3. THE CONNECTION PRINCIPLE

Any correct generalisation of special relativity must be consistent with the principle of connection to the gravitational medium, which I will summarise in the following points: Any dichotomy between possible observers in nature according to the concepts of inertiality and the non-inertiality is illusory: all the laws of physics are the same —invariable—for all possible observers in nature: All possible observers in nature are non-inertial—connected to the gravitational medium—and equivalent amongst themselves. Inertial systems do not exist. The connection principle eliminates the existence of any privileged or absolute referent and, if we start from the premise that special relativity is a valid theory in the absence of gravitation, then it gives rise to the following corollary: for any possible observer, completely independently of whether this observer exhibits a free falling movement with respect to this or that source, its metric is exactly flat, or that of Minkowski, at the precise point—and only at this point or at points that are situated at the same gravitational potential—at which it is found; which is precisely the metric that special relativity postulated. This eliminates any type of privilege among possible observers in nature at the same time as it establishes the bases on which a coherent generalisation of special relativity should be constructed, one that is valid for providing a description without the gaps of gravitation.

“Observer connected to the gravitational medium” refers to a non-inertial observer having a four-dimensional metric—a connected metric—whose matrix elements contain variables characteristic of the gravitational phenomenon, such as masses and distances with respect to the sources of each spacetime point under consideration. Similarly it will also contain information with respect to the concrete point at which the observer itself may be found. Thus it is precisely at this point where the connected metric will be reduced exactly—therefore also “locally”—to a

Minkowski metric. All observers, and not just the “privileged” observers of the equivalence principle, have the right to be considered “flat”. (All members of a mountaineering group have the same right to consider that the surface is locally flat precisely at the point where each one is located, irrespective of their individual movement status and no matter how curved the surface of the mountain is.) On the other hand, it should be noted that it would be inappropriate to continue referring to observers whose corresponding connected metric never coincides as “inertial”, except at the precise point at which the observer is found, or at points that are situated at the same potential, with that of Minkowski. Moreover, due to the historic origin of its meaning described above, the term “inertial” is inadequate if we want to proclaim the equality of

all possible observers in nature, that is, the universal invariability of physical laws.

General relativity, constructed on its equivalence principle—which resurrects the old inertial–non-inertial dichotomy and which still believes in privileged observers—is refuted by its own manifest incompatibility with the connection principle. Thus, is refuted the description of movement of bodies by means of geodesics Einstein's equations that generated a curved spacetime. This description is incompatible with the true universal invariability of physical laws and with the equality of all possible observers in nature. And no one should think that the above is merely a “conceptual game” having no real influence on the equations. However unfamiliar the reader may be with the equations of general relativity, he or she will undoubtedly realise that Einstein’s equations, to cite just one example, are incapable of generating a metric that is “locally” flat for a stationary observer situated at a finite distance from the source (the Schwarzschild metric is only flat at infinity); they are then incapable of meeting the requirements of the connection principle, according to which said observer has every right to find that the precise point where which he or she is located has a flat metric independently of whether it exhibits a free falling movement with respect to said source. I repeat that this is not a mere conceptual game, but rather that the same equations of general relativity are manifestly incompatible with the inviolable connection principle. They are refuted! How does one construct a coherent generalisation of special relativity that manages to pass the three classic tests, to describe gravitation while being consistent with the connection principle?

“Observer connected to the gravitational medium” refers to a non-inertial observer having a four-dimensional metric—a connected metric—whose matrix elements contain variables characteristic of the gravitational phenomenon, such as masses and distances with respect to the sources of each spacetime point under consideration. Similarly it will also contain information with respect to the concrete point at which the observer itself may be found. Thus it is precisely at this point where the connected metric will be reduced exactly—therefore also “locally”—to a

Minkowski metric. All observers, and not just the “privileged” observers of the equivalence principle, have the right to be considered “flat”. (All members of a mountaineering group have the same right to consider that the surface is locally flat precisely at the point where each one is located, irrespective of their individual movement status and no matter how curved the surface of the mountain is.) On the other hand, it should be noted that it would be inappropriate to continue referring to observers whose corresponding connected metric never coincides as “inertial”, except at the precise point at which the observer is found, or at points that are situated at the same potential, with that of Minkowski. Moreover, due to the historic origin of its meaning described above, the term “inertial” is inadequate if we want to proclaim the equality of

all possible observers in nature, that is, the universal invariability of physical laws.

General relativity, constructed on its equivalence principle—which resurrects the old inertial–non-inertial dichotomy and which still believes in privileged observers—is refuted by its own manifest incompatibility with the connection principle. Thus, is refuted the description of movement of bodies by means of geodesics Einstein's equations that generated a curved spacetime. This description is incompatible with the true universal invariability of physical laws and with the equality of all possible observers in nature. And no one should think that the above is merely a “conceptual game” having no real influence on the equations. However unfamiliar the reader may be with the equations of general relativity, he or she will undoubtedly realise that Einstein’s equations, to cite just one example, are incapable of generating a metric that is “locally” flat for a stationary observer situated at a finite distance from the source (the Schwarzschild metric is only flat at infinity); they are then incapable of meeting the requirements of the connection principle, according to which said observer has every right to find that the precise point where which he or she is located has a flat metric independently of whether it exhibits a free falling movement with respect to said source. I repeat that this is not a mere conceptual game, but rather that the same equations of general relativity are manifestly incompatible with the inviolable connection principle. They are refuted! How does one construct a coherent generalisation of special relativity that manages to pass the three classic tests, to describe gravitation while being consistent with the connection principle?

4. THE CORRECT GENERALIZATION OF SPECIAL RELATIVITY AND NEWTON'S LAWS: THE CONNECTED THEORY

Theory of relativity basically consists of two sets of equations: the movement equations and the field equations. Let’s start with the first of these: Having discarded relativistic gravitational geodesics, the connected theory starts with a fundamental movement equation that is supposedly applicable to any interaction. This equation is nothing more than a four-dimensional mathematical extension of Newton's second law. It can be said very synthetically that it is nothing more than “force equals mass multiplied by acceleration” but formulated within a spacetime having four dimensions, one temporal and three spatial. The new fundamental equation represents a basic postulate of connected theory and, once the gravitational geodesics of relativity have been refuted, the most reasonable and simplest (in addition, it can be demonstrated that this implies the only route that remains constant for a particle in a gravitational fall in a stationary field, the temporal contravariant four momentum component). By virtue of its tensorial formulation, it is applicable to any system of coordinates. Starting from the fundamental equation of the connected dynamic, the principle of generalised inertia can be deduced, as stated below: a four-dimensionally free particle (in which the net four force that acts

upon it is null) moves along the geodesics of spacetime. Therefore, as the solution to the geodesic equations depends on the metric, this is the same as saying that: a fourdimensionally free particle remains in repose or in rectilinearly uniform movement (Minkowski metric), or can even accelerate (other metric types). Thus, it is not restricted to moving according to the dictates of the classic principle of inertia. It is allowed to accelerate. It is no longer necessary to invent “fictitious four forces” with which to justify the possible accelerations (it can be demonstrated that these will depend on the derivations of the components of the metric with respect to the coordinates) that the four-dimensionally free particles can exhibit. It is no longer necessary to hypostatise a dichotomy of the referents in nature according to the concepts of inertiality and noninertiality.

The new principle of generalised inertia is what permits the elimination of this dichotomy and it is, of course, consistent with the connection principle. But it only refers to four-dimensionally free particles. A particle that gravitates is not a four-dimensionally free particle. It does not move along the geodesics of spacetime. “Gravitational geodesics” do not exist. For connected theory, if a particle gravitates it is because it is subject to the action of a gravitational four force. And the result is that this four force is described through a law that is written in function of a connected gravitational potential, represented by a symmetric tensor of the second order that does not coincide with the metric (otherwise it would be impossible to construct a theory whose equations agreed with an absolutely relative conceptualisation regarding the movement). Substituting the law of gravitational four force—it can be demonstrated that it is the only possible law that meets certain inalienable conditions—in the fundamental equation of the connected dynamic, the equations of movement in a gravitational field are obtained. In summary, it can be said that connected theory defends a four-dimensional extension of Newton's

laws for the purpose of obtaining a new formulation that is truly consistent with universal invariability of physical laws, in other words, stripped of the inertial–noninertial dichotomy, the existence of privileged or absolute observers. Newton's inertial systems disappear. Einstein's gravitational inertial systems disappear. The sun moves...

It is sufficient to require that the equations of connected movement for relatively weak gravitational fields be reduced approximately to their Newtonian homonyms to obtain, without even having yet postulated any field equations, results that pass the famous three classic tests without any problem. Afterward, the connected field equations are postulated in order to be coherent with the entire conception regarding movement that has been succinctly explained here. They should also be consistent, of course, with the connection principle. And thanks to them it is possible to know the exact mathematical expression of all the formulas of the connected theory of gravitation. In particular, for a stationary observer (who therefore is not in a free fall) they permit the deduction of a connected metric—whose temporal matrix element is approximately the mathematical inverse of what appears in the Schwarzschild metric— that is precisely flat at the point in which this observer may be situated. Finally, the movement equations together with the field equations resolve any phenomenology related to gravitation in an exact manner by using a metric whose spacetime does not break, does not predict the theoretical existence of black holes. Both equations constitute the only coherent four-dimensional generalisation of special relativity —which is considered valid in the absence of gravitation (for observers that conserve a constant speed between them)—that is consistent with the connection principle and which provides, apart from a true explanation of the aforementioned three classic tests, other new predictions. (Solution to the problem of DARK MATTER)