THE END OF SCHWARZSCHILD METRIC AND GENERAL RELATIVITY (II)

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?

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)

THE END OF SCHWARZSCHILD METRIC AND GENERAL RELATIVITY (I)

THE END OF BROKEN SPACETIME

Xavier Terri Castañé

TRACTATUS PHYSICO-PHILOSOPHICUS

ABSTRACT: Any tetradimensional theory of gravity that was a consistent generalization of special relativity and no tolerating outstanding observers will get superior speeds to the local light speed in void. In this article the theory of general relativity from Einstein is refused because it is unable of carry out this requirement, therefore a new alternative generalization is propose, the connected theory that eliminates the black holes. A different light raises where the absolutely darnkness was before.
KEYWORDS: special relativity, general relativity, connected theory, black hole, inertial, inertial-no inertial dichotomy, connected system, metric, equivalence principle, connection principle, gravitational redshift, invariance, fundamental equation of the connected dynamic, Einstein's field equations, generalized principle of inertia, 'c' constant, Newton, Einstein, Hawking.

1. Widespread official of special relativity: general relativity

Our current vision of the cosmos is based on two theories: quantum mechanics, which constructs an entire microcosm, and general relativity, which is supposedly valid for describing macrocosms. The latter was a product of the urgent historical imperative to generalise special relativity, a theory presented in 1905 as an alternative to the Newtonian theories but which, in contrast to these, had the serious drawback of being unable to describe gravitational interaction. It was only applicable to reference systems or very special observers, ones which would today be called “inertial” observers, whose corresponding spacetime metric is characterised by a four-dimensional “flat”, or Minkowski, metric.
According to general relativity, gravitational forces curve spacetime. The metric ceases to be that of Minkowski—such as it is determined by field equations or Einstein equations—and falling bodies simply follow the shortest route in this curved spacetime—with movement, or geodesic, equations, determining the route. With this four-dimensional description of gravitation, it is not only possible to predict already known events, those that the Newtonian theories already predicted, but also to explain three “anomalies” or predictions that escaped the comprehension of the Newtonian theories: gravitational redshift, the residual advance of the perihelion of Mercury and the deflection of light rays that tangentially affect the edge of the solar disk. Such anomalies, corroborated in experiments, demand a high level of precision from any new theory. They represent a difficult test, known as the three classic tests, that any good theory of gravitation must be capable of passing. General relativity is generally said to have passed these tests successfully, but this article will explain that: 1) it is not true that it predicts gravitational redshift coherently, which leads to all types of contradictions and singularities, event horizons and black holes, points where spacetime breaks (points here the Schwarzschild metric breaks); 2) general relativity is a theory that contradicts the thesis that privileged observers do not exist; and 3) an alternative to general relativity exists which truly passes the three classic tests without singularities, does not contradict general relativity, is the only coherent four-dimensional generalisation of special relativity for the purpose of making it compatible with gravity and gives rise to new predictions, the theory of reference systems connected to the gravitational medium, or simply connected theory.
Einstein, faced with the historic imperative of generalising special relativity, applicable only to inertial systems and in the absence of gravitation, tried to find a bridge to connect the gravitational field with the concept of the inertial observer. This conceptual bridge is what as known as the equivalence principle. It establishes the following: “an observer in a gravitational free fall is locally inertial”. Something which seems to mean that the timespace metric for that observer alone will be an infinitesimal spacetime environment (locally), a flat, or Minkowski, metric, precisely as postulated by the special relativity theory for its inertial observers. The equivalence principle establishes, therefore, a relationship between the gravitational phenomenon, symbolised by an observer in a gravitational free fall, and special relativity, whose domain of applicability, although very restricted, seems to be assured by decreeing the existence of locally inert observers. Einstein attempted to generalise special relativity using this bridge. Ten years later he presented his theory of gravitation: general relativity, inspired by the equivalence principle and, as the name itself indicates, supposedly valid —invariable—for all possible observers in nature (that a theory is applicable to the entire system of mathematical coordinates is a merit inherent in the mathematical calculation instruments—tensorial calculus—that it uses. But it does not necessarily mean that it is in agreement with the invariability of physical laws for all possible observers in nature). But, what is the exact meaning of the “inertial” concept that appears in the logic of the equivalence principle? Does it not seem strange that a theory that aspires to surpass special relativity, a theory that had in turn already surpassed the theories of Newton, is based on a concept that originates in theories that are already obsolete? Is general relativity the fruit of precipitation and of historic urgencies?

2. The dichotomy inertial vs non-inertial

The old second Newtonian law, which is the fundamental equation of the classic dynamic and which establishes a relationship between force and three-dimensional acceleration, is nothing more than a simple generalisation of the classic principle of inertia. From here it can be deduced that a three-dimensionally free body (the net force that acts upon it is null) remains in repose or uniform rectilinear movement. But this only occurs when it is observed from a “privileged” system of inertial reference. To try to explain the accelerations that three-dimensionally free bodies sometimes exhibit, Newtonian mechanics finds itself obliged to introduce the opposite concept: that of the non-inertial observer. According to these types of observers, certain specific forces would exist that are called fictitious, apparent or inertial, which would be held responsible for causing the accelerations of three-dimensionally free bodies (those that do not seem to exhibit any “real” interaction with their environment). Adding “fictitious” forces to Newton's second law—which then ceases to be an invariable equation—is the means of trying to justify the accelerations of three-dimensionally free bodies. Succinctly, accepting Newtonian ideas implies accepting the existence of a dichotomy within the class of all possible observers in nature. the inertial–non-inertial dichotomy. It is in this dichotomy where one finds the historic origin of the “inertial” concept on which the equivalence principle rests. Paradoxically it is this old absolute principle that becomes the key when it comes to generalising special relativity for the purpose of arriving at a theory that is applicable to any possible observer.
It is necessary to generalise special relativity. It is necessary to construct, in effect, a new theory of gravitation which is consistent with the universal invariability of physical laws, which is also applicable to those observers that cease to be inertial because of gravitation. But it does not seem very sensible to have to build this new theory on a principle, the equivalence principle, that does nothing more than re-establish the existence of “privileged” inertial observers. General relativity is undoubtedly applicable to the entire system of mathematical coordinates (thanks to tensorial calculus), but it violates the universal invariability of physical laws. It violates the equality of all possible observers in nature from the moment that it “locally” resurrects the inertial–non-inertial dichotomy through the equivalence principle that sustains it.
We will have made no progress if, having refuted the absolutism of Newton’s theories, we then resurrect it by declaring the existence of absolute observers. We will have made no progress if, having separated the earth from its privileged place as the “centre of the universe”, we then make the sun the new centre. (In fact, general relativity even contradicts itself, with its equivalence principle and its geodesics, as it is not difficult to demonstrate that, for a relativistic stationary observer, the acceleration—the second derivation of the radial coordinate with respect to the “coordinated time”—of a falling body depends on its speed. Then it would not even be true for relativity itself that a free falling—“inertial”—observer would locally nullify the gravitational field: strictly speaking, bodies with different speeds will exhibit different accelerations, pág. 3 )

Solution to the relativistic problem of dark matter and Einstein's field equations