The Special Theory of Relativity asserts that the mass of a fixed quantity of matter, spatially in motion with a constant rectilinear velocity in Pseudo - Euclidean Space - Time, is greater than when it is at rest. For this to be so the increase in mass can only take place during the time that spatial acceleration is in effect. This process can be investigated by treating mass as a variable when analysing the change in the Existence Momentum of a point mass subjected to spatial acceleration in D₀ .

If m is the mass of the point mass with Existence Velocity V in D₀ , then its Existence Momentum will be given by:

where V has been substituted from (2.7) . If F is the force applied to effect acceleration then:

Showing that there are four kinetic reaction terms involved in this process.

If F is purely spatial, the temporal component of (3.2) will be zero, whereby:

so that upon separating variables and integrating

The constant of integration is obtained from initial conditions viz: when v = 0, m = m_o, the mass of the point mass when spatially at rest in D₀ , i.e. the 'rest mass'. Then:

which gives in (3.4)

as asserted in the Special Theory. However, it is clear from the above development that, in addition to a constant spatial velocity, (3.6) is also valid during the time that spatial acceleration of a point mass is in effect. For reasons that will be discussed later m will be referred to as Energy Mass.

Substitution of (3.6) into (3.3) yields:

which thus represents the time rate of change of mass subjected to spatial acceleration in D₀ . This can also be obtained by simply differentiating (3.6) with respect to the time t. These last two terms, (3.6) and (3.7), may now be inserted into (3.2), whereupon the temporal component vanishes and, if rectilinear motion only is being considered, F can also be reduced to a spatial vector, F, so that (3.2) yields, after reduction:

As F is arbitrary, (3.8) represents the spatial rectilinear equation of motion of a point mass in D₀ . (Non rectilinear motion is examined in Sections 4 and 5). Note that the right hand side of (3.8) is the product of the spatial acceleration and a mass term.

Putting

then m_a is, from (3.8), synonymous with inertial mass. Three values of mass have thus been identified for the same point mass i.e.

m_o - Rest Mass

m =

m0 æ è 1 -  v2 c2 ö ø 1/2

  - Energy Mass

ma =

m0 æ è 1 -  v2 c2 ö ø 3/2

  - Inertial Mass

The latter two are sometimes referred to in the literature [1],[2],[4] as 'transverse' and 'longitudinal' mass, (see Section 4) .

For interpretation of these results, reference is made to Fig. 3.1. where it is shown that as a consequence of the rotation of V in D_o, the applied spatial vector force F may be resolved from two spatial - temporal vector components, F_a normal to V and F_e parallel to V.

The component F_a is proportional to the change in Existence Velocity, while the component F_e is proportional to the change in the mass. This is clear from (3.2) where by inspection:

and

From these equations, (3.2) can now be interpreted kinetically. From (3.10) and (3.12) it can be seen that the kinetic reactions to the two components of F, each comprise a spatial and temporal term. If the balanced force vector F is diagrammatically represented as in Fig. 3.2, the four kinetic reaction terms of (3.2) can be interpreted as follows:

(i) The spatial term, mdv/dt is the reaction force of the energy mass to spatial acceleration.

(iii) The temporal term,  jc ( 1 - v²/c² )^1/2.dm/dt is a reaction force generated by the combination of mass rate and temporal velocity and acts in opposition to the term in (ii).

(iv) The spatial term, vdm/dt is a reaction force generated by the combination of mass rate and spatial velocity and acts as an additional reaction to spatial acceleration, thereby causing the apparent mass to increase, from m to m_a, during the period of acceleration.

A consequence of this, is that this last term must be related to the difference between inertial and energy mass, by the product of that difference and the spatial acceleration. This may be shown as follows.

From (3.6) and (3.9).

which with insertion of (3.7) gives the required relationship:

Finally, a comment upon two additional points that emerge from the above analysis. Firstly the fact that the two temporal reaction terms, items (ii) and (iii) above, are, as shown by (3.2) and in Fig. (3.2), to be equal in magnitude but opposite in sign, does not mean that they do not separately exist. Whilst they do in the above example, cancel, they are quite different in nature, the first being a mass reaction to temporal deceleration and the second a mass rate reaction to temporal velocity. They are equal in magnitude but opposite in sign solely because, in this case, there is no net temporal component of impressed force, i.e. F is purely spatial.

The second point concerns the inertial mass m_a. While m_a can be expressed solely as a function of the spatial velocity, as is apparent from (3.9), it is equally apparent from (3.15) that its existence is entirely dependent upon the spatial reaction term vdm/dt. As this term only exists while spatial acceleration is taking place, so then can m_a only exist during this period. When spatial acceleration ceases, the term vdm/dt vanishes and, the value of the mass instantly reverts to that of the energy mass, m.

3.2  Energy