Mathematics in physics

MATHEMATICS IN PHYSICS INDEX …REMEMBERING Symmetry transformations The three postulates of temporal space symmetry 1. MATHEMATICS IN PHYSICS Relationship between Mathematics and physical paradigms Galilean inertial system and the weak equivalence principle Lorentz transformations Rupture of the idea of simultaneity 2. THEORY OF SPECIAL RELATIVITY Postulates of special relativity Maxwell and the wave nature of light Renewal of the Newtonian invariants Double substitution of invariant: mass for energy and speed of light Substitution of invariants: force by linear momentum

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3. MINKOWSKI'S FOUR-DIMENSIONAL SPACE 14 The new geometry of special relativity 16 Dilation of time and contraction of space 20 Four classic examples of spatial contractions and temporal dilatations 20 a) Life of muons produced by cosmic rays 20 b) Paradox of the two twins 21 c) Trajectory of a ray of light projected in a train car 22 d) Temporal dilation and spatial contraction on the planet Mercury 23 From the curved space of Riemann to the curved space-time of Minkowski 23 Covariant and contravariant of Lorentz 24 Gravity as curvature of space-time 26 Recent verifications of the principle of weak equivalence 27 4. THEORY OF GENERAL RELATIVITY Einstein field equations General relativity and principle of strong equivalence Gravitational force as an expression of the curvature of space-time Anomalies of Newton's three laws solved by the relativistic paradigm Law of inertia Fundamental principle of the dynamics Principle of action and reaction

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5. BIBLIOGRAPHY 35

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When prostrate upon earth lay human life visibly trampled down and foully crushed beneath religion s cruelty, who meanwhile forth from the regions of the heavens above showed forth her face, lowering down on men with horrible aspect, first did a man of Greece l dare to lift up his mortal eyes against her; the first was he to stand up and defy her. Him neither stories of the gods, nor lightnings, nor heaven with muttering menaces could quell, but all the more did they arouse his soul’s keen valour, till he longed to be the first to break through the fast-bolted doors of nature. Therefore, his fervent energy of mind Prevailed, and he passed onward, voyaging far beyond the flaming ramparts of the world, ranging in mind and spirit far and wide throughout the unmeasured universe; and thence a conqueror he returns to us, bringing back knowledge both of what can and what cannot rise into being, teaching us in fine upon what principle each thing has its powers limited, and its deep-set boundary stone. Therefore, now has religion been cast down beneath men’s feet, and trampled on in turn: ourselves heavenhigh his victory exalts1.

Titus Lucretius ‌REMEMBERING We define as scientific paradigm all universal law of nature empirically contrastable2. It is an objective principle, which does not depend on individual opinions or any concrete culture. But above all, the paradigm is a safe antidote against any pretension to establish dogmatic and perpetual truths. The rejection of science to any dogma commits it to keep a critical position with itself and to submit any scientific law to the continuous experimental test. The substantial phase of the paradigm is its empirical verification, which must be fulfilled in any experimental context as a guarantee of the invariance of a law. The empirical test can serve as proof that confirms the validity of a current law, or a counter test, for its recognition as an anomaly. If the set of anomalies is relevant and someone proposes a new universal law that resolves them and that also ratifies all the historical empirical heritage accumulated until then, a new paradigm can be established3.

Through this method that turns around the empirical test, it is constituted an evolutionary instrument of successive and renewed paradigms that assures the progress of science4.

This progressive evolution depends substantially on the precept of universality of the laws of physics, also known as the principle of uniformity or invariance, which rests on those mathematical transformations that do not alter the physical magnitudes. By invariance is

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Lucretius pays homage in The Rerum Natura to Epicurus, his teacher, whom he considers the founder of science. Trevelyan, R.C. (1920) Translation from Lucretius, London Allen & Unwin., book I, pages 60-79.

2 The empirical test is determinant in Science. In 1912, Wilhelm Wien, Nobel Prize in Physics of 1911, proposed to Lorentz and Einstein for this award for the theory of relativity, a theoretical exposition that has been one of the most important scientific advances in history. However, neither did they obtain it then.

Lorentz had already achieved it in 1902, together with Zeeman, for his experimental investigations of the influence of magnetism on radiation phenomena. Einstein also won it later, in 1921, for his empirical work on the photoelectric effect.

3 Karl Popper poses a stricter paradigm idea. He affirms that experiments can never prove a theory; they can only deny it. His epistemological critique appeals to the falsifiability criterion, which says that for science to recognize a new theory it has to explain all the facts without failing any proof, expose itself to its own refutation and overcome the predictive power of the old theory.

4 In Science, the explanation always follows a universal pattern in only one direction, but not in the opposite direction. The paradigm of Einstein's general relativity also explains that of Newton, just as Newton's is based on and explains Kepler's third law, similarly as Copernicus displaces the convoluted model of Ptolemy. But neither Newton's paradigm can explain that of Einstein, nor that of Kepler of Newton, nor is Ptolemy clearer than Copernicus.

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understood the extension of a physical law to include the entire observable Universe5 without undergoing any alteration in all the spatial and temporal coordinates in which any experiment is designed. The basic objective of the principle of uniformity is the functioning of the laws of physics, always and everywhere, no matter the spatial and temporal context of the Universe in which they are applied, independently of the reference chosen by the experimental observer and the relative displacements that may occur between the observer and the observed.

Therefore, we must ensure that the experimental results are not altered when the position of observers and reference systems undergo changes caused by movements or rotations.

Then it is convenient to resort to the equivalent reference systems, which are sets of spatial temporal coordinates related by mathematical transformations that preserve the physical magnitudes without changes. These transformations are specific, and are known as symmetry transformations.

The importance of equivalent reference systems and of symmetry transformations is mainly of practical sense. Suppose two observers located at two different points (generally belonging to different reference systems), taking measurements on the same object located at a third point. The problem arises when the two observers want to transmit each other the results. As the measurements have been made in two different reference systems, to be able to compare them, they need the help of mathematical transformations that maintain the results of both measurements when changing the reference system. The equivalent reference systems, which are defined as those to which mathematical transformations of symmetry are specifically applied, guarantee the inalterability of results.

Symmetry transformations The invariance of physical laws is solved by applying specific symmetry transformations.

This application obeys the need to compare the results of the measurements obtained in different reference systems. The chained phases of obtaining the results of the measurements and application of symmetry transformations must complete the cycle with the formulation of a mathematical invariance, in the form of an equation, for example, that defines the physical law6. In this way, each law and each mathematical equation that defines it will give meaning to the results of the measurements, composing a double invariance (of the physical law and the mathematical equation) consistent.

It only remains to specify what types of transformations articulates this double invariance.

We are obviously referring to what types of symmetry transformations. For classical mechanics, for example, in any time interval the energy of a gravitational mass suspended above the earth's surface is not altered. Therefore, the magnitude of the energy of the suspended mass is conserved regardless of whether the time of the experiment is delayed or advanced as much as desired. We can conclude then that a temporal translation of symmetry maintains the magnitude of the energy and the equation that symbolizes the law of conservation of energy.

There are also spatial symmetries, which include transformations that do not alter the physical property of an element for a change of position. The conservation of the linear momentum, for example, does not vary with the location of the experiment. Finally, we 5

Every experimental measurement requires the previous existence of an observable field. For example, the opaque universe before the Big Bang, referred to in the cosmological literature, is a speculative universe, not empirically testable. The transparent universe offers the only field of experimental observation.

6 Noether's theorem states that any symmetry described by continuous or continuously differentiable mathematical functions has its corresponding law of physical conservation, and vice versa.

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refer to the conservation of angular momentum as an invariant prototype of rotation symmetries.

The three postulates of temporal space symmetry The classical physics of Galileo and Newton has been developed on three implicit spatialtemporal postulates: homogeneity, isotropy and its extension in all directions of space, also the uniformity of time7.

The first postulate of the homogeneity of space enunciates the equivalence of all its coordinates when it´s about formulating the laws of physics. According to him, there are no singular points8 in the whole Universe where the paradigms vary or work differently.

Therefore, every physical law maintains its mathematical structure, no matter the spatial scope where it is applied.

The second postulate of the spatial isotropy establishes that the spatial directions are equivalent, that is to say symmetrical, at the moment of constructing formulations.

Therefore, there are no preferred addresses where the laws of Physics manifest themselves in a different way to the rest of the directions. Nor is any privileged spatial origin admitted. In conclusion, all directions are equivalent.

Finally, the postulate of uniformity of time lays out that the laws of physics are the same in a fixed time as in any preceding time, and that they will remain the same in the future without any origin of time prevailing. Applied in a strict sense, this postulate of temporal uniformity requires that the formulation of the laws of Physics work in the past tense and in the present tense, whatever the stage of evolution in which the Universe was found in the past.

The postulate of uniformity of time poses the problem of empirical contracting in a nonobservable scenario. When it is about to experimentally verify the operation of a law in the opposite direction to that of the arrow of time9, from the future to the past passing through the present, the irreversible sense of time constitutes an insurmountable asymmetry of experimental limitation. The difficulty of overcoming the time barrier for the observation of phenomena in the sense of the future to the past is insuperable. In cases of experimental blockage like this, science often deposit its trust in Mathematics, which can easily transgress the border established by the observable Universe for physical phenomena.

Knowledge = Empirical Data X Mathematics10 Yuval Noah Harari 1. MATHEMATICS IN PHYSICS The internal consistency of all physical laws requires that their formalization be determined reliably and precisely. Then there are three characteristics that the system that formally defines the physical world must gather: consistency, reliability and precision.

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On these postulates repose the theorems of the classical mechanics of the conservation of energy, impulse and kinetic momentum.

8 Example of singular points are the zeros and the infinities that appear when applying the laws of classical physics and that eases off quantum mechanics. The existence of the quantum of action is incompatible with a zero energy of the vacuum, and in quantum mechanical the extension of a physical point is never zero.

9 Expression coined by the British astrophysicist Arthur Eddington, experienced mathematician who weighed the stars based on their luminosity and who trained in the pursued theory of the unification of quantum mechanics, relativity and gravitation, where Einstein himself failed.

10 Harari, Y.N. (2015) Homo Deus: A Brief History of Tomorrow, Penguin Random House, page 142.

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Historically, the formalization of Physics has been based on Mathematics, a key to understanding Nature.

The connection between physical and mathematical forms of existence is mysterious and profound, but the functioning of the physical world is, in fact, permanently governed by mathematical equations and developments. Concepts and mathematical demonstrations are timeless forms that exist independently of the moment they are perceived for the first time. They are based on purely logical arguments and the validity of mathematical affirmations is deduced from the validity of other mathematical affirmations. It is a chain of affirmations whose end point are the axioms, valid statements that do not need proof by evident. In this way a real structure is built by itself, whose impulse sprouts from the mathematical itself and that seems destined to reflect consistently, reliably and precisely the behaviour of the physical world. The bricks of the mathematical building are the equations, which seem to always emerge opportunely to lead and explain the deep principles that govern the Universe.

Mathematical structures are not mere formal instruments of the laws of physics, but reflect them more penetratingly and more widely than those structures from which we started. It is as if the Universe had chosen the mathematical language to express itself in an absolute way, as if Nature itself were guided by the same criteria as mathematical thought. We can even associate advances in Mathematics that correspond to advances in specific theories of Physics. For example, the use of infinitesimal calculus in the 18th century has been basic to understand and expand notions such as speed, momentum and energy. Later, at the beginning of the 20th century, complex variables have been key in the interpretation of Minkowski space, the activity of electrical circuits or the wave function of Schrödinger, and all relativistic physics has advanced thanks to the tensor and matrix calculations. Finally, already advanced the XX century and to overcome the difficulties that the continuous fields in the representation of the quantum reality offer, Einstein proposed "an algebraic theory based on the discrete [spatial] partition towards the future physics"11.

Relationship between Mathematics and physical paradigms The relationship between Mathematics and Physics is fully realized when, in perfect synchrony, we discover the geometric spaces adapted to the successive physical paradigms that emerge occasionally. Penrose exposes a complete study of the different geometries linked to the three main paradigms of Physics 12. The geometries that he presents offer different perspectives of one of the pillars of modern physics, the equivalence principle initially formulated by Galileo13 and that has evolved considerably over time.

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Einstein, A. (1955) Relativistic theory of the non-symmetric field. Appendix II of The Meaning of Relativity, 5ªedición, Princeton University Press, pages. 133-166.

12 Penrose links the Galilean system to the Euclidean space; the Newtonian to the space linked to the spatial density of Cartan, and the relativist to the space of Minkowski. Penrose, R. (2006) Th road to the reality, eight printing, Publisher Alfred A. Knopf.

13 In an experiment attributed to Galileo, the Italian scientist dropped weights from the highest part of the leaning tower of Pisa. Although the weights were of different sizes and materials, they all reached the ground at the same time, which showed that gravity accelerated all bodies without distinguishing their size or nature. Thus, it was demonstrated that Aristotle's belief that heavy objects fell faster than light objects was false The Budapest experiment, realized by Baron Roland von Eötvös, confirmed the principle of equivalence three centuries later.

In 1971, Dave Scott, astronaut of Apollo 15, confirmed the result of Galileo on the lunar soil. Scott took a hammer in one hand and a feather in the other at shoulder height, and released them at the same time. The feather did not swing when it fell, and reached the ground at the same time as the hammer.

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As has been repeatedly said, the resolutive phase of the physical paradigm is its empirical verification, which strictly limits the scientific competence to that of the observable experimental field. It has also been anticipated in the previous section that the laws of physics must be universal, for all the spatial and temporal coordinates in which any experiment is designed and for any position occupied by the observer. Thus, they must work always, in every place and time, without presenting empirical anomalies that refute their content.

The requirement to guarantee the universality of the principles of Physics independently of the experimental framework reinforces the widespread opinion that finding a suitable geometry for each paradigm represents a main application of Mathematics in Physics 14.

The exigency of universality obeys a preferential practical need: that of accrediting a physical law without being altered by the position held by the observer. For this reason, when we speak of geometries we are referring to the experimental field as a whole, that is, to the totality of positions and instants occupied by any experimenter. It is also necessary to include in this experimental frame the displacements to which the observer and the observed during the observation can be submitted. In short, we must think of a geometry that solves the problem of kinematic variations that may occur in each test or trial.

For example, Newton adopted Galileo's inertial reference systems: one in uniform rectilinear motion, the other at rest. From either of the two systems it was impossible distinguish between the physical magnitudes tested. Both from the coordinates in uniform rectilinear motion and from the coordinates at rest, the observer kept the result of the measurements.

The inertial systems fixed by equations represented for Newton's laws the best guarantee to rigorously abide the precept of universality. Mathematics in this case fulfilled the necessary function of defining the transformations of the reference systems so that in any spatial and temporal coordinate in which an experiment was performed, the invariance of its results would be assured.

Einstein needed to have reference systems in which the speed of constant light was maintained. He then resorted to a universal system, which included the Galilean inertial as a particular case. He opted for Lorentz transformations as equations defining this universal system, able to relate the measurements of a physical magnitude made by two observers in different reference systems without changing the speed of light in any of them.

Galilean inertial system and the weak equivalence principle Galileo is distinguished as the founder of the equivalence principle that Newton associated with the law of inertia. The principle recognizes that the laws of dynamics are exactly the same for observers located in any reference system in uniform rectilinear motion or at rest. In other words, the laws of physics do not distinguish the state of rest from the uniform rectilinear movement in any reference system. With this simple enunciate, contrasted by coincidental experimental observations, the law of inertia is expressed. By extension, reference systems in uniform rectilinear motion or at rest are known as inertial.

Newton based on Galileo's inertial reference systems to support his three essential laws of motion. In his magna work The Mathematical Principles of Natural Philosophy15, the 14

The growing importance of Topology as a branch of Mathematics that studies the geometric properties that remain unchanged by continuous transformations reinforces this opinion.

15 Newton, I. (1968) The Mathematical Principles of Natural Philosophy, Dawson of Pall Mall.

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English physicist proposed three principles (that of inertia, that of dynamics and that of action and reaction). All three were elevated to the category of universal invariants 16 for two observers located in the coordinates (x0, t0) y (x1, t1), provided that the equations [M1] of Galileo's inertial system were met. x1= x0 - vt; t1 = t0 [M1] v is the constant speed17 of the observer that moves from (x0, t0) to (x1, t1).

Newton referred to privileged rest masses or uniform speed mass, and extended the concept of inert mass or passive mass to the gravitational mass or active mass18. To this first enunciate of the principle of equivalence, known as weak equivalence, Newton arrived as a corollary of his three laws. The successive experiments of the abovementioned Baron Roland von EĂśtvĂśs, a prestigious physicist famous for his work in gravitation and for perfecting the Cavendish torsion balance, meant the definitive empirical proof to guarantee its acceptance. The aim of von EĂśtvĂśs was to consolidate the principle of equivalence by measuring the acceleration of objects of different composition and in different places on Earth. All results confirmed that the inertial mass was equal to the gravitational mass, regardless of the nature of the object. The accuracy of the proof19 conclusively endorsed the principle of weak equivalence and marked the starting point of the theory of relativity.

Lorentz transformations The Lorentz transformations, deduced in Annex 1 Lorentz transformations, have solved the limitations presented by Galileo's inertial system, valid exclusively for uniform displacements. The equations [M2]20 define the new systems that guarantee that every experiment manifests in the same way for two observers located in the coordinates (x0, t0) y (x1, t1). đ?‘Łđ?‘Ľ đ?‘Ą0 − 2 đ?‘Ľ0 − đ?‘Łđ?‘Ą đ?‘? đ?‘Ľ1 = ; đ?‘Ą1 = [đ?‘€2] 2 2 đ?‘Ł đ?‘Ł √1 − 2 √1 − 2 đ?‘? đ?‘? c is the speed of propagation of light, fixed for any reference system21.

Lorentz was the first to find the equations of special relativity, although Einstein get to deduce them from two simple principles: the laws of Physics are the same from the reference system searched; the speed of light is a universal constant22.

By proposing an equation that transformed time as a function of position and velocity, the Lorentz transformations [M2] represented a complete revolution. The position of classical mechanics was thus revoked, which treated time as an absolute variable. This new approach allowed to develop the concept of space-time, on which the theory of special relativity was based.

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An invariant guarantees identical formulation of the law in any reference system. The mathematician Henri PoincarĂŠ bet by the inertial system of Galileo that accredited the invariance of the laws of the Nature.

17 The condition of constant v is very restrictive. Ensures the invariance of the observation result when the reference system undergoes a rotation, but not when v is variable.

18 The inert mass is defined in Newton's second law by the resistance offered by a body to change its state of motion when a force is applied; the gravitational mass, by the law of universal gravitation, as property that has a body to attract another.

19 The maximum error of the von EĂśtvĂśs experiment at the end of the 19th century was 5∙10-9. Later, the same experiment has been repeated several times. In 1963, Robert H. Dicke got some results with an error of 10-11; in 1971; Braginsky and Panov minimized the error until 10-12.

20 In Annex 1 we get [M2] starting from the constant light speed in two alternative reference systems.

21 The Galileo and Lorentz reference systems are identical for v << c.

22 The constancy of c had been demonstrated empirically by Michelson Morley.

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Space-time definitively solved the problem of symmetry posed by movements at high speeds23. But above all, the equations [M2] ratified the empirical results in any experimental context: of two different experiments perceived by the same observer in the same place at different times, or in different places, or by two observers in different places, in the same moment or at different times.

Rupture of the idea of simultaneity The new concept of space-time broke with the idea of simultaneity associated with the concept of absolute time. This conceptual transmutation impeded to assure with absolute sense if two events had happened at the same time in different places.

If spatial and temporal intervals are taken, the transformation of time [M2] can be written like this đ?&#x2018;Łâ&#x2C6;&#x2020;đ?&#x2018;Ľ â&#x2C6;&#x2020;đ?&#x2018;Ą â&#x2C6;&#x2019; 2 đ?&#x2018;? â&#x2C6;&#x2020;đ?&#x2018;Ąâ&#x20AC;˛ = 2 â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2018;Ł2 đ?&#x2018;?

If â&#x2C6;&#x2020;t = 0, but â&#x2C6;&#x2020;x â&#x2030; 0 â&#x2020;&#x2019; â&#x2C6;&#x2020;t 'â&#x2030; 0, which means that if two events occur at two different points (Î&#x201D;x â&#x2030; 0) simultaneously (Î&#x201D;t = 0) for an observer located in a system of ordinates S (x, t); for another located in another system S '(x ', t'), which moves with respect to the first one, the events occur at different instants (Î&#x201D;t 'â&#x2030; 0).

In general, any event has three potential positions with respect to another reference: past, future or neither past nor future. This classification is valid for any physical paradigm.

However, classical and relativist mechanics differ in the way of establishing it; for the first, the position is absolute; for the second, relative. In particular, the position without past or future is defined by classical mechanics as simultaneous events; On the contrary, relativistic mechanics identifies them as causally not related to each other events.

I spent ten years of my life testing that 1905 equation of Einstein's, and, contrary to all my expectations I was compelled in 1915 to assert its unambiguous experimental verification in spite of its unreasonableness24 Robert Millikan 2. THEORY OF SPECIAL RELATIVITY25 In the reference system defined by [M2] Einstein supported to establish the first theory of relativity, or special relativity, applicable in a restricted way for the case of practically zero gravity26.

Recall that the Lorentz equations endorse the general principle of symmetry, that is to accredit that for two observers located in the coordinates (x0, t0) and (x1, t1) in the

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In order that the same laws could be applied in the same way for high speeds, the invariant of the speed of light had to be integrated into the new reference system defined by the Lorentz transformations.

24 Quote taken from Albert Einstein on his 70th birthday (2004) APS News January Vol 13 nÂş 11, the back page.

25 Einstein did not like the name chosen by Planck for his theory, because it could lead to confusion. He would have preferred to call it the principle of invariance, or to title it with a name that alluded to "the independence of the laws of nature from the point of view of the observer" Arnold Sommerfeld, quoted

by Robinson, A. (2010), in Einstein. Cien aĂąos de relatividad, Blume, page 136.

26 Hence, Einstein ended up calling the theory of restricted or special relativity, limited to spacetime without the presence of gravitational matter, that is, typical of the empty universe.

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reference system [M2], any mathematical equation or magnitude remains invariant although the relative positions of the two observers vary.

The importance of special relativity resides in the fact that physical properties do not change shape, and therefore the equations that define them. Its essence is that the conservation of properties and equations is maintained in any frame of reference. We are not talking about restricted conservation for a particular change of coordinates, but about the universal conservation of physical properties.

In explaining the theory of special relativity, Einstein and other authors used coordinate systems related by Lorentz transformations. For general relativity, it was necessary to work with non-inertial reference systems, which complicated the mathematical treatment27.

Postulates of special relativity The theory of special relativity developed in Annallen der Physik in 190528 was composed of a kinematic and an electrodynamic part: the first radically changed the perception of space and time29 by a new concept of space-time; in the second part a universal limit of the speed was established. The equations presented in the article announced a pair of phenomena shocking with sensory intuition: the relative spatial contraction, or decrease of a body in movement with respect to another one at rest, and the relative dilation of time, or lengthening the duration of the events that affected a body in movement with respect to a body at rest. The spatial contraction and the dilation of time were deduced directly from the constancy of the speed of light. These surprising results were confirmed by experimental observations.

Einstein formalized with the Lorentz transformation equations the new laws starting from the following postulates: • The laws of Physics are the same in the reference systems defined by the Lorentz transformations. In other words, there is no privileged reference system that can be considered absolute.

• The speed of light in vacuum is a universal constant c that is independent of the movement of the light source.

• The clocks of the observers in movement go slower than the clocks of the observers in rest.

• Moving bodies contract with respect to bodies at rest.

• Inertia is a function of energy and not of mass, as Newton supposed.

According to the first postulate, all the laws of Physics are always invariant with respect to the Lorentz transformations. Two conclusions emerge from the second postulate: Maxwell's equations, the cornerstone of electromagnetism, are invariant under the Lorentz transformations, and time and space cannot be absolute. The third and fourth principles imply that the time interval between two events and the distance between two points depends on each observer.

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Later, in section 4, Theory of General Relativity, frames of reference are established that require the use of vectors and tensors, mathematical operators necessary to guarantee the invariance of the field equations of general relativity.

28 Einstein, A. (1905) Zur Elektrodynamik bewegter Körper, Annallen der Physik, Vol. 372, 10: 891-921.

29 Even today there are doubts about the original idea of special relativity. In the lecture pronounced in 1904 and entitled On the Present and Future State of Mathematical Physics, Poincaré spoke of the principle of relativity, and he then expounded the concept of relative time in terms of the reference system of the observer. In the historical article by Einstein, there is no reference to Poincaré's works.

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Maxwell and the wave nature of light Maxwell's equations are simply adjustments of the classical model to adapt it to empirical results. They form a set of four equations that describe electromagnetic phenomena after unification of the electric and magnetic fields30.

The first equation [A2.1] from Annex 2 Maxwell equations â&#x192;&#x2018; .đ?&#x2018;Ź â&#x192;&#x2018;â&#x192;&#x2018; = đ??&#x2020; đ?&#x203A; [A2.1]31 đ?&#x153;ş đ?&#x;&#x17D;

is Gaussâ&#x20AC;&#x2122;s law of electric fields, which the classical model presents as the Coulomb law for point charges (see Table A2.1 of Annex 2). The law of Gauss applies, in a dynamic scheme, the concept of electric field and defines the electric flow that crosses a surface in the same way that is used in the mechanics of fluids. Thus, the stationary scheme presented by Coulomb is overcome.

So that, Maxwell's first equation is the generalization of Gauss's law, enunciated in a localized way. He formulates it with a differential operator, while Gauss uses an integral operator defined on a closed surface with the same result. â&#x192;&#x2018; = đ?&#x153;&#x152; đ?&#x2018;&#x2018;đ??´ = 1â &#x201E;đ?&#x153;&#x20AC;0 â&#x2C6;Ťđ?&#x2018;&#x2030; đ?&#x153;&#x152;đ?&#x2018;&#x2018;đ?&#x2018;&#x2030; â&#x2020;&#x2019; â&#x192;&#x2018;â&#x2C6;&#x2021; . đ??¸ [A2.4] â&#x2C6;Žđ?&#x2018; â&#x192;&#x2018;â&#x192;&#x2018;đ??¸ â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018; đ?&#x153;&#x20AC;0 Maxwell's first equation [A2.1] indicates that the net flow of an electric field across a surface depends on the density of the electric charge Ď that encloses that surface and the dielectric constant in the vacuum Îľ0.

The second Maxwell equation expresses that all the lines of a magnetic field always form a closed loop, that is, they start and end at the same point. It is deduced therefore that there is no magnetic monopole, which agrees with the empirical evidence of the spontaneous appearance of two magnets if one breaks. It coincides with the experimental law of Gauss for magnetic fields, which states that this type of fields, unlike electric fields, do not start and end in different charges (Figures M132 y M233). The lines of the magnetic fields are closed, which prevents the existence of magnetic monopoles. That is, if a dipole is available on a closed surface, no magnetic flux comes out or enters, therefore, the magnetic field B does not diverge. In other words, the divergence of the lines of the magnetic field is always zero, as Maxwell's equation [A2.5] of Annex 2 shows: â&#x192;&#x2018; .đ??ľ â&#x192;&#x2018; =0 â&#x2C6;&#x2021; [A2.5]

Figure M1 30

In Annex 2 Maxwell equations, the four equations are developed in detail from the current laws until before the unification of electromagnetic phenomena.

31 Here the correlative numbering of equations of the series [M] is interrupted by the insertion of equations of the Annexes.

32 Figure taken from https://upload.wikimedia.org/wikipedia/commons/b/be.

33 Figure taken from http://electric1.es/cm/lineasdefuerza.html.

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Figure M2 The third Maxwell equation [A2.6] of Annex 2 â&#x192;&#x2018; â&#x192;&#x2018;â&#x2C6;&#x2021; Ă&#x2014; đ??¸â&#x192;&#x2018; = â&#x2C6;&#x2019; đ?&#x153;&#x2022;đ??ľ

[A2.6] is the differential expression of Faraday's law, whose enunciate says that a variable magnetic field B originates an electric field E. Under the form of integral, Faraday proposes in equation [A2,7] of Annex 2 that the variation with respect to the time of the magnetic flux that crosses a surface is equal and of opposite sign to the circulation of an electric field that passes through a conductor located on the edge of the surface. â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018; = â&#x2C6;&#x2019;đ?&#x2018;&#x2018;â &#x201E; â&#x2C6;Ť đ??ľ â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018; â&#x192;&#x2018; đ?&#x2018;&#x2018;đ??´ [A2.7] â&#x2C6;Žđ?&#x2018;? đ??¸â&#x192;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x2018;&#x2020; [A2.6] indicates that any non-stationary magnetic field B produces a rotational electric field E of opposite sign, trying to compensate the variation of the magnetic flux. In other words: a non-stationary field B causes the circulation of E along closed lines (see Figure A2.1 of Annex 2).

Finally, Maxwell's fourth equation [A2.8] in Annex 2 affirms that any electric field that varies with time produces a rotational magnetic field, as reflected in Figure A2.2 of Annex 2. â&#x192;&#x2018; â&#x192;&#x2018;â&#x2C6;&#x2021; Ă&#x2014; đ??ľ â&#x192;&#x2018; = đ?&#x153;&#x2021;0 â&#x192;&#x2018;đ??˝ + đ?&#x153;&#x2021;0 đ?&#x153;&#x20AC;0 đ?&#x153;&#x2022;đ??¸ [A2.8] đ?&#x153;&#x2022;đ?&#x2018;Ą

đ?&#x153;&#x2022;đ?&#x2018;Ą

J is the current density and Âľ0 the magnetic permeability in vacuum.

[A2.8] therefore represents a complementary equation to [A2.6] and a generalization of Ampère's law, formulated in equation [A2.9] of Annex 2 for a static magnetic field B and an electric current I constant. â&#x192;&#x2018; â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;&#x2122; = đ?&#x153;&#x2021;0 đ??ź [A2.9] â&#x2C6;Žđ??ľ But for a mobile magnetic field the electric field varies through time and the equation [A2.9] breaks the principle of conservation of the charge. For this reason, Maxwell had to correct it by adding the displacement current. đ?&#x153;&#x2022;đ??¸â&#x192;&#x2018; đ?&#x153;&#x2021;0 đ?&#x153;&#x20AC;0 đ?&#x153;&#x2022;đ?&#x2018;Ą Therefore, Maxwell's differential equation [A2.8] is but an adaptation of Ampère's integral equation [A2.9] for non-stationary fields.

The four Maxwell equations meant the conclusive proof of the universality of special relativity. It had been shown that both the electric and magnetic fields moved as wave functions, which propagated in vacuum at the speed indicated in [A2.1.8] of Annex 2.1 Magnetic field of a linear conductor.

1 đ?&#x2018;?= đ?&#x153;&#x20AC;đ?&#x153;&#x2021; [A2.1.8] â&#x2C6;&#x161; 0 0

The immediate conclusion derived from [A2.1.8] was that the velocity of a wave did not depend on the velocity of its source; in particular Maxwell set the constancy of the speed of light, confirmed by Einstein. This radical novelty put an end to the hypothesis of the 11

existence of the ether34. Although, as has been said above, background of the invariance of the speed of light already existed since 188135, definitive experimental tests took a long time to complete.

The annulment of the ether hypothesis proved the self-propelling nature of the electromagnetic field, where the continuous presence of accelerations made the mediation of any fluid superfluous. Thus, special relativity assumed the free propagation of the electromagnetic field through space without the need for the existence of a diffusing source. This new conception solved the difficulties of the Newtonian field to explain Maxwell's wave theory, difficulties that Galileo's inertial system taken as reference by Newton could not remedy.

But there still had to be resolved the question of analysing the electric field and the magnetic field as a single electromagnetic field. The Lorentz transformations [M2] corrected the constraints of the equations of [A2.6] and [A2.8] of Annex 2, which prevented unifying the experimental results of the electromagnetic phenomena. Wheeler presents the indistinct phenomenon of the variation of an electric field of the change of a magnetic field in two experimental scenarios: "When the north pole of a bar magnet is thrust through a coil of wire, a momentary voltage is generated between the two ends of the wire. The magnitude of this voltage is predicted by a set of equations formulated byJames Clark Maxwell and Michael Faraday [equation A2.6]. On the other hand, when the coil is thrust with equal speed over the bar magnet, the same voltage is predicted -and found- but a quite different equation [equation A2.8] has to be called on to do the predicting.

Any sensible way if looking at this electromagnetic induction, Einstein reasoned, should not make this distinction between which component â&#x20AC;&#x201C; the magnet or the coil- was moving and which was stationary. Only relative motion [between the magnet and the coil] should matter"36.

Einstein solved the dilemma that electric and magnetic fields depended on reference systems in his famous 1905 paper on the electrodynamics of moving bodies37.He proposed in this work that the electromagnetic induction should be the same, both from a reference system where the coil was kept at rest and the magnet in motion and from a system where the magnet was at rest and the coil in motion.

The acknowledgment of the wave identity of light represented for Einstein the accolade of greater prestige for the theory of special relativity. In effect, the third and fourth equations of Maxwell [A2.6 and A2.8] reveal that an electromagnetic wave is composed of the joint action of an electric field E and a magnetic field B. Figure M338 shows the direction of propagation of the two waves (red), which is perpendicular to the plane of oscillation of the electric (green) and magnetic (blue) fields.

Both fields also advance in orthogonal directions to each other, coinciding their crests and valleys.

34

The ether theory was based on the presence of a weightless and elastic fluid throughout the space, where the electromagnetic waves propagated as mechanical oscillations and to which the light passed through when moving, which is why the measurement of the speed of the light was supposed relative to the ether.

35 In 1881 the experiment of Michelson and Morley tried to measure the speed of the movement of the Earth in relation to the ether. Paradoxically, it confirmed a law that did not exist yet (the law of special relativity was published 24 years later), and erased the ether as a reference for the cosmic movement. His confirmation had to wait until the Kennedy-Thorndike experiment in 1932.

36 Wheeler, J.A. (1990) A journey into gravity and spacetime, Scientific American Library, page. 8.

37 Paper already mentioned in note 28.

38 Figure taken from http://1.bp.bolgspot.com.

12

Figure M3 In Einstein's chimerical dream going after a ray of light, he would see something similar to the electric (green colour) and magnetic (blue colour) vector fields of the electromagnetic wave of Figure M3. He imagined himself contemplating a wave of the light beam, and expected to see how the lengths of the electric and magnetic vector fields would increase from zero to their maximum intensity, and then decrease again to zero on the next wave.

Renewal of the Newtonian invariants Until the exposure of special relativity, nobody had doubted that the electrical, optical and thermal properties of matter were due to Newton's mechanics applied to molecular motion and Maxwell's electromagnetic field theory. But in 1905, the introduction of the space-time concept represented a renewal of the invariants of the old Newtonian paradigm.

Double substitution of invariant: mass for energy and speed of light Einstein refuted Newton's law of conservation of mass by establishing that mass and energy were interchangeable units. From the postulates of special relativity, enunciated above, transcended multiple consequences. The most popular in the history of physics was that the mass represented accumulated energy. The equivalence between mass and energy was formulated as the general Einstein energy equation [A3.4] of Annex 3 Einstein's general equation of energy.

E = mc2 [A3.4] [A3.4] represents the double substitution of mass m for the invariants speed of light c and energy E. The new variable condition of m established definitively by Einstein through the equation is: đ?&#x2018;&#x161;0 đ?&#x2018;&#x161;= [M3]39 2 â&#x2C6;&#x161;1â&#x2C6;&#x2019;đ?&#x2018;Ł2 đ?&#x2018;?

m0 is the rest mass and v the velocity of m. According to [M3], as v grows the value of m increases. When v = c, m â&#x2020;&#x2019; â&#x2C6;&#x17E;.

The loss of the invariance of the mass m forced to revise the Newtonian equations of the kinetic energy and the linear momentum or amount of motion. According to the new definition of mass [M3], the kinetic energy Ec will be: đ??¸đ?&#x2018;? = đ?&#x2018;&#x161;đ?&#x2018;? 2 â&#x2C6;&#x2019; đ?&#x2018;&#x161;0 đ?&#x2018;? 2 = đ?&#x2018;&#x161;0 đ?&#x2018;? 2 [

1 2

â&#x2C6;&#x161;1â&#x2C6;&#x2019;đ?&#x2018;Ł2 đ?&#x2018;?

39

From here, the correlative numbering of the series [M] is recovered.

13

â&#x2C6;&#x2019; 1]

[M4]

[M4] represents an important evolution of the law of Newtonian inertia. From here, the inertia ceases to depend on the mass m and is a function of the energy Ec. On one hand, to the Newton's passive mass corresponds the second summand of the second member of the equation, and to the active mass, the first summand of the second member.

On the other hand, the first summand of the bracket of the equations [M4], known as the Lorentz factor Îł, can be developed using the classic Taylor series: â&#x2C6;&#x2019;1â &#x201E; â&#x2C6;&#x2019;3â &#x201E; đ?&#x2018;Ł 4 đ?&#x2018;Ł 2 â&#x2C6;&#x2019;1 đ?&#x2018;Ł2 [1 â&#x2C6;&#x2019; đ?&#x2018;? 2 ] â &#x201E;2 = 1 + 1â &#x201E;2 đ?&#x2018;? 2 + 22! 2 đ?&#x2018;? 4 + â&#x2039;Ż [M5] Introducing [M5] in [M4], we obtain the equation of kinetic energy for any reference system. đ?&#x2018;Ł4 đ?&#x2018;Ł6 đ??¸đ?&#x2018;? = 1â &#x201E;2 đ?&#x2018;&#x161;0 đ?&#x2018;Ł 2 + 3â &#x201E;8 đ?&#x2018;&#x161;0 đ?&#x2018;? 2 + 5â &#x201E;16 đ?&#x2018;&#x161;0 đ?&#x2018;? 4 + â&#x2039;Ż [M6] The first term of [M6] is the energy at rest, identical to the expression formulated by Newton, which represents in the relativistic paradigm only one particular case for v<<c.

Substitution of invariants: force by linear momentum From the new mass definition of [M3], another fundamental expression of Newtonian mechanics had to be revised. In effect, Newton's momentum or linear momentum (p = mv) stayed modified by the variable state of m, according to the following equation: đ?&#x2018;&#x161; đ?&#x2018;Ł đ?&#x2018;?= 0 2 [M7] â&#x2C6;&#x161;1â&#x2C6;&#x2019;đ?&#x2018;Ł2 đ?&#x2018;?

When v â&#x2020;&#x2019; c; p â&#x2020;&#x2019; â&#x2C6;&#x17E;. Therefore, at the limit of a mass at the speed of light, an immobile observer would perceive that the inertia of m0 would increase indefinitely.

From [M7], it is obtained: đ??š=

đ?&#x2018;&#x2018;đ?&#x2018;? đ?&#x2018;&#x2018;đ?&#x2018;Ą

â&#x2030; đ?&#x2018;&#x161;0 đ?&#x2018;&#x17D;, ya que

đ?&#x2018;&#x2018; đ?&#x2018;&#x2018;đ?&#x2018;Ą

{

đ?&#x2018;&#x161;0 đ?&#x2018;Ł 2 â&#x2C6;&#x161;1â&#x2C6;&#x2019;đ?&#x2018;Ł2 đ?&#x2018;?

} â&#x2030; đ?&#x2018;&#x161;0

đ?&#x2018;&#x2018;đ?&#x2018;Ł đ?&#x2018;&#x2018;đ?&#x2018;Ą

= đ?&#x2018;&#x161;0 đ?&#x2018;&#x17D;

[M8]

[M8]40 induces the replacement of the Newtonian force F by the invariant linear momentum p, a substitution necessary to correct the malfunction of the fundamental principle of Newtonian dynamics, only acceptable for v << c.

Space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality41 Hermann Minkowski 3. MINKOWSKI'S FOUR-DIMENSIONAL SPACE The radical change of paradigm provoked by the theory of special relativity required the fusion of space and time42, units considered a priori to any experiment, and therefore 40

As stated above, the linear momentum is associated with the infinitesimal calculus. Newton dominated what he called fluxions and that they were only derivatives. He applied them in each case as a master craftsman. But the doubt remains as to whether he was the pioneer in discovering the conceptual connection between derivative and integral and whether he gauged the deep scope of this relationship. His fierce struggle with Leibniz for the original idea of infinitesimal calculus has been narrated in detail by DurĂĄn, J. A. (2006) Isaac Newton & Gottfrield Wilheim Leibniz. La polĂŠmica sobre la invenciĂłn del cĂĄlculo infinitesimal, CrĂ­tica.

41 From the speech delivered by Minkowski at the German General Assembly of Natural and Physical Scientists of 1908. https://wikivisually.com/wiki/Minkowski_space.

42 The impossibility of defining time as an absolute quantity Einstein based on inherent relationship between time and the signal speed of a gravitational wave produced by an accelerated mass.

14

independent of each other, by classical Physics. Minkowski, professor of Einstein, found the new space-time defined by the equations [M2] of Lorentz, where for the first time was included as a fourth dimension. With the addition of the imaginary dimension cit to the three real dimensions of classical mechanics (x, y, z), the new space-time of 6 planes was shaped: 3 pure spatial planes (xy, xz, yz) and 3 temporal-space planes (ctx, cty, ctz).

The space of Minkowski arose opportunely, as it rained from heaven, to harmonize the mathematical formalization to the new physical paradigm of special relativity. The Lorentz transformations were directly applicable to the Minkowski space, as shown in Annex 4 Minkowski's four-dimensional space, where we arrive at the equations [M2] with the simple rotation of an angle of the axes.

Minkowski showed that if space and time ceased to be separate dimensions and were integrated geometrically into a four-dimensional space-time43, relativistic transformations were reduced to simple rotations in the new four-dimensional space-time.

From the mathematical plane, this fusion facilitated the good functioning of the new paradigm of special relativity with the transformations already established by Lorentz, which was decisive for the radical substitution of the Galilean inertial reference systems for the four-dimensional ones. From the plane of Physics, space-time ceased to have a separate existence from physical reality; the objects were extended in it and the concept of empty space lost its meaning Four-dimensional reference systems permit two observers located in two systems S (x, y, z, t) and S '(x', y ', z', t ') to measure the same space-time interval44 between any two events, independently of S and S'. Or what is the same, they guarantee that the elementary interval ds is an invariant for all transformation suffered by S and S' during the observation. The equation that defines four-dimensional spacetime is: ds2 = dx2 + dy2 + dz2 – c2dt2 [M9] [M9] is a quadratic function, positive in the plane tangent to each point, such as Euclidean geometry, and therefore fully maintains the calculation and measurement capacity of this new geometry. Thus, in four-dimensional space it is possible to calculate the length of a curve, the area of a surface, the angle formed between two curves and the radius of curvature at any point, provided that equation [M9] is fulfilled.

The appearance of a negative quadratic summand in the fourth temporal space dimension of [M9] forces us to introduce the imaginary unit i, which leads to the inevitable use of complex variables. The almost magical45 component of the imaginary unit i reveals us untold properties in the relationship between the four dimensions that shape Minkowski's space. On the one hand, complex variables imply the existence of an angular gap between their imaginary component46 (cidt) and the complex variable itself (ds). On the other hand, its use arises in Physics whenever an equation is required that reflects a phenomenon of harmonic vibrational nature, that is to say every time that it is necessary to handle variables function of the pulsation ɷ or of the frequency ν. These variables appear as an imaginary component of the equation; in [M9], the imaginary component is a function of ɷ or of the frequency ν, since c = λɷ/2π = λν, where λ is the wavelength.

43

At first, Einstein did not give importance to Minkowski's geometrical interpretation, taking it merely as a mathematical formality without physical meaning, but eventually changed his attitude by adopting the fourdimensional geometrical point of view he would use for the postulation of the General Theory of Relativity.

Published by Martín, A., blogspot la teoría de la relatividad, https://es.scribd.com/doc/256309409/LaTeoria-de-La-Relatividad-Armando-Martinez-Tellez.

44 The interval in four-dimensional space corresponds to the distance in the three-dimensional.

45 Usual adjective conferred by Penrose to complex numbers in general.

46 See below Figure M4, where the four-dimensional light cone and the angle formed by the generatrix of said cone and the vertical axis of coordinates are presented.

15

The second member of the equation [M9] is a function of the strictly spatial component of Newton's previous paradigm, which comprises the three classical dimensions dx, dy and dz, and other temporal kinetics, cdt, imaginary. The difference in metrics between the old Euclidean geometry and the new one is radically contrasted when integrating ds. đ??ľ In three-dimensional space, â&#x2C6;Ťđ??´ đ?&#x2018;&#x2018;đ?&#x2018; represents the minimization of the distance traveled between two extreme positions, A and B; in four-dimensional space, the integral means the maximization of the interval elapsed between A and B. According to Hawking, the renounce to consider cdt as a real magnitude guarantees the existence of a finite space without singularities.

"The singularity theorems of classical general relativity showed that the universe must have a beginning, and that this beginning must be described in terms of quantum theory. This in turn led to the idea that the universe could be finite in imaginary time, but without boundaries or singularities. When one goes back to the real time in which we live, however, there still appear to be singularities. The poor astronaut who falls into a black hole will still come to a sticky end. It is only if he could live in imaginary time that he would encounter no singularitiesâ&#x20AC;?. â&#x20AC;&#x153;This might suggest that the so-called imaginary time is really the fundamental time, and that we call real time is something we create just in our minds. In real time, the universe has a beginning and an end at singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities and boundaries"47.

The four-dimensional character of ds allows any space-time interval to be absolute and its measurement is kept as a reality independent of the reference system. This solves the classical problem of the double trajectory followed by a stone dropped from a train in Galileo's inertial system, where the magnitudes of space and time remain independent.

Indeed, within the system defined by the equations [M1], two observers perceive two different trajectories: a straight line from the train, and a parabola from the outside to the train. Nevertheless, in the space-time defined by the equations [M2], the stone describes a single geodetic trajectory48, which Minkowski calls the universal line.

The space of Minkowski allows to establish the composition of speeds in different systems of reference. Equation [A4.3] of Annex 4 Minkowski's four-dimensional space relates the velocity v' in the system S' with the velocity of the mobile v measured in S with u being the speed with which the system S' moves away from the system S. đ?&#x2018;Ľ â&#x20AC;˛ +đ?&#x2018;Łđ?&#x2018;Ąâ&#x20AC;˛

đ?&#x2018;Ľ

đ?&#x2018;Łâ&#x20AC;˛ = đ?&#x2018;Ą =

đ?&#x2018;Ł đ?&#x2018;?

đ?&#x2018;Ą â&#x20AC;˛ + 2đ?&#x2018;Ľ2

=

đ?&#x2018;˘+đ?&#x2018;Ł đ?&#x2018;˘đ?&#x2018;Ł

1+ 2 đ?&#x2018;?

[A4.3] 49

On the one hand, for prior to relativity physics v << c â&#x2020;&#x2019; v' = u + v. On the other hand, in [A4.3] the constancy of the speed of light is fulfilled: if v = c â&#x2020;&#x2019; v' = c.

The new geometry of special relativity The paradigm of special relativity does not recognize space permanently at rest nor the identical passage of time for all observers of the same phenomenon, as Newton supposed.

In relativistic physics, the clock of an immobile observer marks a different time than that of another that moves at a relative speed with respect to the first. This double 47

Hawking, S. W. (2005) The theory of everything. The origin and fate of the Universe, Phoenix Book, page 100.

48 The geodesic is the line of minimum length that connects any pair of points in a curved space. In the particular case of a sphere, it is a maximum circle arc on its surface and is included in a plane that always passes through the center of the sphere.

49 Here the correlative numbering of equations [M] is interrupted again.

16

characteristic, a dynamic space and a time with a differentiated rhythm for each pair of observers that moves closer or farther away from each other, offers great advantage to the geometry of special relativity with respect to the Galilean inertial. The simple fusion of space-time into a four-dimensional flat dimension represents greater guarantee of invariance of physical laws for the geometry of special relativity over Newtonian.

The new geometry is based on the Lorentz [M2] equations, which work for every movement that the object and the observer experience within the reference system without altering the magnitude of the observed. The constancy of the measurement within the system defined by the Lorentz transformations confirms the first postulate of special relativity, which says that the laws of physics must be the same in all systems of reference.

The conical surface of Figure M450 represents the new geometry, where all events occur within the two light cones.

Figure M4 The graphic scheme has been divided into four quadrants and, for greater simplification, the three spatial dimensions (x, y, z) have been reduced to two (x, y), and c = 1 has been assumed, until finally staying a three-dimensional space-time (x, y, t). The conical surface of Figure M4 limits the horizon of an observer located at its vertex, a horizon defined by making ds = 0 in equation [M9]. dx2 + dy2 + dz2 â&#x20AC;&#x201C; c2dt2 = 0 [M10]51 The fundamental concept of the new flat space-time geometry is that ds no longer represents the distance that measures the separation between two points in space, but the interval that separates two events in spacetime. dx, dy, dz and cidt are respectively the separations longitudinal, transverse, vertical and temporal. In this scheme, events are separated in space and time; this joint separation is called an interval, which for some a pair of events can be spatial or temporal, but can never be spatial-temporal.

[M10] defines the set of all points of all null interval events (ds = 0) 52, which is an invariant linked to the universal invariant c, established between near and far events independently of the observer. This characteristic of the temporal-space geometry is represented in the cone of light of the future, projected along the positive semi-axis of time, and the cone of light from the past, prolongation of the previous along the semi-axis negative. On the outside of the two light cones, there are no observable fields53, and therefore no fields that allow empirical experimentation and that endorse verifiable results of any scientific paradigm. In this way, the observable field is reduced to the interior 50

Figure taken from https://commons.wikimedia.org/wiki.

From here, the correlative numbering of the series [M] is recovered.

52 The null geodesic (ds = 0) are also known as vision horizon or cone of light.

53 It is inadmissible for any theory to deal with elements that can never be observed.

51

17

limited by the conical surface, which is defined by the angle formed by the generatrix of the cone and the time axis itself54. Since (dx2 + dy2 + dz2)1/2 = cdt, this angle measures π/4.

Before special relativity, it made sense to talk about a past and a common future, since time was absolute and universal for any place in the Cosmos. But after Einstein's theory, for each observer there is a past, a present and a future delimited by the light cones of Figure M4. The common vertex of the two cones delimits the past and the future of the observer. Since no one can overcome the speed of light, the only way to reach the present from the past is by moving in the enclosure limited by the lower cone. And the only trajectory of arrival at a point in the future is moving within the contour bounded by the upper cone. Thus, there are only five relations between events within the scheme of Figure M4, always assuming a fixed one as reference in the common vertex of the two cones: • Event located below the lower cone → Past: events separated by a time interval.

• Event located on the lower cone → Past: null interval between events.

• Event located between the lower and upper cone → Neither past nor future: events separated by spatial intervals.

• Event located on the upper cone → Future: null interval between events.

• Event located above the upper cone → Future: events separated by a time interval.

It is now assumed that the relative velocity between two observers is v, and that the common x-axis is maintained and two kinetic-temporal axes, ct and ct ', belonging to two different reference systems, S (x, ct) and S' (x ', ct')55, according to Figure M556.

Figure M5 If the relative velocity between the two observers is v = 0,4c and the first observer moves the distance x1 = 1 in a temporal kinetic interval t1c = c, the second observer will need a kinetic temporal interval ct2 = cx2/v = 2.5c to travel the same distance x2 = 1. If the first observer moves a distance x’1 = 2 in a temporal kinetic interval t’1c = c, the second observer will use in the same distance x’2 = 2 an interval ct’2 = cx’2/v = 5c. According to these calculations, for Δx1 = (x’1 – x1) = 1 → Δx2 = (x’2 – x2) = 1 and for Δct1 = (ct’1 – ct1) = 0 → Δct2 = (ct’2 – ct2) = 2,5 c Therefore, the points of the ordinate axis ct' of the reference system S' will be aligned in the blue line of Figure M557. When the relative velocity v of the two observers decrease 54

Recall that the angular gap is associated with the imaginary nature of the time axis.

The origins of S and S 'in our scheme we make them coincide in a single one for simplifying.

56 Figure taken from http://2.bolgspot.com.

57 Again, the angular gap associated with the imaginary component of the time axis arises. In Figure M5, the gap shift is the angle formed between the half axes ct and ct'.

55

18

Δct2, it will increase for the same Δx2, ie the blue line ct’ will approach the vertical axis ct, and vice versa, when v increases, the blue line will move away from the ct axis. In the limit, the ct axis and the line ct' will coincide when v = 0, and for v = c, the ordinate axis of S' will coincide with the bisector of the first quadrant of the Cartesian plane x-ct (yellow line of the Figure M558).

The yellow line in Figure M5 is the geometric verification test of constant c that marks the barrier limits v ≤ c. The space covered between the semi axis ct and the yellow line bounds the observable field of existence of physical phenomena within the first quadrant, which corresponds to the observable field defined by the light cone of Figure M4.

How do two observers 1 and 2 perceive the tracking of the trajectory of a ray of light?

For the observer 1 that moves in the reference system S '(x'-ct'), the ray of light projected from the point E (x’ = 0, ct’ = -a) reaches the point P (x' = a, ct' = 0), where a mirror reflects it towards R (x' = 0, ct' = a). It is obvious that the path EP followed by the ray of light in Figure M659 is parallel to the bisector of the first quadrant, and the trajectory PR, parallel to the bisector of the third quadrant.

Figure M6 On the other hand, within the reference system S (x-ct), the stationary observer 2 sees the trajectory of the light beam according to Figure M7. Starting from the same point E (x '= 0, ct' = -a), the path EP is parallel to the bisector of the first quadrant of the reference system S; the point of arrival of the reflected ray is R (x '= 0, ct' = a). Drawing from R the parallel to the bisector of the third quadrant of the system S, P is at the cut point of the two yellow bisectors of Figure M7. The blue dotted line that joins the common origin of the systems S and S’ with the point P marks the direction of the axis X', which forms an angle with the X axis equal to that formed by the semi axis ct and ct'.

Figure M7

58 59

On his fantastic journey up in a ray of light, Einstein would follow the direction of the yellow line M5.

Figures M6 and M7 have been taken from http://2.bp.bolgspot.com.

19

Between the two observers that follow the trajectory of the light beam, for the observer 1, which moves in the reference system S', the points E and R are kept in the same ordinate ct' = 0 of Figure M6, which means that he perceives them simultaneously. On the contrary, the stationary observer 2 of the system S perceives E and R as successive events separated by the interval Î&#x201D;ct of Figure M7. In the latter case there is a temporal dilatation60, which corresponds to a contraction Î&#x201D;x, while Î&#x201D;x' remains constant along the experiment.

Dilation of time and contraction of space From the constancy of the speed of light established in the second postulate of special relativity, two verifiable consequences emerge, resorting to the Lorentz transformations: the contraction of space and the delay of time. Both magnitudes, space and time, vary depending on the speed. Thus, for example, an observer located within a moving reference system S' undergoes a spatial reduction and a delay in its clock with respect to another observer within a system S at rest, or whose relative velocity is lower than that of the first. Therefore, the distances will be shortened more and the days will be longer within S than in S'.

Four classic examples of spatial contractions and temporal dilatations Next, four experiments are presented where spatial contractions and temporal dilatations are manifested: a) Life of muons produced by cosmic rays In the famous experiment performed by David Frisch and James Smith in 1963, the number of muons61 that reached Earth was measured. In the recorded data at different heights by the measurements, Frisch and Smith detected that the intensity of muons/hour measured at sea level was much lower than expected. The calculation had predicted 412 muons/hour in the lowest level of measurement and counted only 27. In the balance they deducted from the 568 muons / hour registered in the highest level of the experiment, the top of Mount Washington, those lost by disintegration to sea level62.

The anomaly that revealed the experiment of Frisch and Smith could not explain why the speed of the muons that came to touch earth before disintegrating was greater than that assumed in principle. The definitive explanation was found in the dilation of the time predicted by the special relativity, equivalent to the equation [M2] of Lorentz: đ?&#x2018;Łđ?&#x2018;Ľ đ?&#x2018;Ąâ&#x2C6;&#x2019; 2 đ?&#x2018;? đ?&#x2018;Ąâ&#x20AC;˛ = 2 â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2018;Ł2 đ?&#x2018;? v = 0.9978c and t = 2â&#x2C6;&#x2122;10-6 s are typical values of a muon, to which corresponds the distance x = 0.9978â&#x2C6;&#x2122;3â&#x2C6;&#x2122;108â&#x2C6;&#x2122;2â&#x2C6;&#x2122;10-6m â&#x2030;&#x2C6; 600 m The time delay t' calculated by substituting these values in the previous equation is: 60

Once the speed of the constant light c has been established, we will refer to components and temporal rather than temporal kinetic variations hereafter.

61 The muon is an unstable particle, generated by the interaction of cosmic rays with the atmosphere, traveling at a speed close to light, negatively charged as the electron, but about 200 times heavier. It was detected in 1936 by Nobel laureate Carl Anderson while studying cosmic rays 62 The balance had taken as average life of the muon, that is to say the time it took to disintegrate half of initial particles, 2 microseconds.

20

đ?&#x2018;Ąâ&#x20AC;˛ =

2 â&#x2C6;&#x2122; 10â&#x2C6;&#x2019;6 0,99782

â&#x2030;&#x2C6; 30 â&#x2C6;&#x2122; 10â&#x2C6;&#x2019;6 đ?&#x2018;

â&#x2C6;&#x161;1 â&#x2C6;&#x2019; t' = 30 Âľs is the time spent traversing the earth's surface for an observer moving with the muons. In the calculation of the path, t' is the reference time, instead of the 2 Îźs for an observer at rest. The dilation of the time predicted by the special relativity is then the calculated path for 30 Îźs instead of 2 Îźs, that is: x' = 0,99780.9978â&#x2C6;&#x2122;3â&#x2C6;&#x2122;108â&#x2C6;&#x2122;30â&#x2C6;&#x2122;10-6m â&#x2030;&#x2C6; 9.000 m >> x = 600 m In the 600 m not calculated in the experiment, 385 muons were lost, difference between 412 predicted and 37 registered. b) Paradox of the two twins Paul Langevin, organizer of the historic Solvay congress, presented the following mental exercise: an astronaut begins a long journey in a spacecraft at speed close to that of light, while his twin brother remains on Earth. At the end of the trip, the two twins meet again, but one of them barely recognizes the other.

According to special relativity, the astronaut returns younger than the terrestrial twin. The reason is that the time in the spacecraft go slower than on Earth. The Lorentz transformations explain that dilation of time.

The paradox originates when the movement is posed from the perspective of the traveling twin, who sees the twin get away from the Earth. The first expects his brother on Earth to age less, when moving with respect to the spacecraft at speed close to light. Undoubtedly, the singularity is not that one twin gets older more than the other, but that each brother concludes that it is the other who will age less.

The paradox posed by Langevin was solved by the Hafele-Keating experiment, which tested the temporal dilation predicted by special relativity. The test consisted of uploading two atomic clocks to separate planes and comparing their readings with another identical clock on Earth and synchronized with the first two at the beginning of the flights.

A plane took off Eastbound, following the direction of the Earth's rotation; the other plane followed the opposite direction, towards the West.After a long journey of 40 hours, the two planes landed at the same point of departure, and by comparing their atomic clocks with that of the Earth they were no longer synchronized. The clock of the plane heading eastward registered less time than the corresponding one on board the plane heading towards the West. Therefore, the first plane had invested less time than the second on the route, which was equivalent to higher apparent speed, or an apparent delay with respect to the other two clocks, of the plane heading East. Published in the journal Science63, the results were the following: Direction of the plane East West

Nanoseconds expected earned General Special Total Nanoseconds relativity relativity measured earned +144Âą14 -184Âą18 -40Âą23 -59Âą10 +179Âą18 +96Âą10 +275Âą21 +273Âą7

63

Difference 0,76Ď&#x192; 0,09 Ď&#x192;

The table corresponds to the one published in the Science article and includes the expected earned time by general relativity. Hafele J.C, and Keating R.E. (1972) Around the world atomic clocks: predicted relativistic time gains, Science 177 (4044): 166-168.

21

c) Trajectory of a ray of light projected in a train car The typical example that explains the lack of sense of absolute simultaneity is the synchronized emission of two rays of light from two lanterns placed at the ends of a moving train car using synchronized clocks. A static observer, within the reference system S, with the X axis next to the railroad tracks and parallel to them and whose origin coincides with the midpoint of the wagon at the time of light emission, will notice the emission of the two rays at the same time from the two ends of the wagon. It will therefore conclude that the two blinks are simultaneous.

Another observer within the reference system S', whose axis X' is the longitudinal axis of the moving wagon, will first notice the emission of a ray of light followed by the emission of the second. It will conclude that the two emissions are not simultaneous. If the rays of light are projected onto the roof of the wagon and reflected in a mirror, the observer at the origin of S', who travels inside the wagon, will perceive that the reflected ray returns to the starting point; the observer placed next to the tracks, on the origin of S, will perceive that the point of departure of the ray and the arrival point, once reflected, are different.

In Figure M8, E1 and E2 are the emission points of the light ray and the return point, once reflected by the mirror. Both are on the abscissa x’ = 0, but in different ordinates. For the mobile observer, located in the reference system S’, that is to say inside the wagon, the emission and reflection of the ray will occur in the same place after a time interval between both. However, from the reference system S, occupied by the stationary observer close to the train tracks, the time interval will be perceived less than from S’ (Δct < Δct’), and also the emission and reflection events of the beam of light will occur at different locations (x1 < x2).

Figure M8 Figure M964 demonstrates the relevance of the concept of relative simultaneity.

In this Figure, three general cases are represented: 1. Two simultaneous events A and B, , t1 = t2, in two different places, x1 ≠ x2, in the reference system S; the same events occur at different times, t’1 ≠ t’2, and in different places, x’1 ≠ x’2, in the reference system of S’. Therefore, A and B happen simultaneously within S, but not within S’.

2. Two events C and D, simultaneous, t’3 = t’4, in the reference system S’ happen in different places, x’3 ≠ x’4; the same events occur at different times, t3 ≠ t4, and at different places, x3 ≠ x4, in the reference system S. 3. Finally, events E and F are not simultaneous either within S or S’.

64

Figures M8 and M9 have been taken from http://1.bp.bolgspot.com.

22

Figure M9 d) Temporal dilation and spatial contraction on the planet Mercury An observer located on a planet influenced by a gravitational force of the Sun, greater than the one supported on Earth, will see time pass slightly more slowly than on our planet. From the Earth, the measurement of the time used in a complete return of the planet Mercury, for example, will be higher than that obtained by an observer located on Mercury. Therefore, any speed observed from the planet closest to the Sun will be less than that observed from Earth. This difference in speeds is equivalent to a relative temporal dilation in Mercury.

Another way to focus the experiment is to compare the orbital paths. When it is finished a complete Mercury orbit from Earth, it will not be completed for the Mercury observer, which means that the space travel has been less than expected (the full turn) in this second observation. The empirical test will end for the observer of the Earth completed the turn, but from Mercury less than one turn will be computed, that is to say from the Earth it will be observed to rotate to Mercury more quickly. The discrepancy between the two observations is caused by the spatial contraction originated by the higher relative velocity of Mercury relative to Earth, caused by the greater gravitational force in the proximity of the Sun From the curved space of Riemann to the curved space-time of Minkowski In the mid-nineteenth century, the great mathematician Bernhard Riemann had elaborated the geometry of curved space65 thanks to which, said Einstein, the space was free of its rigidity and got to participate in physical events not as a simple witness "great masses struggling to bend the motion of other massesâ&#x20AC;Ś Spacetime guide the battle and itself participatesâ&#x20AC;?66.

In this new geometry Minkowski based himself to develop the multiple dimension spacetime, characterized by its irregularity in the small distances and its uniformity in the ordinary distances.

Einstein took the definitive step by associating Minkowski's space-time with the curved space of Riemann. For this he needed a quadratic form defined at any point of a curve that worked like the equation [M9] of flat spacetime.

65

The geometry of Riemann started from the geometry of Gauss and Lobachevsky. It satisfied all the axioms and postulates of the Euclidean geometry except the postulate of the parallel lines.

66 Wheeler, J.A. (1990) work cited, page. 2.

23

The equation searched by Einstein for the mathematical treatment of general relativity is of the quadratic form [A5.14] of Annex 5 Operators. Metric and quadratic forms of a tensor. Covariant and contravariant of Lorentz. Its gmn components are the coordinates of a tensor67 that establishes the universal metric that is verified for the general coordinates xm and xn of a curved spacetime of dimension mn. đ?&#x2018;&#x2018;đ?&#x2018; 2 = đ?&#x2018;&#x201D;đ?&#x2018;&#x161;đ?&#x2018;&#x203A; đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;&#x161; đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ?&#x2018;&#x203A; [A5.14]68 gmn is the metric that guarantees finding the space-time curvature independently of the coordinates that we use. It is the interval measurement pattern of the equation [A5.14], which expresses that the square of the difference of the interval of the position of a vector is an invariant. From its always positive sign, the existence of a tangent plane at each point of curved space-time is deduced. The tangent plane guarantees the constancy of the measurements in any reference system, which mathematically means that the covariant derivative of the metric is zero. đ?&#x203A;ťgmn (x) = 0 [A5.15] [A5.15] fully maintains the measurement capacity of the Euclidean geometry: length of a curve, area of a surface, angle formed between two curves and radius of curvature at any point.

The gmn component tensor defines the curved spacetime of Minkowski, so that it allows measurement along the ds lines, tangent to the curved surface. The interval Ď&#x201E; between two points A and B of a curve drawn on the surface is: đ??ľ

đ?&#x153;? = â&#x2C6;Ť đ?&#x2018;&#x2018;đ?&#x2018; đ??´

after replacing ds from [A5,14].

Covariant and contravariant of Lorentz 69 In the treatment of more complex magnitudes than scalars, such as vectors and tensors, you need to use a complex mathematical apparatus Specifically, it is necessary to resort to the use of the Lorentz covariant and contravariant tensor, which are vector elementary metrics that guarantee that the equations do not change the shape even though coordinate changes occur in the different frames of reference.

As has been repeated above, the essential function of the metric is to maintain the objective that two observers, installed in the reference frames S and S´, have absolute certainty that the measurements of their experiments are comparable and that the equations they handle define the same physical properties identically. This finality is met when the coordinates of S 'versus S are related through equation [A5.5] of Annex 5. đ?&#x153;&#x2022;đ?&#x2018;Ś đ?&#x2018;&#x161; đ?&#x153;&#x2022;đ?&#x2018;Ś đ?&#x2018;&#x203A; đ?&#x2018;&#x2021;đ?&#x2018;&#x161;đ?&#x2018;&#x203A; (đ?&#x2018;Ś) = â&#x2C6;&#x2018;đ?&#x2018;&#x;đ?&#x2018; đ?&#x153;&#x2022;đ?&#x2018;Ľ đ?&#x2018;&#x; đ?&#x153;&#x2022;đ?&#x2018;Ľ đ?&#x2018; đ?&#x2018;&#x2021;đ?&#x2018;&#x;đ?&#x2018; (đ?&#x2018;Ľ) [A5.5] [A5.5] is the contravariant transformation of a frame tensor of reference x and of dimension rs, in a tensor of reference y and of dimension mn. Reciprocally, the coordinates of S versus S´ are combined through the equation [A5.6].

67

The tensor is an operator that defines linear relationships between vectors, scalars and other tensors. In engineering, where it comes from, it is applied to calculate the mechanical stress in a material point on a continuous surface. Hence its name. Einstein used complex multidimensional operators to universalize calculations in all curved spacetime. The Annexes 5, Operators. Metric and quadratic forms of a tensor.

Covariant and contravariant of Lorentz, 6, Tensor Ricci and curvature and 7 Einstein field equations, present definitions and applications with tensors, essential for the development of the general theory of relativity.

68 Here the correlative numbering of equations [M] is interrupted again.

69 In Annex 5, the two Lorentz transformations have been developed widely and in detail in two frames of reference.

24

đ?&#x2018;&#x2021;đ?&#x2018;&#x161;đ?&#x2018;&#x203A; (đ?&#x2018;Ś) = â&#x2C6;&#x2018;đ?&#x2018;&#x;đ?&#x2018;

đ?&#x153;&#x2022;đ?&#x2018;Ľ đ?&#x2018;&#x; đ?&#x153;&#x2022;đ?&#x2018;Ľ đ?&#x2018;

đ?&#x2018;&#x2021; (đ?&#x2018;Ľ) đ?&#x153;&#x2022;đ?&#x2018;Ś đ?&#x2018;&#x161; đ?&#x153;&#x2022;đ?&#x2018;Ś đ?&#x2018;&#x203A; đ?&#x2018;&#x;đ?&#x2018;

[A5.6]

[A5.6] is the covariant transformation of the same tensor Tmn (y), inverse of [A5.5].

For the practical calculation, the contravariant transformations [A5.5] and covariant [A5.6] can be presented in matrix form. The components the metric gmn of [A5.14], valid for any reference system, are: â&#x2C6;&#x2019;đ?&#x2018;? 2 0 0 0 1 0 0) đ?&#x2018;&#x201D;đ?&#x2018;&#x161;đ?&#x2018;&#x203A; = ( 0 0 0 1 0 0 0 0 1 Its inverse is: â&#x2C6;&#x2019;1 0 0 0 đ?&#x2018;?2 đ?&#x2018;&#x201D;đ?&#x2018;&#x161;đ?&#x2018;&#x203A; = 0 1 0 0 0 0 1 0 ( 0 0 0 1) The coordinate transformations between the reference systems S and S' are given by the covariant Lorentz tensor. In its matrix form, equivalent to the equation [A5.6], it is given by Î&#x203A;. đ?&#x203A;ž â&#x2C6;&#x2019;đ?&#x203A;ž/đ?&#x2018;? 0 0 â&#x2C6;&#x2019;đ?&#x203A;žđ?&#x2018;? đ?&#x203A;ž 0 0 đ?&#x203A;Źđ?&#x2018;&#x161;â&#x20AC;˛ ) đ?&#x2018;&#x203A; =( 0 0 1 0 0 0 0 1 m' indicates the row and n column of the system S', and Îł is defined by: đ?&#x203A;ž=

1 2

â&#x2C6;&#x161;1â&#x2C6;&#x2019;đ?&#x2018;Ł2 đ?&#x2018;?

v is the linear displacement speed.

The transformation of gmn from system S to S' through the covariant tensor of Lorentz in its matrix form Î&#x203A; is: đ?&#x2018;&#x203A;â&#x20AC;˛ đ?&#x2018;&#x201D;đ?&#x2018;&#x161;đ?&#x2018;&#x203A; = đ?&#x2018;&#x201D;đ?&#x2018;&#x161;â&#x20AC;˛đ?&#x2018;&#x203A;â&#x20AC;˛ đ?&#x203A;Źđ?&#x2018;&#x161;â&#x20AC;˛ đ?&#x2018;&#x161; đ?&#x203A;Źđ?&#x2018;&#x203A; In a reciprocal way, the contravariant elements are transformed according to the following relationship: đ?&#x2018;&#x203A; đ?&#x2018;&#x201D;đ?&#x2018;&#x161;â&#x20AC;˛đ?&#x2018;&#x203A;â&#x20AC;˛ = đ?&#x2018;&#x201D;đ?&#x2018;&#x161;đ?&#x2018;&#x203A; đ?&#x203A;Źđ?&#x2018;&#x161; đ?&#x2018;&#x161;â&#x20AC;˛ đ?&#x203A;Źđ?&#x2018;&#x203A;â&#x20AC;˛ Finally, to change the coordinates between the reference systems S and S', we operate as follows: đ?&#x2018;Ą đ?&#x2018;Ąâ&#x20AC;˛ đ?&#x203A;ž â&#x2C6;&#x2019;đ?&#x203A;ž/đ?&#x2018;? 0 0 đ?&#x203A;žđ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x203A;žđ?&#x2018;Ľ/đ?&#x2018;? đ?&#x2018;Ľ đ?&#x2018;Ľâ&#x20AC;˛ â&#x2C6;&#x2019;đ?&#x203A;žđ?&#x2018;? đ?&#x203A;ž 0 0 đ?&#x203A;žđ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x203A;žđ?&#x2018;?đ?&#x2018;Ą đ?&#x2018;&#x161;â&#x20AC;˛ đ?&#x2018;&#x203A; ( ) = đ?&#x203A;Źđ?&#x2018;&#x203A; đ?&#x2018;Ľ = ( )( ) = ( ) đ?&#x2018;Ś đ?&#x2018;Śâ&#x20AC;˛ đ?&#x2018;Ś 0 0 1 0 đ?&#x2018;§ đ?&#x2018;§ 0 0 0 1 đ?&#x2018;§â&#x20AC;˛ The geometry of Riemann offers enormous advantages because of its global nature. It includes the three types of geometry that comply with the spatial isotropy and homogeneity required by the postulates of spacetime symmetry: Euclidean, hyperbolic and elliptical. But the most important utility it offers is as a starting point to develop the geometry of curved spacetime. This is what Einstein and Grossmann did, associating gravitational force with curvature70 for the first time.

70

Einstein, A, & Grossmann, M (1913) Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation, Zeitschrift fßr Mathematik und Physik, 62: 225-265.

25

Gravity as curvature of space-time The concept of gravitation as a space-time curvature closed the transitory period opened in 1905 by the theory of special relativity. With the presentation in the Prussian Academy of Sciences of the equations of general relativity, a ten-year stage ended in which Physics worked on standby, waiting to completely replace Newton's paradigm. In 1915, the spatiotemporal curvature upon which Einstein constructed the theory of general relativity replaced flat spacetime.

As stated above, the conservation of the speed of light caused in classical mechanics the change of invariant force for the momentum linear. For electromagnetism, the constancy of c represented the conservation of its laws with universal character. But until 1915, doubts remained about Newton's law of universal gravitation, enunciated according to the equation: đ?&#x2018;&#x161; đ?&#x2018;&#x161; đ??š = đ??ş đ?&#x2018;&#x;1 2 2 [M11]71 F is the force acting between two masses m1 and m2 separated by r. According to Newton's second principle, F causes an acceleration on the mass m2. But [M11] does not work for the mass of the photon m2 = 0. It can be reformulated like this: đ?&#x2018;&#x161; đ?&#x2018;&#x161; đ??š = đ?&#x2018;&#x161;đ?&#x2018;&#x2013; đ?&#x2018;&#x17D;; đ?&#x2018;&#x17D; = (đ??ş 21 ) 2 [M12] đ?&#x2018;&#x;

đ?&#x2018;&#x161;đ?&#x2018;&#x2013;

Doing m2 = mi in [M12], the acceleration results: đ?&#x2018;&#x161;1 đ?&#x2018;&#x17D; = (đ??ş 2 ) đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x2018;&#x161;2 = đ?&#x2018;&#x161;đ?&#x2018;&#x2013; đ?&#x2018;&#x; Newton himself experienced the measurement of the period of a pendulum built with different materials and found that this magnitude depended only on the pendulum length and on the acceleration of gravity, which meant that m2 = mi. Therefore, for Newton, as for Einstein, the trajectory in a gravitational field was independent of the nature of the mass. The principle of weak equivalence between inertial and gravitational mass was closed, so that the gravitational force was equal to the resistance force that the mass offered to an applied force.

The equality between gravitational and inertial mass led Einstein to the mental experiment of the elevator in free fall (see box below).

From the mental experiment it is concluded that the acceleration of the elevator is equal to the acceleration of the mass in relation to the Earth, đ?&#x2018;&#x17D; = đ?&#x2018;§â&#x20AC;˛Ě&#x2C6;, always what m2 = mi and that the gravitational field is uniform. This last assumption is not fulfilled, since the intensity of gravity is proportional to 1/ r2, according to [M12]. However, if one considers the fall of the elevator of short duration and that its size is small, the reference system xz of the box can be considered an inertial system, where the laws of special relativity would be fulfilled for an enclosure in free fall, without rotation and occupying a small region of spacetime.

The concept of inertial mass began to open the gap between Newton and Einstein: for the first, inertia was the invariance of movement of bodies; for the second, the dynamic interaction between the bodies. This conceptual variation was incorporated naturally in the relativistic description of the gravitational force, which was proposed as a manifestation of the curvature of space-time, curvature caused by the presence of matter.

The radical change that the curved space-time of Minkowski caused in the law of gravitation can be summarized in the following idea: matter is made of movement, which is the basis of space-time. In spacetime there are particles of different sizes. The photon, for example, generates an electromagnetic wave of pulsation ɡ, which moves the space71

From here, the correlative numbering of the series [M] is recovered.

26

time itself at the same time it generates it72, and the same movement forms a point of dynamic vacuum similar to an infinitesimal black hole73.

If an elevator rises at uniform speed, with an inert mass mi located at the z' height, at the moment it has travelled L (t), the equations of motion will be: z' = z - L(t) = z - vt; đ?&#x2018;&#x17D; = đ?&#x2018;§â&#x20AC;˛Ě&#x2C6; = đ?&#x2018;§Ě&#x2C6; .

The force to which a mass is subjected mi inside the elevator is đ??š = đ?&#x2018;&#x161;đ?&#x2018;&#x2013; đ?&#x2018;§Ě&#x2C6; â&#x20AC;˛ = đ?&#x2018;&#x161;đ?&#x2018;&#x2013; đ?&#x2018;§Ě&#x2C6; z

z'

L(t) x

In free fall with acceleration of gravity g, the equation of motion is z' = z - L(t) = z - 1/2g t2; đ?&#x2018;&#x17D; = đ?&#x2018;§â&#x20AC;˛Ě&#x2C6; = đ?&#x2018;§Ě&#x2C6; â&#x2C6;&#x2019; đ?&#x2018;&#x201D; As đ?&#x2018;§Ě&#x2C6; = 0, the force that the mass m2 suffers then results: đ??š = đ?&#x2018;&#x161;2 đ?&#x2018;§â&#x20AC;˛Ě&#x2C6; = â&#x2C6;&#x2019;đ?&#x2018;&#x161;2 đ?&#x2018;&#x201D; â&#x2020;&#x2019; m2 = mi Esta masa m es la masa gravitacional.

The enormous range of particles present in space-time is characterized by its pulsation ɡ, which determines the diverse spatial-temporal scales existing in the Universe. But in turn, each scale is fixed by its density, on which depends the relative correspondence of space and time with c. Thus, different ɡ and density belongs to each particle, as well as a different way of measuring space-time. The measurement depends on the size and pulsation of the particle that generates the scale (star, black hole, atom, etc.). It also depends on the space-time density, which varies significantly between the core of a planet and its surface.

Two examples can clarify how space and time correspond and how space-time density evolves to maintain correspondence: â&#x20AC;˘ â&#x20AC;˘

An accelerated spacecraft increases its size, since the spatial expansion induced by the acceleration causes a decrease in the temporal density, that is, the clock of the spacecraft is slower.

A black hole generated by "tremendous mass clashes. It crunches the space in the [its] interior ", and its extreme density stops the light and puts an end to time74.

Recent verifications of the principle of weak equivalence The latest verifications of the equivalence between the inertial and gravitational mass have been made by observations in our own solar system. Periodically, a group of New Mexico astronomers has launched pulses of light from a powerful laser source towards the reflectors installed on the Moon since 1969 by the Apollo 11, 14 and 15 missions, as 72

Einstein said that light was a self-sufficient oscillation of the electric and magnetic fields.

The existence of black holes in spacetime presents the problem of its discontinuity, which breaks the postulate of homogeneity symmetry.

73

74

Wheeler, J.A. (1990) work cited, page 2.

27

well as by a couple of Russian missions. The objective was to check if there were orbital anomalies due to differences in the acceleration between the Earth and the Moon with respect to the Sun. In particular, the series of experiments tried to find out whether the theory of a fifth force of nature75 of gravitational origin broke the principle of weak equivalence. The measurements have been accurate enough to confirm the equality between the gravitational and inertial mass, with a precision of 10-13. To date, there has been no indication that gravitational energy contributes to the gravitational mass and not to the inertial mass and, therefore, there is no difference between the acceleration of the Earth and the Moon related to the Sun. Matter tells Spacetime how to curve, and Spacetime tells matter how to move John Archibald Wheeler76 4. THEORY OF GENERAL RELATIVITY General relativity broke the status quo maintained by special relativity with Newton's law of gravitation. Its appearance supposed a radical change of the principle of equivalence, which jumped from the exegesis of weak equivalence to strong equivalence.

Until 1905, flat spacetime had been sufficient to explain special relativity; Einstein's acceptance of curved space-time occurred between 1908 and 1915. From this date, the principle of strong equivalence prevailed, and on a deep similarity between the force of gravitation and the curvature of space-time settled the new geometry of general relativity.

Wheeler explained the process of how the distance effect of gravity provided a local curvature: " mass there bends local spacetime there. This faraway spacetime forces a slightly lesser bending on the local spacetime around it -even though this space is free of mass. This curvature, still faraway, in turns imparts a curvature -a still smaller one-to the local space still farther out from the mass. And so on, all the way out to the here"77.

The concept of strong equivalence would finish strengthening the theory of general relativity, whose notion of gravity is no longer expressed as the interaction of bodies through a law of forces, but as a direct effect of the space-time curvature. This idea culminated in the original thought that the main orders of the movement of objects originated from spacetime itself and from nobody else.

The new interpretation of general relativity ditched the anomaly unresolved by the Newtonian paradigm of the deviation of the light produced by gravitation. Empirically, the effect of gravity on the light emitted from the Sun was proved by the decrease in the frequency of the radiant energy coming from our star. Frequency descent was checked by the relative displacement of sunlight to red78 with respect to that arising from interstellar space. Thus, the conclusive empirical test of the curvature of light projected by the Sun 75

New theories about gravity pointed to the principle of equivalence had to break. The existence of a fifth force, opposite to gravity and that only acted at short distances, was based on the lack of precision of the measurements, but all the experiments have confirmed to date that the gravitational and inertial mass are equivalent.

76 Wheeler, J.A., and Kenneth, F. (2000) Geons, Black Holes, and Quantum Foam: A Life in Physics, W.W. Norton & Company, page 235.

77 Wheeler, J.A. (1990) Work cited, page 68.

78 The shift of sunlight to red was demonstrated empirically by Pound and Rebka in 1960 and by NASA's artificial satellite, Gravity Prove B, launched into space in 2004, whose high-precision gyroscopes it transported measured the gravitational distortions caused by Earth.

28

was the displacement of its frequency towards red, a conclusive phenomenon that showed the validity of the paradigm of general relativity.

The other anomaly solved by the new paradigm was mismatch in the calculation of Mercury's perihelion. Although Newton predicted that his orbit was a non-closed ellipse79, the equations of the movement of the planet closest to the Sun started from Kepler's third law, which formulated the constancy between the square of the period of revolution of the orbit and the cube of the semimajor axis. of the ellipse. Obviously, the law does not recognize the incidence of the curvature of space, which caused an anomaly in the calculation of the precession of the perihelion of Mercury. Already in the Introduction of his paper80, Einstein boasted one of the most valued tests for the validity of his theory of general relativity: the ability to correctly estimate the precession of Mercury's perihelion. In effect, applying the equations of general relativity, the perihelion rotational value of 43Ęş of an angle per century81 was obtained, equivalent to more than 300,000 years for the perihelion to end a complete revolution.

Einstein field equations As has been said above, the covariant concept is necessary to use in order to all physical laws have the same form in any reference system. These laws have to be expressed through tensors, because they are the invariant when the Lorentz transformations are applied. Therefore, to construct the equations of the gravitational field it is necessary to find the covariant form of the tensor source of the field. Since the source of the gravitational field is the way in which matter manifests in spacetime, the tensors considered are the energy tensor and the momentum tensor. So that the equations with which Einstein formalized the theory of general relativity tried to relate the curvature of space-time with the energy tensor and the momentum tensor.

As a precondition, Einstein's field equations had to satisfy the following requirements: â&#x20AC;˘ Pass the experimental tests. â&#x20AC;˘ Reduce Newton's theory of gravity for the particular case of weak gravitational fields and low speeds. â&#x20AC;˘ Staying universal for any reference system. â&#x20AC;˘ Result consistent with the conservation of the energy momentum.

Gravity is the curvature of local spacetime, according to general relativity; gravity is the action at a distance of a force, according to Newton. Both perceptions are complementary.

From the Poisson equation [A7.2] of Annex 7 Einstein's field equations is deduced that Newtonian gravity ceases to be consistent with special relativity. In effect, [A7.2] does not depend on time. Therefore, the potential Ď&#x2022; responds instantaneously to any variation of the density of the gravitational mass Ď . In this way, the relativistic principle that no physical signal can propagate at a speed greater than c is broken. â&#x2C6;&#x2021;2 â&#x2C6;&#x2026; = â&#x2C6;&#x2019;4đ?&#x153;&#x2039;đ??şđ?&#x153;&#x152;(đ?&#x2018;Ľ) [A7.2] 82

79

The orbit of the planet Mercury is not a closed ellipse. In each orbit of the planet it rotates, so that the point closest to the Sun (perihelion) moves an angle in the precession movement.

80 Einstein, A. (1917). Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie. Sitzung der physikalisch-matdmatischen Klasse vom 8: 142-157.

81 The incidence on the calculation of orbitals is minimal for our solar system. The 43" of angle per century of displacement of the perihelion of Mercury calculated by Einstein are practically the difference between the experimental measurements and the calculation obtained from Newton's equations.

82 Here the correlative numbering of equations [M] is interrupted again.

29

On the other hand, the density Ď is not a scalar invariant but a vector, which means that its magnitude depends on the reference system, depending on the relative speed that the observer maintains with the observed.

Einstein's field equations try to relate the concepts of force and curvature, so that force is the manifestation of the curvature of space-time. They are expressed through the formalism of tensors. The reason for using such an uncomfortable mathematical device is that the handling of the curvature of a four-dimensional space is very complicated. For example, in an orbital motion around the Earth, the gravitational field caused by the space-time curvature manifests itself through accelerations in a different direction. These complex effects require complex operators.

Annexes 5 Operators. Metric and quadratic forms of a tensor. Covariant and contravariant of Lorentz, 6 Ricci tensor and curvature and 7 Einstein field equations present, in detail, the conceptual and mathematical formalization until finally arriving at the equation of the Einstein field [A7.14] non-linear of Annex 7.

8đ?&#x153;&#x2039;đ??ş đ?&#x2018;&#x2026;đ?&#x153;&#x2021;đ?&#x2018;Ł â&#x2C6;&#x2019; 1â &#x201E;2 đ?&#x2018;&#x201D;đ?&#x153;&#x2021;đ?&#x2018;Ł đ?&#x2018;&#x2026; + đ?&#x2018;&#x201D;đ?&#x153;&#x2021;đ?&#x2018;Ł đ?&#x203A;Ź = đ?&#x2018;? 4 đ?&#x2018;&#x2021;đ?&#x153;&#x2021;đ?&#x2018;Ł [A7.14] The first member of [A7.14] refers to the curvature of spacetime; the second, to the mass and energy; Âľ and Ę&#x2039; are the dimensions of space time; RÂľĘ&#x2039;, Ricci curvature tensor; gÂľĘ&#x2039;, the tensor metric; R, the curvature scalar; Î&#x203A;, the cosmological constant and TÂľĘ&#x2039;, the forceenergy-momentum tensor, and the sum of the two first terms of the first member is the so-called Einstein tensor.

[A7.14] expresses, read from left to right, the control of spacetime (first member of the equation) on motion (second member of the equation). But it also describes, read from right to left, that matter and energy (second member of the equation) curves spacetime (first member of the equation).

The cosmological constant Î&#x203A; was introduced by Einstein to compensate for the effect of the cosmological space. Its insertion modified the original equation for a static universe, which had only two summands in the first member. Einstein added the third summand to show up the cosmic acceleration. In 1929, Hubble experimentally discovered the expansion of the universe by the redshift of external galaxies; then the need for Î&#x203A; disappeared. However, in recent years it was observed that the acceleration of the universe was growing and some physicists resurrected the cosmological constant. But the doubt of including or not the term Î&#x203A; in [A7.14], that persecuted to Einstein during good part of its life, has subsisted in the time. In any case, the curvature of Î&#x203A; is very small and tends to zero as the observable radius of the Universe increases83.

General relativity and principle of strong equivalence The theory of general relativity represented a radical change in Einstein's analysis, which replaced the reference system of flat inertial space-time of special relativity by curved space-time deformed by gravity. With this change, general relativity was linked to the most universal reference systems, which admitted all kinds of movements: produced by accelerations caused by variations of the module, or of the speed direction of the observed object, or of the rotation of the reference system itself.

From this new scheme, an observer falling in a gravitational field was identical to an observer outside of gravity. Einstein himself declared that the â&#x20AC;&#x153;greatest idea of my lifeâ&#x20AC;? was that, in a gravitational field, things behave â&#x20AC;&#x153;as they do in a space free of

Interpreted as density, Î&#x203A; cannot be more than two or three times the present average density of our universe â&#x2030;&#x2C6; 10-27 kg / m3. Data obtained from Penrose, R. (2006) work cited, page. 1012.

83

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gravitationâ&#x20AC;?84. He saw the floating as the natural state of the movement, from where the simplicity of space-time, which seemed so flat far away from the planets, the Sun or the galaxy, was best captured. This intuition of free floating led him to the theory of general relativity85 and to rectify Newton's idea that the origin of gravity was at the centre of the Earth.

General relativity maintains an embryonic relationship, in its origin, with the principle of equivalence between the inertial mass and the gravitational mass. But his bet goes further than the classics and proposes a principle of strong equivalence, very evolved with respect to Newton's equivalence principle. In fact, it confirms with more emphasis than the classics that gravity and acceleration are the same thing, at the same time that it disagrees with the defining Newtonian equations of inertial and gravitational forces. Einstein uses the equality between inertial mass and gravitational mass to offer an absolutely original approach: he announces that with measurements made within any reference system the observer cannot distinguish whether he is under the action of a gravitational field or a uniform acceleration generated by another force. This transcendental enunciated is known as the principle of strong equivalence and is the basis for the description of gravity as curvature in space-time.

The empirical stock of equality between the inertial mass and the gravitational mass is abundant. Centuries of astronomical observation and experiments in elementary physics and basic practices of daily life offer a continuum of confirmations of the principle of equivalence. An astronaut floating inside a space station or inside a spacecraft, or landing with the same acceleration as the spacecraft without hitting the ground or the walls, is undoubted proof of the equality of its inertial mass and its gravitational mass. In all the cases of the example, the astronaut, the space station and the spacecraft support the same acceleration, in spite of the fact that the station and the spacecraft are constructed of materials different from the astronaut's body.

Gravitational force as an expression of the curvature of space-time The step added to the equality of mass inertia and the gravitational mass is given by the general theory of relativity, which drastically changes the concept of gravity until concluding that it is an attribute of the geometry of space-time. The approach of gravity as curvature of space-time automatically explains two phenomena contrasted experimentally: the effect of curved light in the proximity of a gravitational field 86 and the geodesic line drawn by a body in free fall.

Among all the lines of space that connect an initial position with another final one, the geodesic is the minimum length interval. And among all the temporal lines that connect an initial event with a subsequent one, the geodesic is the one for which the elapsed time proper87 is maximum. It is also the trajectory that all ships and planes follow in their displacements by sea and air and that which traces the curved matter of space-time influenced by gravity, that is, induced by gravitational waves propagated at the speed of light.

84

The idea of Einstein that in freefall cannot detect any effect of the gravity of the environment is collected in Wheeler, J.A. (1990) work cited, page. 11.

85 The name of general relativity replaced the primitive idea that gravitation was actually free floating. IbĂ­d. page. 11.

86 For a photon of zero mass, the gravitational force calculated according to Newton's equation [M11] is zero, which leaves the phenomenon of the curvature of light without explanation.

87 Proper time is the total time from the beginning to the end through all the geodesic segments. It takes place in a reference system installed in the same centre of the observed phenomenon, as if the particle moving through the geodesic would carry its own clock.

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The geodetic trajectories represented the clear proof of the necessary change of the principle of weak equivalence. Einstein exhibited continuous ingenious demonstrations throughout his life that showed the depth of the concept of strong equivalence: the elevator experiment in descent, where the weight loss increased with its acceleration up to the limit of reaching zero weight if the elevator cable was broken; the impossibility of discerning whether the parabola traced by a falling ball inside an accelerated elevator (more open as the elevator accelerated) was due to gravity or the acceleration of the elevator, etc.

In the experiment imagined by Einstein of an elevator in free fall, a ray of light penetrated by a lateral hole and hit the opposite wall in a point slightly superior to the one of entrance.

An observer outside the elevator would see a soft upward curve; in uniform motion, the ray trajectory would be straight.

All these demonstrations have consolidated the following principle of equivalence: Acceleration ≡ Gravity The principle of equivalence of general relativity took hold by the identity of acceleration and gravity and in the empirical test that it deformed space-time. Wheeler took the three physical units involved (gravity, curvature and spacetime) and fitted them perfectly into the new concept: “Gravity is not a foreign and physical forced transmitted through space. It is instead a manifestation of curvature of that four-dimensional union of space and time that we call spacetime"88.

The manifestation of the curvature is progressive and in it the element that affects and is affected by gravity intervenes: the mass. The mass is felt in the curvature that imposes on spacetime, but the curvature89 at the same time is discovered in the increase of the inclination of two nearby geodesics, i.e. in their relative movement. The manifestation of the curvature is progressive and in it intervenes the angles of curvature. Between two close geodesics, the momentum and energy contained in a space-time differential element is zero, but the sum of the angles of curvature associated with the differential element is not much less zero. And this addition of angles is the curvature or rotation of the direction of a geodesic in relation to another geodesic near and almost parallel90, i.e. gravity.

Therefore, the key to the action of mass over spacetime is not the rotation or curvature, but the momentum of curvature taking as centre of rotation the centre of the space-time differential element. This momentum is the resultant of the products of the open curvatures between each pair of close geodesics by the distance they maintain with the center of rotation. That is, it is the integral of the momentum extended to all the spacetime differential element.

Equation [A7.13] of Annex 7 Einstein's field equations, which includes the Einstein field equations expresses that the linear momentum or amount of motion equals the sum of the momentums of rotation by the constant 8π.

Gμν = 8πGTμν [A7.13] [A7.13] is the clearest image of gravity. Read from left to right, it indicates that the mass controls the space-time and that the curved spacetime with greater mass density contracts the adjacent spacetime of lower density and curves it. In this way, a region of vacuum, where the linear momentum, the energy and the momentum of rotation are zero, ends up being curved by the effect of the curved adjacent spacetime. Read from right to left, [A7.13] tells us that the momentum of rotation governs the control of space-time over the 88

Wheeler, J.A. (1990) work cited, page XI. The curvature is positive when the nearby geodesics curve toward each other; it is negative when the two geodesics move away from each other; is zero when the relative velocity of both is zero.

90 Wheeler, J.A, (1990) work cited, page 116.

89

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mass. From both readings it follows therefore that the control of the mass over spacetime has the counterweight of space-time control over the mass. In brief, gravity works through double control of mass over spacetime and spacetime over mass.

Gravity is a manifestation of the curvature of the environment, which is proportional to the mass, but decays in inverse proportion to the cube of the distance. However, the Moon causes on Earth the effect of the tides away from it about 400,000 kilometers If space-time were not curved, Wheeler continues: "Every object in free float would move in a straight line with uniform velocity forever and ever, [but then] the Earth and the other planets would not enjoy the companionship of the Sun"91.

The trajectory of the Earth is not straight nor its movement is uniform away from the Sun. On the contrary, the Sun keeps the Earth in its orbit, although it is space-time away from Earth, with little curvature then, who acts on it. Wheeler argues that this is why Earth takes 365 days to complete its orbit.

Anomalies of Newton's three laws solved by the relativistic paradigm Einstein did not consider Physics to be based on the concept of the Newtonian continuous field. Their conceptual revolution of the new idea of the gravity reached the principles of Newton, considered by the relativist paradigm as particular laws that worked only for spaces at rest or at low speeds. In Newton's three laws anomalies were discovered.

Law of inertia Newton's first law, also known as the law of inertia, said that if a body did not act any force it would remain indefinitely at a constant speed in rectilinear motion. Newton was referring to an inert mass92, which resisted to be put in motion in a state of rest, or to change the magnitude or direction of its speed in the state of motion. The relativistic theory reduced the fulfilment of this law to the concrete case of a flat space, where the particular trajectory of the straight line coincided with the universal trajectory of the geodesic93 in curved spacetime.

Fundamental principle of the dynamics The second law, known as the fundamental principle of dynamics, has ended up included by relativistic mechanics. Given the mass m and the acceleration a measured in Galileo's inertial system, Newton defined the force as đ??š = đ?&#x2018;&#x161;đ?&#x2018;&#x17D;. Einstein replaced it with the equation [M13], where v is the velocity and m0, the rest mass. đ??š=

đ?&#x2018;&#x161;0 đ?&#x2018;&#x17D;â&#x192;&#x2018; 2 â&#x2C6;&#x161;1â&#x2C6;&#x2019;đ?&#x2018;Ł2 đ?&#x2018;?

+

â&#x192;&#x2018; â&#x2C6;&#x2122;đ?&#x2018;&#x17D;â&#x192;&#x2018;)đ?&#x2018;Ł â&#x192;&#x2018; đ?&#x2018;&#x161;0 (đ?&#x2018;Ł 3 đ?&#x2018;Ł2 đ?&#x2018;? 2 â&#x2C6;&#x161;[1â&#x2C6;&#x2019; 2 ] đ?&#x2018;?

[M13]94

â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018; , and operating according to the [M13] is reached starting from the equality đ??š â&#x2C6;&#x2020;đ?&#x2018;Ą = đ?&#x2018;&#x161;â&#x2C6;&#x2020;đ?&#x2018;Ł attached box.

Whether the trajectory is rectilinear or circular uniform, the direction of movement is parallel to the acceleration. In the first case, the first member of equation [M13] is 91

IbĂ­d., page. 12.

With the replacement of Newtonian movement theory by relativistic theory, the terms inert mass and gravitational mass fell into disuse.

93 The straight line is nothing but a geodesic of zero curvature.

94 From here, the correlative numbering of the series [M] is recovered.

92

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annulled; therefore, the direction of đ??š is the direction of đ?&#x2018;Ł, which is the direction of đ?&#x2018;&#x17D;. In the second case, the second member of [M13] is equal to zero. Therefore, the direction of đ??š is the direction of đ?&#x2018;&#x17D;. When v << c, the equation [M13] is reduced to the principle of Newton, đ??š = đ?&#x2018;&#x161;đ?&#x2018;&#x17D; .

The fundamental principle of Newtonian dynamics starts from the equation đ??š â&#x2C6;&#x2020;đ?&#x2018;Ą = â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018; , whose first member is the impulse and the second the amount of motion. By đ?&#x2018;&#x161;â&#x2C6;&#x2020;đ?&#x2018;Ł deriving both members with respect to t, the fundamental principle of the dynamics is reached đ??š = đ?&#x2018;&#x161;đ?&#x2018;&#x17D; , where the mass m is treated as invariant, and therefore constant, when deriving the amount of movement. The relativist paradigm replaces mass by đ?&#x2018;&#x161;0 đ?&#x2018;&#x161;= 2 â&#x2C6;&#x161;1 â&#x2C6;&#x2019; đ?&#x2018;Ł2 đ?&#x2018;? and part of the equation: đ?&#x2018;&#x161;0 â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018;â&#x192;&#x2018; đ??š â&#x2C6;&#x2020;đ?&#x2018;Ą = â&#x2C6;&#x2020;đ?&#x2018;Ł [M14] 2 â&#x2C6;&#x161;1â&#x2C6;&#x2019;đ?&#x2018;Ł2 đ?&#x2018;?

The key to the universality of equation [M13] is the invariance of c. It is reached by deriving [M14] from t and keeping m0 and c constant.

[M13] is the general expression of the fundamental principle of the dynamics when v and F are not parallel. The first member is the perpendicular component of the force with respect to the movement; if the force is orthogonal to the velocity, the velocity is also orthogonal to the acceleration; therefore, the scalar product đ?&#x2018;Ł â&#x2C6;&#x2122; đ?&#x2018;&#x17D; = 0. The second member is the parallel component of the force, since it maintains the direction of the velocity.

In conclusion, the second law of Newtonian mechanics is but a particular case included in relativistic mechanics.

Principle of action and reaction The empirical anomaly discovered on the third law of Newton, or principle of action and reaction, represented in its moment the definitive confirmation to the relativistic physics.

Their enunciated says that if a force đ??š12 acts on an object, another force đ??š21 appears, of equal magnitude and direction but opposite to the first direction. Equality đ??š12 = â&#x2C6;&#x2019;đ??š21 is then fulfilled.

The anomaly occurs in the field of electromagnetism. In particular, the Lorentz equation95 â&#x192;&#x2018;1 đ??š12 = đ?&#x2018;&#x17E;2 đ?&#x2018;Ł2 Ă&#x2014; đ??ľ [M15] determines the value of a magnetic force F12 acting on an electric charge q2 moving at the velocity v2 within a magnetic field B1.

Substituting in [M15] B1 for the Biot Savart law of a magnetic field, â&#x192;&#x2018; 1 = đ?&#x153;&#x2021;đ?&#x2018;&#x17E;1 (đ?&#x2018;Łâ&#x192;&#x2018;1 Ă&#x2014;đ?&#x2018;˘2â&#x192;&#x2018; 12 ) đ??ľ [M16] 4đ?&#x153;&#x2039;

đ?&#x2018;&#x2018;

is obtained: đ??š12 =

â&#x192;&#x2018; 2 Ă&#x2014;(đ?&#x2018;Ł â&#x192;&#x2018; 1 Ă&#x2014;đ?&#x2018;˘ â&#x192;&#x2018; 12 ) đ?&#x153;&#x2021;đ?&#x2018;&#x17E;2 đ?&#x2018;&#x17E;1 đ?&#x2018;Ł 4đ?&#x153;&#x2039;

95

đ?&#x2018;&#x2018;2

[M17.1]

Hendrik Antoon Lorentz, Nobel Prize in Physics of 1902, is the same scientist who proposed the Lorentz transformations.

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q1 is the electric charge generating the magnetic field in its environment, Îź the magnetic permeability, d the distance separating the two charges and đ?&#x2018;˘ â&#x192;&#x2018; 12 the unit vector that goes from q1 to q2.

Analogously, the force caused by the electric charge q2 on q1 is: â&#x192;&#x2018; 21 )) â&#x192;&#x2018; 2 = đ?&#x153;&#x2021;đ?&#x2018;&#x17E;2 đ?&#x2018;&#x17E;1 đ?&#x2018;Łâ&#x192;&#x2018;1 Ă&#x2014;(đ?&#x2018;Łâ&#x192;&#x2018;2 Ă&#x2014;(â&#x2C6;&#x2019;đ?&#x2018;˘ đ??š21 = đ?&#x2018;&#x17E;1 đ?&#x2018;Ł1 Ă&#x2014; đ??ľ [M17.2] 2 4đ?&#x153;&#x2039;

đ?&#x2018;&#x2018;

B2 is the magnetic field caused by the electric charge q2 and đ?&#x2018;˘ â&#x192;&#x2018; 21 the unit vector that goes from q2 to q1.

Comparing the expressions [M17.1] and [M17.2], it is shown that they are neither of equal direction nor of equal magnitude, as Newton's third law establishes.

Solving the double vector product of equation [M17.1], we obtain: đ?&#x2018;Ł2 Ă&#x2014; (đ?&#x2018;Ł1 Ă&#x2014; đ?&#x2018;˘ â&#x192;&#x2018; 12 ) = (đ?&#x2018;Ł2 â&#x2C6;&#x2122; đ?&#x2018;˘ â&#x192;&#x2018; 12 )đ?&#x2018;Ł1 â&#x2C6;&#x2019; (đ?&#x2018;Ł2 â&#x2C6;&#x2122; đ?&#x2018;Ł1 )đ?&#x2018;˘ â&#x192;&#x2018; 12 As the scalar product đ?&#x2018;Ł2 â&#x2C6;&#x2122; đ?&#x2018;Ł1 is annulled for being both orthogonal vectors, đ??š12 will be in the plane formed by the vectors đ?&#x2018;Ł1 and đ?&#x2018;˘ â&#x192;&#x2018; 12 .

We solve in the same way the double vector product of [M17.2]. đ?&#x2018;Ł1 Ă&#x2014; (đ?&#x2018;Ł2 Ă&#x2014; (â&#x2C6;&#x2019;đ?&#x2018;˘ â&#x192;&#x2018; 21 )) = (đ?&#x2018;Ł1 â&#x2C6;&#x2122; (â&#x2C6;&#x2019;đ?&#x2018;˘ â&#x192;&#x2018; 21 )đ?&#x2018;Ł2 â&#x2C6;&#x2019; (đ?&#x2018;Ł1 â&#x2C6;&#x2122; đ?&#x2018;Ł2 )(â&#x2C6;&#x2019;đ?&#x2018;˘ â&#x192;&#x2018; 21 ) Therefore, đ??š21 is in the plane formed by the vectorsđ?&#x2018;Ł2 y đ?&#x2018;˘ â&#x192;&#x2018; 21 .

In conclusion, in general đ??š12 and đ??š21 are not coplanar and |đ??š12 | â&#x2030; |đ??š21 |, which implies a noncompliance of Newton's third law and accredits the substitution of the paradigm of classical mechanics for the relativistic.

5. Bibliography Albert Einstein on his 70th birthday (2004) APS News January Vol 13 nÂş 11.

DurĂĄn, J.A. (2006) Isaac Newton & Gottfrield Wilheim Leibniz, La polĂŠmica sobre la invenciĂłn del cĂĄlculo infinitesimal, CrĂ­tica.

Einstein, A. (1905) Zur Elektrodynamik bewegter KĂśrper, Annallen der Physik, Vol. 372, 10: 891-921.

Einstein, A. & Grossmann, M (1913) Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation, Zeitschrift fßr Mathematik und Physik, 62: 225-265.

Einstein, A. (1917). Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie.

Sitzung der physikalisch-matematischen Klasse vom 8: 142-157.

Einstein, A. (1955) Relativistic theory of the non-symetric field, Apendix II of The Meaning of Relativity, 5ÂŞediciĂłn, Princeton University Press, Princeton.

Hafele J.C., and Keating R.E. (1972) Around the world atomic clocks: predicted relativistic time gains, Science 177 (4044): 166-168.

Harari, Y. N. (2015) Homo Deus: A Brief History of Tomorrow, Penguin Random House.

Hawking, S. W. (2005) The theory of everything. The origin and fate of the Universe, Phoenix Book. http://electric1.es/cm/lineasdefuerza.html. http://1.bp.bolgspot.com. http://2.bolgspot.com. https://commons.wikimedia.org/wiki. https://upload.wikimedia.org/wikipedia/commons/b/be. https://wikivisually.com/wiki/Minkowski_space.

MartĂ­n, A., blogspot la teorĂ­a de la relatividad, https://es.scribd.com/doc/256309409/LaTeoria-de-La-Relatividad-Armando-Martinez-Tellez.

Newton, I. (1968) The Mathematical Principles of Natural Philosophy, Dawson of Pall Mall.

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Penrose, R. (2006) The road to reality, eight printing, Publisher Alfred A. Knopf.

Robinson, A. (2010) Einstein. Cien aĂąos de relatividad, Blume.

Trevelyan, R.C. (1920) Translation from Lucretius, London Allen & Unwin.

Wheeler, J.A. (1990) A journey into gravity and spacetime, Scientific American Library.

Wheeler, J.A., with Kenneth, F. (2000) Geons, Black Holes, and Quantum Foam: A Life in Physics, W.W. Norton & Company.

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