The critical size of a gravitational source is herewith defined as the size at which the internal characteristics change, such that it starts to generate a repulsive type internal gravitational field, (Acceleration Potential), instead of an attractive type field. For the entire Universe this was shown in [2] to be when

(2.1)

where

To determine whether this condition can apply to any other naturally occurring accumulation of matter, [1], Eq.(4.6) is inserted into (2.1), the nomenclature changed to reflect that of a normally sized gravitational source, and the resulting equation re-arranged for the mass, thus

(2.2)

Eq.(2.2) shows the relationship between the mass and the radius of any gravitational source for which an identical criteria to (2.1) is met. Using a value of 2.99 x 10¹⁰ cm/sec for c, [3], and 6.67 x 10 ^-8 cm³/gm.sec² for g , [4], a table of values of m_g versus s _g for this criteria can be constructed from (2.2), as in Table 1 below.

For reasons that will be discussed later, Table 1 has been constructed to cover the smallest composite matter particle for which a dimension is known, the Nucleon, up to the largest known accumulation of matter, the entire Universe. Also note that the above value used for c is the velocity of light at the surface of the Earth. This represents an approximation because the correct value that should be used is the Terminal Velocity in Pseudo-Euclidean Space, which will be slightly larger than the above value because of the absence of a gravitational field, See [1] Appendix D for a further explanation.

Table 1 is now graphed, to LOG-LOG axes, (base 10), in FIG.1, as the straight line relationship denoted A.

To the right of Line A, a gravitational source would be stable and produce internally a normal attractive type gravitational field in which the Acceleration Potential was negative. To the left of Line A the opposite would be true. To determine whether any celestial bodies other than the entire Universe can produce an internal field of repulsive gravity, FIG.1 needs to be augmented with points representing the full range of celestial bodies for which the appropriate parameters are known.

Table 2 has therefore been constructed relating the radius and mass of various objects within and up to the entire Universe. Scientific notation is used.

Insertion of the mass and radius for items 1 - 8 and 15 from Table 2 into FIG.1 then produces the individual points so labelled. The other entries in Table 2 are discussed later. Interpretation of the graph then results in the comments in the following five Sections.

The first observation concerning FIG.1 is that the points designated 1 to 4, appear to lie on a straight line. That line has been designated B. This suggests that the gravitational accumulation of matter where nuclear fusion is present in the core, may conform to a "law" represented by the equation of this line. Note that point 1 for the entire Universe has been included in this trend despite the fact that its radius is a variable. However, the radius of the Universe used in Fig.1 is that estimated via the process described in Note 1 to Table 2, and in Appendix B. This value is sufficiently close to line B in FIG.1 for its inclusion in the trend line not to produce a significant error. This is so in view of the magnitude of the numbers concerned and the relative "slowness " with which the radius of the Universe is changing. This "coincidence" is explained in Section 2.3. Also it is noted that the use of a straight line makes allowance for the possibility that the radius of the largest star, point 3, may still be reducing as the star continues to evolve.

The equation of line B is,

(2.3)

From (2.3), for any such celestial body it is only necessary to determine an estimate of its size in order to obtain an estimate of its mass and/or its average density. However, the axes scales of FIG.1 are extremely coarse and so any prediction using (2.3) is only accurate to about a factor of 4, (except for point 3 which is accurate to a factor of 9). Nevertheless, because the masses of such celestial bodies is so large, the above degree of inaccuracy is acceptable as an initial estimate.

From FIG.1 it is clear that the planetary data points, 5, 8 and 15 do not lie on line B, and therefore (2.3) does not apply to planetary sized bodies. To determine whether an alternative empirical relationship to (2.3) exists for such bodies, FIG.2 has been constructed from all the planetary data of Table 2. Again, although some spread is evident, a clear trend is apparent as shown by the trend line drawn.

The equation of this line is

(2.4)

Predictions using (2.4) are accurate to about a factor of 2. A quadratic law has been chosen here for two reasons. Firstly, it clearly fits all the points reasonably well, and secondly it would be expected that its slope should decrease as it converged with line B in FIG.1. This is so because at the exact point of convergence, the planetary mass would then have become sufficiently large for fusion to ignite in the core, and the mass begin the process of turning into a star. To illustrate this, when (2.3) and (2.4) are equated, i.e. the masses produced by both equations are the same, they reduce to

(2.5)

Eq. (2.5) has roots of

(2.6)

LOG₁₀(s _g) cannot be complex and so it is assumed that the appearance of the small imaginary term in (2.6) is due to the inaccuracies in the data used in the plots. Therefore, assume that this small imaginary term can be ignored. Then (2.6) gives

(2.7)

which therefore yields

(2.8)

Inserting (2.8) into either (2.3) or (2.4) then gives for the corresponding mass

(2.9)

Eqs.(2.8) and (2.9) are then the radius and mass at which a planetary body starts to transform into a star. To provide further support for this hypothesis consider the slopes of (2.3) and (2.4). First for (2.3)

(2.10)

and then for (2.4)

(2.11)

When these slopes are equal, equating (2.10) and (2.11) produces

(2.12)

from which

(2.13)

i.e. virtually the same value as in (2.7). Therefore, the trend line in FIG.2 is tangential to line B in FIG.1 at the point of convergence.

These laws exist because as matter gravitationally accumulates, the external and internal processes governing its formation, result in its mass/radius relationship, i.e. its average density, moving towards conformance with these curves. The external process is the increasing gravitational compression as the mass grows and, the internal process is the pressure and temperature generated throughout the internal structure in response to this compression. Stability is reached when the external and internal processes balance, and there is no further external material from which the mass can grow. When that occurs the accumulated mass remains on the point of the curve, B in FIG.1 or the trend line in FIG.2, it has reached. Thus, as a planetary mass starts to gravitationally accumulate, its mass/radius ratio will move towards and then up the trend line in FIG.2 until its mass reaches the value given in (2.9). At that point the internal temperature and pressure have become sufficiently high for nuclear fusion to ignite in the core, and the accumulated mass begins transformation into a star. Further gravitational accumulation of matter causes the mass/radius ratio to move up line B in FIG.1. It is of course not necessary for stars to always form from planetary bodies in this manner. Provided there is a sufficiently vast amount of closely distributed matter available, gravitational accumulation can lead straight into line B of FIG.1 at any point. It is believed that in this manner the accumulation can proceed all the way up to the size of an entireu universe where, in Phase I of its evolution, complete star systems and galaxies are being gravitationally attracted to the original core.

Note that the mass of the Sun is approximately 1000 times that of (2.9) and its radius about 5.8 times that of (2.8). The interesting point is however, if the radius of Jupiter, the largest planet in the Solar System, were only 1.68 times larger and its mass 1.21 times greater, it would be at the point of transforming into a star.

During the dissertation in Section 2.2 above, it was stated that the gravitational accumulation of matter could proceed all the way up to the size of a universe, following the law represented by line B in FIG.1. A limit is however reached at the coincidence of lines A and B in FIG.1. At this point the radius of the accumulated matter equates to three times its gravitational radius. Temporal flow in the core centre stops, and gravitational reversal takes place. Further gravitational accumulation of matter is progressively halted and all material previously gathered is then gravitationally repelled. This was the detailed subject of [2].

From FIG.1 it can be seen that this co-incidence occurs at a mass of approximately 6.0 x 10⁵³ gm., and a radius of some 1.3 x 10²⁶ cm, (~4.6 x 10⁹ L.Y.). Both of these values are quite close to those estimated for the Universe. Amounts of gravitationally accumulated matter in excess of the above figures, eventually, due to intrinsic gravitational attraction, develop a physical radius that crosses to the LHS of line A in FIG.1 i.e. contracts to a value less than three times the gravitational radius. It thereby, via the process detailed in [2], starts to generate internal repulsive gravity which eventually results in its complete dispersal. The location of the point representing the Universe in FIG.1 is therefore quite in keeping with the above interpretation, being just 100 times greater in mass. This now explains the conformance of the mass/radius characteristic of the Universe to Line B in FIG. 1 as discussed in Section 2.2

In classical Cosmology theory, a black hole is defined as a gravitational source so strong that no material body, nor even electro-magnetic radiation, can escape its gravitational field. In the literature the critical distance from the source below which this occurs is termed the Event Horizon, and from [1], Eq.(4.7) is seen to be when

(2.14)

Thus if the physical radius of a gravitational source is less than, or equal to, twice its gravitational radius, the object is classified as a black hole. However, in the Relativistic Domain Theory of gravitation, it is seen from [2], and this paper, that before a large gravitational source contracts to the "Event Horizon", an earlier critical radius occurs at three times the gravitational radius. At this point it was shown in [2] that internal gravity reverses and becomes repulsive. Also at this point, the temporal rates both inside and outside the source change such that a new critical radius is established at , i.e. a radius equal to the gravitational radius, and the apparent critical radius of (2.14) no longer exists. In a future paper it will be shown that subsequent to the 3a criterion being reached, the source continues to contract to a minimum radius, the point of inflexion, equal to this new criterion, but then immediately starts to expand again under the influence of the repulsive gravity field generated internally. This was partially demonstrated in [2], for the entire Universe. Therefore it is only at the single instant of minimum contraction, when the physical radius of the source is equal to its gravitational radius, will a "black hole" exist. Once the source starts to expand and its physical radius is once again greater than the gravitational radius, light and material bodies will again be able to escape its gravitational field. Consequently, subsequent to the gravitational reversal, and the dispersal of the outer layers, the inner core that would be left would probably be degenerate matter in the form of compacted neutrons, i.e. a neutron star as discussed below. Thus, it is clear that apart from the "brief instant" when the physical radius of a collapsed gravitational source was equal to its gravitational radius, within the Relativistic Gravitational Space-Time Domain D₁, black holes do not exist. Thus, the most compact astronomical objects within a Relativistic Domain Universe, would be neutron stars.

The position of the point for this object in FIG.1 is to the left of the line designated A. Consequently, it would be expected that such an object would exhibit an internal repulsive gravitational field. In accordance with [2] Eq.(3.10) the Acceleration Potential internal to the star would then be

(2.15)

Its external gravitational field would still be attractive with an Acceleration Potential in accordance with [2] Eq(3.13), thus

(2.16)

which is seen to be half the strength of a normal gravitational field.

Neutron stars are purported to be the remnant of a supernova which is said to occur when a large star starts to run out of fusion fuel in the core. Following an initial expansion to a red giant, the ensuing gravitational collapse becomes more and more rapid, as it leads to more and more complex fusion processes, involving the higher elements up to iron. Finally, it is then said to end with the process becoming unstable, thereby culminating in a vast nuclear explosion. The precise reason for the onset of this instability is not clear.

An alternative to the above explanation could be as follows. If the gravitational collapse is sufficiently rapid, and the radius reduces fast enough, as it falls below the 3a limit, the reversal of the gravitational Acceleration Potential could be sufficiently violent that, in addition to the nuclear release of energy in the form of particulate and electromagnetic radiation, the ensuing gravitational expulsion of the outer layers of the star would appear, on an astronomical time scale, similar to a vast nuclear explosion. The final result would be a rapidly expanding shell of high temperature gas, radiation and other debris. In the middle of this would remain the small nucleus of the neutron star now steadily generating the Acceleration Potentials exemplified by (2.15) and (2.16) above. Some or all of the expanding shell of gas etc would then either drift out into inter-stellar space or, under the influence of the reduced gravitational field of the neutron star, slowly return to its surface.

The internal repulsive gravitational field of the star would also act to disperse its remaining matter content into inter-stellar space. However, because such objects are purported to be constructed from "neutrons in contact" the strong nuclear force would oppose this and thereby maintain its structural integrity.

Very little need be said about this end of the scale from the gravitational point of view. Clearly the point on FIG.1 representing this particle is very far from any of the lines on FIG's 1 or 2. This is of course an expected result because gravity plays little if any part in the existence of the individual nucleon. The only significant thing that can be said, and the only reason it has been included, is that from FIG.1 it is clearly impossible for particles of this size to exist with internal gravitational fields of a repulsive nature.

C2 Version 1.1.1
Ó; P.G.Bass March 2006

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