Here below it is shown how those singularities arise from a little mistake due to an inappropriate extension of the special relativity results to limit conditions.
The General Relativity base hypothesis
The base hypothesis in the develop of the general relativity theory is that, whenever the coordinates system is properly chosen, the special relativity equations remain still valid for infinitesimal four-dimensional regions of space-time. At this purpose one need to choose the acceleration of the local coordinate system in such a way he doesn’t see any gravitational field; that is possible for an infinitesimally small region only.
(1) ds2 = dx12 + dx22 + dx32 - dt2 = dl2 - dt2
A free falling observer with a caliper to measure length and a clock, evaluates the distance between two events as dl and dt. Another observer moving fast with respect to the first, and in the same portion of space and at the same time will evaluate the distance between the same two events as dl’ and dt’, much bigger and leaning to infinite with the increase of the relative velocity.
With the introduction of the imaginary time dx4 so that
dx42 = - dt2
In order to express the equation (1) referred to a general coordinates system X1, X2, X3, X4, not local but that cover a finite zone of space of which for example we want to describe the gravitational effect of the bodies inside it, we need to linearize the relations between the local and the general reference systems as follows:
dxv = Ss avs dXs (Ss summatory base s)
and therefore the equation (1) can be expressed with respect to the general coordinate system as:
ds² = Svs gvs dXv dXs
The space-time singularity as consequence
Let’s consider a very massive body in a static condition for which the parameter 2MG/R is just a little more than 1. The general relativity theory overestimates the gravitational potential in the proximity of the surface of the body that would lean to infinite as 2MG/R approaches to 1.
A mass m << M on the surface of M would undergo a gravitational interaction whose potential energy is given by:
Ep = mc²/( Sqrt(1-V²/c²)-1)
Which upshots?
The mass of a body is the parameter that characterizes its interaction with the space-time regarding the gravitational viewpoint as well as the inertial one. Such interaction is not however immediate as it needs to propagate at the finite velocity of light.
And this must be true also regarding the inertial aspect. Hence the mass cannot be consider as a scalar entity equal to the ratio from the applied force and the consequent acceleration, but must be represented in a more complex way. The special relativity describes the behavior of a motion in a steady state and in flat space big enough to have the transient completed, actually as more extended as higher is the velocity involved.
This aspect shall be shown in a forthcoming article where shall be demonstrated that taking into account this aspect of the inertial interaction of a body with the space-time one can give a physical account of the quantistic behavior of the matter and the so much searched connection between the two theories.
For the time being, as a demonstration of the potentiality this little mistake can have, we shall report only the following considerations regarding some cosmological aspects.
- The space at the border of a gravitational system applies an expansive action as bigger as 2MG/R approaches the value 1, with all the cosmological aspects this can originate.
- Common matter can scatter from a missed gravitational collapse. Let’s consider a gravitational system with the value 2MG/R close to 1. As said before a small black hole cannot exist, but only a big one. Such a system must be of a very important mass and with internal pressure that determines very condensed state of the matter (like neutrons or quarks density). When the mass increases again also the dimension R of the system increases proportionally. The density inside becomes consequently lower and lower until these condensed states of the matter can decay into common matter. This could be the case of galaxies and clusters.
- Also regarding the formation of the solar system many possibilities come out. The most accredited theory regarding the formation of our solar system from the rests of a supernova explosion shows some difficulties:
- A disk or lens shape of the system is does not conform to an origin by aggregation. Besides some left portion of the supernova core should still exist not so far from us.
- The mass is nearly all made of hydrogen and helium in the proportion actually identical to the primordial one.
- The sun represents more than the 98% of the total mass of the system, and in spite of this has only the 2% of the total angular moment. Actually it should spin much faster if deriving from the aggregation of supernova rest.
A neutron star with the mass of our sun has a ratio 2MG/R = close to 0,5. As the mass increases the general relativity theory foreseen the gravitational collapse for a mass starting from 1.4 times the sun mass (the rotation is very important at this regard)
In absence of the collapse he mass can be much higher. From two to four times the solar mass the density decreases and the neutron core can decay in proton and electrons and create a normal matter with a big delivery of gravitational energy. The outside layer of the star (mainly made of iron) shall be sprayed around on the rotation plan end it will start to concentrate in planets. The inside core shall become for the most part hydrogen and helium and in a well defined portion. The spinning speed will decrease of about 2000 times the original speed due to the expansion.