In the very early stages of Phase I, a physical boundary will depend upon the distribution of matter in the local vicinity, and would be expected to be at best very diffuse and ragged. However, as more and more matter was gravitationally drawn from the surrounding space, such material would become increasingly sparse. This would gradually result in a better defined boundary emerging as the Universe grew. Firstly, as a result, a firm initial value for the gravitational radius would be established. This would in turn establish the 3a_u criterion although, at this early stage, the gravitational radius would be so small compared to the physical radius, that the latter would be far greater than the 3a_u criterion.

Secondly, despite the increasing sparseness of outlying galactic masses as the Universe grew, it would still gravitationally draw from this material albeit at a reduced rate. The resulting question as to the relative rate of growth of the physical and gravitational radii, is examined in the next Section. Prior to that however, to show how the gravitationally migrating galactic material would arrive at the location of the core in Phase I, consider the equation of motion applicable to celestial bodies in the vicinity. This is given by the rectilinear version of [2], Eq.(3.18), thus

Eq.(2.1) was obtained from [2], Eq.(3.18) by putting w = 0.

The first integral of (2.1) is obtained from [2], Eq.(B4), again by putting w = w₀ = 0, thus

Substitution of (2.2) into (2.1) yields, after minor reduction

u is the temporal rate at the location of the gravitationally migrating body.

u₀ is the temporal rate at the initial location of that body.

Now, substitution of [2], Eq. (4.7) into (2.4), with the nomenclature selected to represent the Universe, gives after reduction

This is the distance from the centre of the core, at which the net acceleration towards that centre, of a random celestial body, becomes zero. It is at this distance that the positive acceleration produced by the velocity of the mass, as it moves through the spatial gradient of the Universe's temporal rate, exactly balances its negative Acceleration Potential. Clearly this distance will be different for each body depending upon s₀. For the case where s₀ >> 2a_u then (2.5) may be approximated by

and is thereby solely a function of the gravitational radius. The primary consequence of the above result is as follows.

Once a mass gravitating towards the core passes the point represented by (2.5), or as approximated by (2.6), it starts to be decelerated by the positive acceleration resulting from its velocity through the spatial gradient of the temporal rate generated by the core. This effect, at distances from the core below the value of s in (2.5), is greater than the Acceleration Potential at such distances. Consequently, when such masses approach the core they will have been greatly decelerated and their arrival will be more of a "gentle congregation" rather than a violent collision. In this way the core will grow progressively but more as a close proximity of a large number of masses rather than one single large entity. In this latter stage, the normal individual gravitational field of each core mass will also become significant in this process.

To finalise the development of Phase I, it is necessary to analyse the dynamic relationship between the physical and gravitational radii. This will show that a_u will grow faster than s_u, so that the 3a_u criterion is subsequently realised.

2.2  In Phase I - The Dynamic Relationship Between the Physical and gravitational Radii.

As the Universe begins to form, initially its gravitational radius will be very small. Subsequent to the emergence of the boundary, the gravitational radius can be expressed as

au = gmu c2 = 4 3 pgru su3 c2

(2.7)

where

r_u is the average density of the Universe

Although initially the average density could be quite high, the smallness of m_u coupled with the known values of g and c, will ensure that a_u << s_u. As these parameters grow, as more and more material is accumulated, the relative rate of growth can be determined by taking the differential of (2.7) with respect to s_u, thus

dau dsu = 4pg su2 c2 æ è ru + su 3 dru dsu ö ø

(2.8)

In (2.8) both s_u and r_u must be positive, but the sign of the term dr_{u /} ds_u will depend upon the density of the matter continually being accumulated. Also, the manner of accumulation near the core as discussed in Section 2.1 above, will also affect the sign of dr_{u /} ds_u. As the term involving dr_{u /} ds_u will predominate, the sign of (2.8) will therefore be primarily determined by it. Conjecture about this point can be resolved via reference to [3] Fig.1. There it is clear that as matter is accumulated and the mass of the Universe increases, the mass to radius relationship will eventually conform to [3], Eq.(2.3), and therefore using this equation, a_u can be empirically expressed as

au = gmu c2 = 5.36 x 1017g su1.38 c2

(2.9)

Consequently

dau dsu = 7.4 x 1017g su0.38 c2

(2.10)

Clearly here the change of a_u relative to s_u is positive and therefore, as the Universe evolves in Phase I the physical radius and the gravitational radius slowly converge until, at the centre, the 3a_u criterion is reached, and Phase II is initiated.

From (2.8) and (2.10), an empirical relationship for dr_{u /} ds_u can be expressed as

dru dsu = 3 su æ è 5.19 x 1016 su1.62 - ru ö ø

(2.11)

So that for dr_{u /} ds_u to always be positive so that (2.8) is always positive

ru < 5.19 x 1016 su1.62

(2.12)

As one example, when s_u = 10¹⁰ L.Y., from (2.12)

ru < 2.8 x 10-29grms/cm3

(2.13)

Compared with the estimates of r_u in [1], Section 2.3.4 for the home Universe, this inequality is certainly of the right order of magnitude.

The consequence of the above results, i.e. the faster growth rate of a_u compared to s_u, is that the 3a_u criterion will always eventually be realised. This will be so even if the galactic material in the surrounding space is near to inexhaustible. Of course, if this material is limited, the gravitational radius will become constant while the physical boundary of the Universe will start to shrink under the gravitational influence of the core. In this case the 3a_u criterion will thereby be realised earlier.

The final significant question concerning the evolution of the Universe as depicted in [1], and this paper, is, once Phase II has started, to what radius does the boundary fall before expansion starts. i.e. what is the radius of the point of inflexion. This is the subject of the next Section.

2.3  In Phase II - The Radius of the Point of Inflexion.

After Phase II has been triggered via the 3a_u criterion, the celestial bodies making up the boundary will continue to migrate towards the centre via momentum gained during Phase I.

The question analysed here is to what value does the physical radius fall at the point of inflexion. Subsequent to gravitational reversal, (s_u = 3a_u), the spatial/temporal flows are as shown in Fig.2.1 below, (repeated from [1], Fig.3).

Fig.2.1 - Spatial/Temporal Flows Subsequent to Gravitational Reversal.

In Fig.2.1 the flows along the Spatial Plane of Existence are the linear contraction velocities of the now contracting spatial dimensions. The transverse terms, (the j terms), are the corresponding temporal flow velocities.

From Fig.2.1 it is clear that a new criterion at the boundary, ((iii) in the Figure), has become established due to the transformation from Phase I to Phase II. That criterion would be realised should s_u shrink to become less than a_u. However, if the migration of all the matter towards the centre is arrested while s_u ³ a_u, then Phase II of the evolution, as described and developed in [1], will proceed normally. If not, and the collapse continues so that s_u < a_u, the Universe would not expand again. It is therefore very important to determine the radius of the Universe at the point of inflexion.

To investigate this question, it is necessary to first establish the velocity of the boundary in Phase I as it passes the critical point 3a_u.

In Phase I the velocity of a gravitating body is given by (2.2). Inserting [2], Eq. (4.7) into (2.2), i.e. expanding u and adopting the nomenclature to represent the Universe, gives

× s   u  = c æ è 1- 2au su ö ø 1/2   é ê ë 1- 1- 2au su u02 ù ú û 1/2

(2.14)

and substitution of su = 3au then reduces (2.14) to the initial condition at the boundary for the start of Phase II, call this

× s

0B

, then

× s   0B  = c Ö3 æ è 1- 1 3u02 ö ø 1/2

(2.15)

The total contraction of s_u can now be obtained as follows. The net acceleration experienced by a gravitating body in Phase II of a Relativistic Domain, is given by (2.1) with the sign of the Acceleration Potential term reversed. The solution of the resulting equation, with a non zero initial condition, for a gravitating mass on the boundary of the Universe, is then given by

× s   u  = cu æ ç è -1+ u2 u0B2 + × s   2 0B  u2 c2u0B4 ö ÷ ø 1/2

(2.16)

In (2.16) u is the temporal rate in Phase II for a gravitating mass at the boundary and is obtained from Fig.2.1(iii) as

u = æ è 1- au su ö ø 1/2

(2.17)

u_0B is the temporal rate in Phase II at the location of the boundary when s_u = 3a_u and from (2.17) is

u0B = æ è 2 3 ö ø 1/2

(2.18)

× s

0B

is the initial velocity of the gravitating mass in Phase II when su = 3au and is given by (2.15). Thus substitution of (2.15), (2.17) and (2.18) into (2.16) gives

× s   u  = cu æ è 1- au su ö ø 1/2   æ è 5 4 æ è 1- 1 5u02 ö ø - au su æ è 9 4 - 1 4u02 ö ø ö ø 1/2

(2.19)

The roots of (2.19) give the point of minimum radius of the boundary resulting from normal gravitational contraction in Phase I, and the deceleration in Phase II. Clearly the first root occurs when

su = au

(2.20)

and the second root occurs when

su = au ( 9u02 - 1 ) (5u02 - 1 )

(2.21)

The second root is larger than the first, and because u₀ must be close to unity, (2.21) is approximately s = 2a_u. This is the point of inflexion, and expansion in the remainder of Phase II starts from that radius.

The Acceleration Potential at the boundary in Phase II is, from [1], Eq.(2.12), with s_i = s_u

Au = gmu su2

(2.22)

and with s_u = 2a_u = 2gm_u / c² this becomes at the point of inflexion

Au = c4 4gmu

(2.23)

This is the Acceleration Potential on all matter at the boundary of the Universe after it has contracted to the point of inflexion in Phase II. This is clearly positive and initiates the ensuing expansion as developed in [1].

C3 Version 1.4.0
Ó P.G.Bass, May 2010