)
v is spatial vector velocity. M is the spatial momentum vector of the particle energy mass.Taking the magnitude of the final expression in (3.2) then gives
M is the magnitude of the spatial momentum.and (3.3) is the desired relationship. With regard to the existence of energy as a spatial - temporal quantity, it is seen from the central expression in (3.2), that the temporal component is the rest mass energy while the spatial component is the spatial term in (3.2). However, this raises an apparent anomaly in that the spatial term in (3.2) does not appear to approximate to the classical expression for kinetic energy. The validity of (3.2), and therefore its use in the above derivation of Einstein's energy-momentum equation, can be verified in two ways. First take the positive root of (3.3) thus
Thus containing the rest mass energy and the classical expression for kinetic energy, (the remaining terms are relativistic additions to the kinetic energy). A second verification of (3.2) can be effected as follows. In artificially induced motion it is well known that the rate of change of momentum is equal to the spatial distribution of energy. Thus taking spatial-temporal terms
s is a distance along the Existence Velocity Vector path.Thus inserting the momentum expression in (2.2) for M gives
Converting this as follows
But by definition, (see [1])
So that
Both the spatial and temporal terms in (3.13) are exact integrals, so that
as per (3.2). Its use in the above new simplified method of derivation of Einstein's relativistic mass - momentum relationship is therefore a valid one. Another point which arises from the derivation in (3.2) is that the final result, (3.3), is the result of taking the magnitude of a spatial-temporal expression. It can therefore only have a positive root. Therefore, this suggests that Dirac's negative energy field does not exist in D0. This point is discussed further in Section 4. 3.2 The Spatial - Temporal Distribution of Energy.Interpretation of the results of the previous Subsection is as follows. Consider a mass of matter in motion with a spatial velocity magnitude v. Clearly, from (3.14) the total energy of the mass, is directed along its Existence Velocity Vector. This can be pictorially represented as in Fig. 3.1 below. 
This representation is clear from (3.3), (3.6) and (3.7) where the magnitude of spatial - temporal terms about the Existence Velocity Vector have been taken to obtain the total energy. The spatial terms are therefore the projection of the total energy into the spatial dimension, i.e.
as in (3.2), (note that the vector designation in (3.2) has been dropped as unnecessary in this representation). The temporal terms are in concert with the spatial terms a projection of the total energy into the temporal dimension i.e.
Thus it is shown that although the accelerative force, and therefore the applied energy in such motion is all derived spatially, the distribution of this energy is shared between the spatial and temporal dimensions. The same is true of the rest mass energy as the spatial velocity increases and the Existence Velocity Vector rotates into the spatial dimension. The projection of the total energy into the temporal dimension contains an element of the kinetic energy, which compensates for the reduction in rest mass energy in that direction, so that the total energy in that dimension remains constant. This is necessarily so because all of the accelerating energy is directed in the spatial dimension, and no energy is either added or subtracted in the temporal direction. The energy projected into the spatial dimension increases as the spatial velocity increases but, as shown consists of both kinetic and rest mass energy elements as in the temporal dimension. It must be noted that despite this spatial-temporal distribution, the energy actually "perceived" in the spatial direction is that of the total energy and not just that reflected into it. That this is so evident from the fact that the total energy of the particle can be derived from just spatial considerations as well as from a combination of both spatial and temporal. However, it is believed that despite this "perception", the energy actually ävailable" in the spatial dimension is only that reflected into it. This is still greater than the energy supplied to cause the motion, and therefore may have interesting consequences at the quantum level, i.e. a different explanation for quantum tunnelling.
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P.G.Bass, October 2006
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