Since the first proposed solutions of the formulation of the Einstein general relativity theory, the singularities on the space-time that were discovered, were considered due to an excessive simplification of the models adopted. Further on, due also to their persistence for more complicated models and some exceptional experimental confirmations of the theory, those singularities have been accepted and besides considered the most sensational results of the theory itself.

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.

However the reference to the extension of the spatial and temporal regions introduces an incongruence, as these extensions are relatives and depend from the motion conditions of the observer. One could think that if a spatial and temporal region is infinitesimal for one observer it should be infinitesimal for all the other too. But it’s not true, because a very small region for one observer can be big and even infinite for another one.

Mathematically the base hypothesis of the general relativity can be expressed saying that the space-time interval between two events, as below defined, remain constant for any local observer for which does not exist any gravitational field.


(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.

In a flat and infinite space the space-time distance shall be the same, but in presence of a gravitational field and velocity approaching c shall be different, and therefore the equation (1) is not valid for all the observers.

With the introduction of the imaginary time dx4 so that

dx42 = - dt2

Minkowski has made the invariant theory for the continuum four-dimensional formally similar to the theory of the invariant for the Euclidean three-dimensional space.

However the fact that x4 is imaginary causes a great difference with respect to the three-dimensional space. Two points (i.e. two events in the space-time) very closed each other, can have the space and time coordinates very far and at limits infinitively far.

This depends on their motion conditions.All the points on a real plan that are at the same distance from the origin of any reference system, are placed on a circle of radius equal to the distance itself. The value of each coordinate remains always less or at limit equal to the distance considered. On the contrary on an imaginary plane dl / idt all the points at the same distance from the origin lay on a hyperbole.

Of course the distance does not vary, by definition, but if you rotate the reference system, the respective coordinates can become very high and also infinitive.In case of extreme velocity and gravitational fields, it's not possible to define the infinitesimal zones where the special relativity remains still valid. Two spatial-temporal events infinitesimally closed, can be, under extreme conditions, finite and also infinitively extended.In those conditions one cannot consider still valid the special relativity theory, being so possible only for infinitesimal zones.

From the mathematical point of view, the mistake is on the following passage:

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 parameters gxx , that depend on X1, X2,X3,X4 describe the gravitational field in the whole finite zone of the general observer.

However the linearization of the relation between the two coordinate systems is valid only for minimal variation of the coordinates dxv, and not for finite or above all infinite variation as happens in the limit cases considered above.

The space-time singularity as consequence

The singularities in the space-time found inside the models based on the general relativity theory derive from this incongruity.

For a free falling system in a gravitational field, without rotations and at very high speed, the special theory of relativity cannot be applied. The energy associated to the motion is less and above all is not infinite for velocity approaching the light speed.

Let’s think to the difference existing between a system traveling at the speed of light in a flat and infinite space, and a system that is reaching the same velocity at the entrance of a black hole.

In the first case, due to the length contraction the observer is at the same time present in the whole infinite space (and therefore has an infinite energy). Not in the second case; the observer is simply on the border of a black hole and it takes to him a finite time to reach it. It’s not at the same time present everywhere (according to his time of course) in an infinite space. He has therefore a finite energy, becoming smaller and smaller as the diameter of the black hole decreases.

The mistake with respect to the predictions of the general relativity, that is very insignificant for even very massive bodies but far from the condition 2MG/R=1, shows a basic difference with respect to the so called gravitational collapse.

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)

The potential energy is therefore less and does not tend to infinite for V -> c. It would tend to infinite only for M-> infinite.

The pressure in the centre of the body is more than the one foreseen by the Newtonian gravity, but does not sort out in a gravitational collapse.

Expressed in different way, the small black holes cannot exist, but only the big ones and without singularities inside.

Which upshots?

This little mistake could seem to regard only a very limited case and not have real consequences on the behavior of the matter. But it’s not true. It involves a big modification in the concept o mass itself, as well as with regards many cosmological occurrences.

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.

1. 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.
2. 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.
3. 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.