Space and spaces

5. Space and spaces

General Relativity has a mixture of 2 things which mathematicians wrongly use the same name:

1. Geometric space
2. Analytical spaces

In the cases where n variables occur, the analysts often name a system of numerical values that are shared with these variables “analytical point” and all these points the name “analytical space”. The number of dimensions of the analytic space envisaged is the number of variables that make it up.

These definitions are purely analytical and independent of the concrete meanings of the given variables.

The geometer’s point of view is different. For him, the number of dimensions is not a property of space, but a property of the space element. This requires an explanation.

The position of a geometric point is determined by three coordinates. The totality of the positions of the geometric points would thus form an analytical three-dimensional space. But a straight line is determined by four numbers, which are also called their coordinates; the position of a solid body is determined by six coordinates, etc. If one regards the straight line as an element, the totality of the possible positions forms an analytical space of four dimensions (Plücker’s ordered space).

The totality of the positions of a solid body would also define a six-dimensional analytic space. For the geometer, the location of the points is the same as that of the straight line or the solid: it is always the same space.

The space considered as a place in the sense of the geometer does not have a certain number of dimensions.

Classical mechanics considers systems whose position depends on any number n of parameters. The totality of the possible positions of this system forms an analytical space of n dimensions; the place of these possible positions always belongs to the same indefinite space of the geometer.

The point of an event in the relativistic sense is determined by three position coordinates that are linked to a time value. Their entirety forms a four-dimensional analytical space. But if the event is composed of the simultaneous consideration of two point positions and a time value, the whole forms an analytical space of seven dimensions.

The totality of the possible connections between two completely independent event points would form an analytical space of eight dimensions.Further examples are superfluous. The ones given here suffice to make it clear which essential difference there is for the geometer between the local space and the total space. They are two different terms that are referred to by the same name.

6. The relativistic spacetime and the analytic space of Newtonian gravity

Relativity has only a four-dimensional spacetime in mind, which it examines in the form of quadratic differentials; this should play a role similar to that of the line element of a surface in geometry.

The force of gravity would hereafter be determined by starting from this square shape. The natural motion of a material point would be represented by a geodetic line of the differential form in question. This geodesic line is his world line. A geodetic line corresponds to every movement.

Something similar can be found in classical mechanics. The principle of the smallest effect leads to the fact that the representation of the motion of a system is based on a geodetic line in the form of quadratic differentials. But one has in the eye the movement of a whole system, which is viewed as a solid whole, and more that of a single element.

The quadratic form then comprises as many variables as are necessary to determine the position of the system, and it is the movement of the whole that is represented by a line from the form in question.

If, for example, one imagines the universe as formed by a total of n mass points, the position of the whole will depend on 3n variables. The corresponding analytical space will have 3n dimensions. Time is not a supplementary coordinate, because the movement of a timepiece of whatever kind leads away from the entirety of the movements of the universe.

The square shape mentioned is as follows:

In it U denotes a function of the coordinates of the system. The calculation involves the introduction of an auxiliary variable t, which is defined by the equation is determined. This allows the geodetic differential equations to be reduced to the usual form of the equations of mechanics. This auxiliary variable t is the canonical time of classical mechanics. The canonical frame of reference is the one for which the kinetic energy of the observable universe is minima.

If one finally determines U according to a minimum requirement for the energy of the accelerations, one finds where mi and mk denote the masses of two elements and rik denote their distance. This is Newton’s first law. The equations of motion then have the form These equations contain not only the coordinates of the point under consideration, but also those of all other points in the system, which gives the whole thing closed 1). An interesting fact of relativity, which classical mechanics reveals but escaped Einstein’s school, is the relative character of the principle of the equality of action and counteraction. This principle does not express a property of matter: it is a property that comes from the choice of the frame of reference.

7. On the impossibility of representing the phenomena of gravity by Einstein’s theory.

It remains to be shown that it is impossible to represent the phenomena of gravity, starting from Einstein’s basic hypothesis.

Let T be a quadratic form of differentials of four variables x1 , x2 , x3 , x4 . The equations of the geodetic lines of this form can be written as follows:

They allow three of the coordinates to be expressed as a function of the fourth and any six integration constants. The only difference between two solutions is the numerical values of these six constants.

Let us consider two solutions that represent the movements of any two material elements. Under y1 , y2 , y3, y4 are to be understood the coordinates of the elements of the first, under z 1 , z2 , z3 , z4 those of the second. You can for example, assume that y1 , y2 , y3 are expressed as a function of y4, and also z1 , z2 , z3 as a function of z4. But there is no necessary relationship between y4 and z4 : there is generally no necessary relationship from element to element between two geodetic lines. Onecould evidently manufacture such a product by i.e. set y4 = z4 = t, where t denotes a time.

However, this agreement is by no means essential. Nothing in the differential equations (2) would be changed if for the first line y4 = t and for the second set z4 = t + α , where α means any constant. The lack of a regular relationship between the points of occurrence of two different geodetic lines is the main reason that Einstein’s theory is unsuitable for representing the phenomena of gravity.

One can derive differential equations from theory which will more or less approximate those of the motion of a single point; but one will never be able to derive the equations for the motion of any solid system from it. It is not the difficulty of the problem or the inability of the authors to blame for the failure of the attempts that have been made in this sense, but rather it is grounded in the essential contradiction that exists between the principle of Einstein’s theory and the fact of unity.

It has not even been possible to set up the equations for the motion of a system of two bodies that are related to a reference system that does not have one of these bodies as its starting point. The secret of this powerlessness lies in the restriction of the analytical space corresponding to the problem of gravitation to four dimensions.

Analytical mechanics, free from the superstition of spacetime, cleanly and accurately solves the problem by introducing the necessary number of variables. Relativistic mechanics stomps in the same place, unable to get out of its four-dimensional prison. The four-dimensional analytic space of Einstein does not contain the 3 n-dimensional analytic points which correspond to each position of a whole of n material elements. While for this reason relativity can only treat the elements individually, classical mechanics treats the whole of the observable universe in its totality.

8. Gravity is a property of the universe considered in its entirety.

Gravity is a law of acceleration or interaction.

But in this way the problem is robbed of its true nature. The so-called Newtonian effect, which is inversely proportional to the square of the distance, only applies to movements related to certain reference systems.

Since these systems are oriented towards the starry sky, they actually depend on the totality of the stars observed.

The wording of the law of attraction also presupposes the choice of a special point of reference for the time so that the acceleration can be determined. This canonical time is also established, theoretically by considering the entire universe, practically by the apparent rotation of the starry sky.

It is always the whole of the universe that comes into its own.

The concept of two equal and directly opposite actions at a distance seems at first to contradict our understanding. However, we prove that:Whatever the nature of a moving whole, whatever the movements of the elements that compose it - there are always systems of reference which are so constituted that the relative movement of the whole with respect to any particular one within it seems to take place solely on the basis of 2 mutual, equal, and directly opposite effects.

The mutual remote effects are therefore essentially a fact of relativity which results from the determination of the reference system.

Einstein’s method did not make it possible to uncover this important result.

In order to finally express the law of mutual effect in a form that is independent of the choice of the reference variable, one would have to use the totality of the parameters that serve to determine the position of the observable universe as a whole.

That too is beyond the capabilities of Einstein’s method.

The results confirmed by the relativistic school only appear satisfactory if they are admitted without criticism. This applies e.g. from the deceptive indication of 42 “for Mercury instead of 374” and the inability of the method to explain the rest.

9. Conclusion.

These general statements make it unnecessary to go into the various irregularities of the method and the pseudo-geometric theories of relativity. One gets the same impression from them and finds the same lack of criticism, combined with some assertions that are downright absurdities.

My very clear conclusion is that Einstein’s RTH does not belong to the field of positive science.