2.3  The Mathematical Development.

This will cover the process after the formation of a definable boundary in the first phase and then proceed through both phases. Throughout, the characteristics of each phase will be derived and the precise mechanism resulting in the change between them demonstrated. The details covering the initial part of the first phase including the dynamic relationship between the physical and the graviational radii will form the subject of a future paper.

2.3.1  The First Phase and the Change to the Second.

The first phase has been described as one in which the Universe is generating a spatial expansion volume such that the associated spatial linear expansion velocity at some random point s_i is of the same form as that of [3], Eq.(3.30). To give this mathematical form, it is expressed as

where

m_u is the mass of all matter in the Universe. s_u is the physical radius of the Universe.

To determine the Acceleration Potential in this phase, (2.1) cannot simply be differentiated with respect to the time. This is because the physical radius, s_u, is now a secondary variable, unlike in most other gravitational sources in which the physical radius is constant. The Acceleration Potential at s_i is solely a function of the spatial distribution of u_is at that point, and includes the magnitude of s_u but not its rate of change. To determine the Acceleration Potential therefore, it is necessary to partially differentiate (2.1) with respect to s_i and then transform the result to a time variable. Thus the partial differential of (2.1) is

and the Acceleration Potential is then given by

Ais = uis ¶uis ¶si = - gmu si su3

(2.3)

and which therefore has the same form as that internal to all other gravitational sources as represented by [3], Eq. (3.35).

The second order rate at which the spatial volume at s_i is expanding in this phase, is from (2.1) and (2.3)

¶2W ¶t2 = 4psi2 ( Ais ) + 8psi ( uis2 )

(2.4)

= 24pgmu si su -12pgmu si3 su3

(2.5)

Which at the edge of the Universe, when s_i=s_u, becomes

¶2W ¶t2 =12pgmu

(2.6)

Eqs.(2.5) and (2.6) are the same as for a normal gravitational source applicable both internally and externally as discussed in [3]. Now consider the temporal rate within the Universe. This, in line with [3], Eq.(3.31), is given by

ui = æ è 1- 3au su + au si2 su3 ö ø 1/2

(2.7)

where

a_u is the gravitational radius of the Universe.

At the very centre of the Universe, i.e. when in (2.7), s_i = 0, u_i becomes

ui = æ è 1- 3au su ö ø 1/2

(2.8)

and is the temporal rate at the centre.

At this point to best illustrate the mechanism by which this first phase of the evolution is transformed to the second, it is useful to introduce a somewhat more detailed version of the spatial/temporal flow diagram first shown in [3], Fig. 3. The diagram presented below for the first phase shows the spatial/temporal flows both internal to the Universe and just beyond its outer periphery.

Fig. 1 Spatial/Temporal Flows in the First Phase of the Evolution of the Universe.

In Fig. 1 the terms along the Spatial Plane of Existence are the applicable spatial linear expansion velocities at (i) the centre of the Universe, (ii) the general point s_i internal to the Universe, (iii) at s_u, the edge of the physical Universe, and, (iv) the general point s external to the Universe. The terms transverse to the Spatial Plane of Existence, (the j terms), are the corresponding temporal velocities. All of these terms were developed in [3] for a single gravitational source.

As all matter in the Universe gravitates towards the centre, eventually its physical radius, s_u, and its gravitational radius, a_u converge to the point whereby the physical radius equates to three times the gravitational radius. When this occurs, it can be seen from (2.8) and Fig.1 that at the centre of the Universe the temporal rate, and therefore the temporal flow, is reduced to zero. At this point therefore the passage of time at the centre has stopped. Also the spatial linear expansion velocity at this point has increased to the maximum possible, the velocity constant c. At this one singular instant the spatial/temporal flows have become as shown in Fig. 2 below.

Fig. 2 Spatial/Temporal Flows at s_u = 3a_u+

As can be seen from Fig. 2, at all points of the Universe other than the centre, the spatial/temporal flows continue to generate a negative Acceleration Potential, as in (2.3), which in turn continue to cause the gravitational migration of all galactic masses towards the centre. This tends to cause the physical radius of the Universe to reduce to less than three times its gravitational radius. As a consequence, at the centre, the temporal flow undergoes the following transformation at the critical point when s_u = 3a_u exactly. Using (2.8),

Transformation from

dx0 dt ê ê su = 3au +  = j c æ è 1- 3au su ö ø 1/2   ê ê su = 3au +  = j 0

(2.9)

To

dx0 dt ê ê su = 3au -  = j c é ë - æ è 3au su -1 ö ø ù û 1/2   ê ê su = 3au -  = -0

(2.10)

Thus the temporal flow at the centre has been transformed to a negative spatial flow, and it is proposed that at that instant, for reasons of flow continuity, this change is promulgated throughout the rest of space. In this way the first phase of the evolution of the Universe is thereby transformed to the second.

2.3.2  The Second Phase.

At the instant of the above transformation, the spatial/temporal flow pattern changes from that in Fig. 2 to that in Fig. 3 below.

Fig. 3 Spatial/Temporal Flows in Phase II

As can be seen from Fig.3, and also implicit in (2.9) and (2.10), both the spatial flow, and the temporal flow in the past region of the temporal dimension, have, in this form of representation, been rotated clockwise through 90 degrees by the transformation.

The main characteristics of this phase are as follows. First, from Fig. 3, it can be seen that the spatial linear expansion velocity is now negative and the spatial dimension is now caused to contract instead of expand as in the first phase. Thus u_is in this phase is a contraction velocity and, from Fig. 3 for the general point s_i, is given by

uis = - æ è gmu si2 su3 ö ø 1/2

(2.11)

From this, the internal Acceleration Potential in this phase is

Ais = uis ¶uis ¶si = gmu su3 si

(2.12)

Eq. (2.12) shows that gravity internal to the Universe in this phase has reversed and is now repulsive. Under this Potential the inward migration of all galactic masses towards the centre is gradually halted, reversed and then, with a steadily increasing velocity, gravitationally accelerated away from the centre. This phase continues until the dispersion of all galactic objects is so great that the Universe no longer retains a unique gravitational capability.

In this phase the second order rate of spatial contraction at s_i is as follows

¶2W ¶t2 = 4psi2 æ è gmu su3 si ö ø + 8psi æ è gmu su3 si2 ö ø

(2.13)

= 12pgmu si3 su3

(2.14)

Which at the edge of the Universe, the "surface" of the gravitational source, reduces to

¶2W ¶t2 =12pgmu

(2.15)

Which is identical to the boundary condition in phase I. The external condition has however changed. Again the spatial flow has become negative in concert with the internal flow. This occurs as a result of continuity across the boundary. The Acceleration Potential outside the Universe is still attractive but the magnitude has halved and from Fig. 3 is given by

dus dt = - gmu 2s2

(2.16)

Accordingly the second order rate of change of spatial volume contraction, while still positive, is also halved and is given by

d2W dt2 = 6pgmu

(2.17)

As a result, from (2.12) and (2.16), in addition to their opposed direction of action, there is a discontinuity in the magnitude of the Acceleration Potentials at s_u, the edge of the universe. This consequently tends to accentuate the density across the boundary.

The effects of the reversed gravity of the second phase on a galactic mass within the Universe can now be derived. This is accomplished in the following Sections.

C1 Version 2.1.1
ÓP.G.Bass March 2006

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