The Fourth Dimension: From its spatial nature in Euclidean geometry to a time-like component of non-Euclidean manifolds
aa r X i v : . [ phy s i c s . h i s t - ph ] F e b A quarta dimens˜ao: da sua naturezaespacial na geometria euclidiana `acomponente tipo tempo de variedades n˜aoeuclidianasThe Fourth Dimension: From its spatialnature in Euclidean geometry to a time-likecomponent of non-Euclidean manifolds
Jos´e Maria Filardo Bassalo; Francisco Caruso; , Vitor Oguri Academia Paraense de Ciˆencias & Funda¸c˜ao Minerva, Bel´em, Par´a, Brasil. Centro Brasileiro de Pesquisas F´ısicas, Coordena¸c˜ao de F´ısica de Altas Energias,22290-180, Rio de Janeiro, RJ, Brasil. Universidade do Estado do Rio de Janeiro, Instituto de F´ısica Armando Dias Tavares,20550-900, Rio de Janeiro, RJ, Brasil.
Resumo
Neste artigo, apresenta-se a evolu¸c˜ao das ideias sobre a quarta dimens˜ao espacial, partindodaquelas que surgem da geometria euclidiana cl´assica e abordando, em seguida, as que resul-tam do ˆambito das geometrias n˜ao euclidianas, como as de Riemann e Minkowski. Particularaten¸c˜ao ´e dada ao momento no qual o tempo real passa efetivamente a ser considerado comouma quarta dimens˜ao, conforme introduzido por Einstein.
Palavras-chave:
Quarta Dimens˜ao; Espa¸co; Geometrias n˜ao Euclidianas; Hist´oria da Geo-metria; Hist´oria da F´ısica.
Abstract
In this article, the evolution of the ideas about the fourth spatial dimension is presented,starting from those which come out within classical Euclidean geometry and going throughthose arose in the framework of non-Euclidean geometries, like those of Riemann and Min-kowski. Particular attention is given to the moment when real time is effectively consideredas a fourth dimension, as introduced by Einstein.
Keywords:
Fourth Dimension; Space; non-Euclidian Geometry; History of Geometry; His-tory of Physics.
The beginning of the history: not more than three
The first step in the history of the fourth dimension was actually an attempt to deny itsexistence. Indeed, the impossibility of a fourth dimension was actually sustained by Aristotleof Stagira (384-322 b.C.). In effect, in his
De Caelo , which consists of Four Books, he treatedthis impossibility just right in the first paragraph of Book 1, saying, in summary, that: “A magnitude if divisible one way is a line, if two ways a surface, and if three a body.Beyond these there is no other magnitude, because the three dimensions are all that thereare, and that which is divisible in three directions is divisible in all” [1].
In this same paragraph, the Stagirite continues giving a cosmological justification of thisnumber three by appealing to its divinization, sustained by the Pythagoreans. Quoting him, “For, as the Pythagoreans say, the universe and all that is in it is determined by thenumber three, since beginning and middle and end give the number of the universe, andthe number they give is a triad. And so, having taken these three from nature as (so tospeak) laws of it, we make further use of the number three in the worship of the Gods”.
Such kind of identification between the tri-dimensionality of space and God’s will is re-current in the history of science. Johannes Kepler (1571-1630), for example, asseverated thatthree is exactly the number of dimensions due to the Holly Trinity [2, 3].The second necessary (but not sufficient) step toward the conception of the fourth dimen-sion has to do with the systematization of the geometric knowledge in Ancient Greece.The Greek mathematician Euclid of Alexandria (c. 323-285 b.C.) wrote his famous treatiseon Geometry,
The Elements , which has influenced the development of Western Mathematicsfor more than 2 000 years. This classical work contains Thirteen Books. From Book 1 to 10just Plane Geometry is considered. In Book 1, Euclid presents 23 definitions including thoseof point, line and surface, as following: 1) “A point is that which has no part”; 2) “A line isbreadthless length”; (...); 5) “A surface is that which has length and breadth only” [4].The third dimension is treated only in the last three Books, when a solid is defined in thefollowing way: “A solid is that which has length, breadth and depth” [5].Throughout the entire compendium, Euclid limits himself to treat Linear, Plane andSpatial Geometry, and he did not even consider the possibility of a fourth dimension.Some centuries later, the Greek astronomer Claudius Ptolemy (85-165), in his (lost) book
On Distance , published in 150 a.C., gave a “proof” about the impossibility of the fourthdimension, based on the very fact that it is impossible to draw a fourth line perpendicular tothree mutually perpendicular lines. This is indeed not a proof, but rather reinforce that weare not able to visualize the fourth dimension from which one cannot conclude about its nonexistence.To the best of our knowledge, speculations and new ideas about the existence of a fourthdimension had to wait for the middle of the 16th century on to be strengthened, when a morepropitious intellectual atmosphere is to be found, as we will see all over this paper.2or the moment, it is important to stress that the two greater synthesis of the ClassicalGreek Philosophy – that of Aristotle and that of Euclid – considered impossible the existenceof more than three spatial dimensions. This very fact is not meaningless, so far the enormousinfluence of these two thinkers is considered. The long period in which there was no discussionwhy space could not have a different dimensionality may be, in part, attributed to Aristotle’sauthority and, on the other side, to the fact that the study of Euclid’s
Elements in the MiddleAges, including different educational levels, was languished or quite neglected [6].The overcoming of the domination of the Aristotelian influence and the abandonment ofits Cosmos must still await the Renaissance [7, 8]. As Koyr´e emphasizes, this implies andimposes the reformulation of the basic principles of philosophical and scientific reason [9]. Incontrast, the revival of interest in Euclid’s
Elements should wait the invention of printingpress by Johannes Gutenberg (c. 1400-1468) [10].It is not out of place to remember that, since the first prehistoric cave painting untilthe medieval period, the World was pictured without any perspective in two dimensionalspaces, probably translating the difficulty to represent the third dimension on a bi-dimensionalcanvas, wall or any other surface. To go further required a good understanding and domainof Geometry.The geometrization of space and the desire to represent it in painting had an enormousimpact on the Italian Art in the end of the
Trecento and during the following centuries.Actually, the first to introduce the tridimensionality in Medieval Art [11] was the Italianpainter Giotto di Bondone (1266-1337). Giotto painted almost all the walls of St. Francis’Lower Church, in Assisi. “The Legend of St. Francis”, whose authorship is traditionallyattributed to him, is the theme of a cycle of 28 frescoes that are found in the Upper Basilicain Assisi, probably painted between 1297 and 1300. The frescoes painted by him in the ArenaChapel at Pauda, about the year 1305, mark an entirely new stage in the development ofempirical perspective [12]. He was also the first artist in that period to paint the Sky in blue,diverting up from the traditional golden Sky characteristic of the Byzantine Art [13, 14].Clearly, he was inspired by St. Francis’ world view, who pointed out emphatically that therewas a second book (not only the Sacred Scripture) able to bring someone to God: it is the
Book of Nature . One should look at Nature as it is. This attitude suggests and anticipatesa transition to the new relation between Man and Nature, which is a prelude to a new manthat is still to be forged in the Renascence [15].The formal discovery of perspective is attributed to the Italian architect Filippo Brunelles-chi (1377-1446), who suggested a system that explained how objects shrink in size accordingto their position and distance from the eye. In 1435, in a book named
On Painting , LeonBattista Alberti (1404-1472) provided the first theory of what we now call linear perspec-tive. The tri-dimensional representations of painting conquered then a scientific aspect whenpainters and architects of the
Quattrocento started to study the relationship between Geo-metric Optics [16] and Perspective in the Euclidean space, as did, for instance, Piero dellaFrancesca [17]. 3
The first ideas of a new dimension in space
Back to the fourth dimension, the idea of a new spatial dimension was revived by thestudies of several mathematicians in the 16th and 17th centuries. Indeed, the Italian physicist,philosopher, mathematician and physician Ge(i)rolamo Cardano (1501-1576) and the Frenchmathematician Fran¸cois Vi`ete (1540-1603) considered such “additional” dimension in theirresearches on quadratic and cubic equations. The same did the French mathematician andphysicist Blaise Pascal (1623-1662) in his study named
Trait´e des trilignes rectangles et leleurs onglets [18], when, generalizing his “trilignes” from the plane to the space and beyond,he wrote: “
The fourth dimension is not against the pure Geometry .”Meanwhile, the French philosopher and mathematician Ren´e du Perron Descartes (1596-1650), as is well known – and following the same pragmatic view of Aristotle, Euclid, andothers concerning space (Section 1) –, holds that “ the nature of matter, or body consists (...)simply in its being something which is extended in length, breadth, and depth ” [19].However, in his
Treatise on Algebra , published in 1685, the English mathematician JohnWallis (1616-1703) condemned again the existence of a higher-dimensional space saying thatit is a “ monster in nature, less possible than a chimera or a centaur ” [20]. And continues:“
Length, Breadth and Thickness take up the whole of space. Nor can Fansie imagine howthere should be a Fourth Local Dimension beyond these Three ” [21], [22], [23].The young Immanuel Kant (1724-1804), in his doctoral thesis (1747), tried to explainwhy is space three dimensional [24]. Actually, he did not succeed [25], but this work hastwo important merits: Kant pointed out that space dimensionality should be understood inthe framework of Physics, which proved to be a fruitful idea in the 20th century [26], and heconcluded his speculations by imagining various types of spaces – which came true later in the19th century – and alluding to them with these words of hope: “
A science of all these possiblekinds of space would undoubtedly be the highest enterprise which a finite understanding couldundertake in the field of geometry. ” [27].During the 18th century, the theme of the fourth dimension was treated again from adifferent perspective, i.e. , by associating it to time no more to space . We are talking aboutthe contribution of the French mathematician Jean le Rond d’Alembert (1717-1783) and hisproposal in the entry “Dimension” wrote for the
Encyclop´edie ou Dictionnaire Raisonn´e desSciences, des Arts, et de M´etiers , published between 1751 and 1772, by Denis Diderot (1713-1784) and himself.Time was considered also as a fourth dimension by the Italian-French mathematician andastronomer Joseph-Louis Lagrange (1736-1813), in his books
M´ecanique Analytique , de 1788,and
Th´eorie des Fonctions Analytiques , de 1797. Later, Lagrange says something like: Onecan consider the Mechanics as a Geometry in four dimensions and the Analytical Mechanicsas an extension of the Analytical Geometry, developed by Descartes in his book
La G´eom´etrie ,published in 1637 [23].In the beginning of the 19th century, more specifically in 1827, in the book
Der Barycen-trische Calcul , the German mathematician Augustus Ferdinand M¨obius (1790-1868) rejectedthe existence of the fourth dimension when he observed that geometrical figures cannot be4uperimposed in three dimensions since they are the mirror images of themselves [28]. Such asuperposition, however, could happen just in a four dimensional space but, “ since, however,such a space cannot be thought about, the superposition is impossible ” [23].The fourth dimension was also proposed by the German physicist and mathematicianJulius Pl¨ucker (1801-1868) in his book entitled
System der Geometrie des Raumes , publishedin 1846, in which he affirm that planes are nothing but collections of lines, as the intersection ofthem results in points. Following this idea, Pl¨ucker said that if lines are fundamental elementsof space, then space is four-dimensional, because it is necessary four parameters to cover allthe space with lines. However, this proposal was rejected because it was saw as metaphysics.But, in any case, it was quite clear for many mathematicians that the three-dimensionalGeometry had to be generalized [29].It is important to stress that before, in 1748, and later, in 1826, the Swiss physicist andmathematician Leonhard Euler (1707-1783) and the French mathematician Augustine LouisCauchy (1789-1857), respectively, had tried to represent lines in space. In 1843, the Englishmathematician Arthur Cayley (1821-1895) had developed the Analytical Geometry in a n -dimensional space, taking the theory of determinants (name due to Cauchy) as a tool. Soon,in 1844, the German mathematician Hermann G¨unter Grassmann (1809-1877) published thebook Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik , in which he thought ona n -dimensional Geometry, stimulated by the discovery of the quaternion, announced by theIrish mathematician and physicist Sir William Rowan Hamilton (1805-1865), in 1843 [21, 23].Actually, the conjectures about the fourth dimension acquire more soundness from thedevelopment of the so-called non-Euclidean Geometries in the 19th century [30]. Let us nowsummarize how it happened. It is attributed to the Greek philosopher and geometer Thales of Miletus (c. 624 - c. 546)the demonstration of the following theorems: In isosceles triangles, the angles at the base areequal to one another, and, if the equal straight lines be produced further, the angles underthe base will be equal to one another [31]; If two straight lines cut one another, they makethe vertical angles equal to one another [32]; Those theorems allow one to prove the so-calledThales Theorem: (...), and the three interior angles of the triangle are equal to two rightangles [33]. This theorem was considered as a divine truth by the influent Italian philosopherand theologian Thomas Aquinas (1225-1274), when, in his famous
Summa Theologica , issuedaround 1265, sustained to have proved that God could not construct a triangle for whichthe internal angles summed up more than 180 ◦ . It is opportune to remember that ThalesTheorem is also a consequence of the famous Postulate 5 of Euclid Book 1: That, if a straightline falling on two straight lines make the interior angles on the same side less than two rightangles, the two straight lines, if produced indefinitely, meet on that side on which are theangles less than the two right angles [34].In 1795, this Postulate number 5 was enounced by the English mathematician John Play-fair (1748-1819) as follow: Through a given point only one parallel can be drawn to a given5traight line. This is known as the Parallel Postulate [35].The Parallel Postulate started to be criticized by the German mathematician and physicistJohann Carl Friedrich Gauss (1777-1855) – who invented the concept of curvature –, in thelast decade of the 18th century, when he tried to demonstrate it by using Euclidean Geometry.In effect, in 1792, when he was fifteen years old, he wrote a letter to his friend the Germanastronomer Heinrich Christian Schumacher (1780-1850), in which he discussed the possibilityof having a Logical Geometry where the Parallel Postulate did not hold. In 1794, he conceiveda new Geometry for which the area of a quadrangular figure should be proportional to thedifference between 360 ◦ and the sum of its internal angles. Later, in 1799, Gauss wrote aletter to his friend and Hungarian mathematician Wolfgang Farkas Bolyai (1775-1856) sayingthat he had tried, without success, to deduce the Parallel Postulate from other postulates ofEuclidean Geometry [36].During the 19th century, Gauss continued the discussion with friends on the plausibility ofthe existence of a Non-Euclidean Geometry. So, around 1813, he developed what he initiallycalled Anti-Euclidean Geometry, then Astral Geometry and, finally, Non-Euclidean Geometry.He was so convinced about the existence of this new Geometry that he wrote a letter, in 1817,to his friend and German astronomer and physician Heinrich Wilhelm Matth¨aus Olbers (1758-1840), stressing the physical necessity of such a Geometry as follow [37]: “I am becoming more and more convinced that the [physical] necessity of our [Euclidean]geometry cannot be proved, at least not by human reason nor for human reason. Perhapsin another life we will be able to obtain insight into the nature of space, which is nowunattainable. Until then we must place geometry not in the same class with arithmetic,which is purely a priori, but with mechanics.” Seven years later, in 1824, answering a letter from the German mathematician Franz AdolfTaurinus (1794-1874) talking about a demonstration he did that the sum of the internalangles of a triangle cannot be neither greater nor smaller than 180 ◦ , Gauss told him thatthere was not geometrical rigor in that demonstration because, in spite of the fact that the“metaphysicists” consider the Euclidean Geometry as the truth, this Geometry is incomplete.The “metaphysicists” quoted by Gauss were the followers of Kant, who wrote, in 1781, inhis Kritik der reinen Vernunft [38], more precisely in its first chapter entitled TranscendentalDoctrine of Elements what follows: a) Space is not a conception which has been derived fromoutward experiences; b) Space then is a necessary representation a priori , which serves forthe foundation of all external intuitions; c) Space is represented as an infinite given quantity;d) Space has only three dimensions; d) (...) possibility of geometry, as a synthetic science apriori , becomes comprehensible [39].Although we owe to Gauss the discovery of Non-Euclidean Geometry, he did not have thecourage to publish his discoveries. Indeed, in a letter sent to a German friend and astronomerFriedrich Wilhelm Bessel (1784-1846), in 1829, Gauss affirm that he probably would neverpublish his findings in this subject because he feared ridicule, or, as he put it, he feared theclamor of the Boetians, a figurative reference to a dull-witted Greek tribe [37].In his research on the existence of a Non-Euclidean Geometry, Gauss figured out hypothe-tical “worms” that could live exclusively in a bi-dimensional surface, as other “beings” could6e able to live in spaces of four or more dimensions [40]. It is interesting to mention that,trying to verify his theory, Gauss and his assistants measured the angles of a triangle formedby the peaks of three mountains, Brocken, Hohehagen and Inselsberg, which belong to theHarz Mountais, in Germany. The distance between two of them were 69,85 and 197 km,respectively. The sum of the internal angles of this triangle was 180 ◦ and 14”,85. This resultfrustrated Gauss since the error were within the errors associated to the instruments he usedto measure the angles [23, 36].Independently of Gauss, the mathematicians, the Russian Nikolay Ivanovich Lobachevski(1793-1856) and the Hungarian J´anos Bolyai (1802-1860) (son of Wolfgang), in 1832, demons-trated the existence of triangles which sum of the internal angles are less than 180 ◦ [41, 42].The German mathematician Georg Friedrich Bernhard Riemann (1826-1866), after thepresentation of his Doktoratsschrift , in December 1851, in G¨ottingen University, about theFourier series and what is now know as Riemannian surfaces, started to prepare himself tobecome
Privatdozent of this same University. So, at the end of 1853, he presented his
Habili-tationsschrif together with three topics for the
Habilitationsvortrag . For his surprise, Gausschoose the third topic entitled “ ¨Uber die Hypothesen, welche der Geometrie zu Grunden lie-gen” (“On the Hypothesis that are on the Base of Geometry”), where he demonstrated theexistence of triangles of which the sum of its internal angles could be greater than 180 ◦ . Thistopic was timidly presented by Riemann in June of 1854, but it provoked a deep impact onGauss, because it was a concrete expression of his previous ideas about a Non-Euclidean Geo-metry (today, Riemannian Geometry) that he was afraid to publish, as previously mentioned.Riemann’s metrical approach to Geometry and his interest in the problem of congruence alsogave rise to another type of non-Euclidean geometry. We are talking about a new geometrythat cames out not by the rejection of parallel axioms, but rather by its irregular curvature.It is important to remember that those geometries, today generically know as Non-Euclidean Geometries [43, 44] influenced physical thought in 19th century [45]. They areconsequences of the observation that the relaxation of the Parallel Postulate could give riseto two new interpretations. One, in the Hyperbolic Geometry of Bolyai-Lobachevsky [46],for which, from a point outside a line, an infinite number of parallels can be drawn, and thesecond, in the Spherical Geometry of Riemann, where from a point outside a line, no parallelcan be drawn to it [23, 47]. Two specific areas of philosophical debate were the initial source of a sui generis publicinterest in the non-Euclidean geometries and in the geometry in higher dimensions: the natureof the geometric axioms and the structure of our space [48]. As time went on, a expressiveinterest of the general public fell on the nature of the space and the number of its dimensions.A historical record of this fact can be found in the accurate bibliography prepared by DuncanSommerville (1879-1934), a Scottish mathematician and astronomer [49]. This history iswell documented in the interesting book of Linda Henderson (b. 1948), historian of art [48].According to her, everything started with a movement to popularize n -dimensional spaces7nd non-Euclidean geometries in the second half of the 19th century. A whole literature wasdeveloped [49] around philosophical and mystical implications in relation to spaces of largerdimensions, easily accessible to a public of non-specialists; in particular, about the imaginationof a fourth dimension, long before Minkowski’s work and Einstein’s Special Relativity and theCubism. The popularization of these ideas contributed, as carefully analyzed in Ref. [48], toa revolution in Modern Art and, in particular, was fundamental to the Cubism, an artisticmovement contemporary to Einstein’s Special Relativity, where also use was made of non-Euclidean geometry, namely Minkowski’s space-time. It was Riemann who generalized the concept of Geometries, by introducing the definitionof metric, that defines how one can calculate the distance between two points, given by (innowadays notation) d s = X i,j = g ij d x i d x j ; ( i, j = 1 , , g ij is the metric tensor of Riemann. In the case of flat spaces and rectilinear coordinates( x, y, z ), g ij = ( e i , e j ) = δ ij where δ ij is the Kronecker delta, e i ( i = 1 , ,
3) are the vector-basis of a particular coordinatesystem and the notation ( e i , e j ) means the scalar product between the two vectors.Thus, the distance can be written asd s = X i,j = δ ij d x i d x j = d x + d y + d z known as the Euclidean metric. This definition is straightforwardly extended to higher n -dimensional spaces just doing i, j → µ, ν = 1 , , , · · · n .The Riemann work about Non-Euclidean Geometry (which easily allows the existence ofmore dimensions than the usual three), was soon recognized and flourish in all Europe, witheminent scientists propagating his ideas to the general public. For example, the Germanphysicist and physiologist Hermann Ludwig Ferdinand von Helmholtz (1821-1894) consideredGauss’ worms leaving now in a Riemannian surface (on a sphere). However, in his bookentitled Popular Lectures of Scientific Subjects , published in 1881, he warned that it is im-possible to represent (to visualize) the fourth dimension, because (...) such a representationis so impossible how it should be a color representation for someone born blind [50].From now on, let us summarize the route of the assimilation of such ideas in Physics.The success of Newtonian mechanicism will be put to the test, at first, by the study of heatmade by the French mathematician and physicist Jean-Baptiste Joseph Fourier (1768-1830).In the Preliminary Speech of his
Analytical Theory of Heat , he states that8
Whatever the scope of mechanical theories, they do not apply to the effects of heat.These are a special type of phenomenon, and cannot be explained by the principles ofmovement and balance”.
The propagation of heat will be described by a partial differential equation and no longerby an ordinary differential equation, as in the case of Newtonian mechanics. It is the beginningof valuing the causa formalis over the causa efficiens as the basis of the causal explanatorysystem in Physics, intrinsic to Newton’s system [51]. It is the beginning of the description ofPhysics by Field Theories [52]. Later, in the second half of 19th century, also electromagnetismwill reaffirm this trend [53]. The discovery of electromagnetic waves by the German physicistHeinrich Rudolf Hertz (1857-1894) will give Maxwell’s theory a new status. However, Maxwelltheory is still a phenomenological theory not able to predict, for example, the interactionof light with matter. One of the first attempts to develop a classical interpretive theorycapable of explaining the interactions of electromagnetic fields with matter dates from 1895and is due to the Dutch physicist Hendrik Antoon Lorentz (1853-1928), who combines theElectromagnetism and Classical Mechanics with an atomistic model of matter, the so-calledDrude-Lorentz model, and initially develops a Newtonian Classical Electrodynamics, knownas Lorentz Electrodynamics.Soon after the electron discovery, it gains a prominent place in theoretical physics [54]. Infact, as we have already mentioned, Lorentz will dedicate himself to include the interactionof this particle with the electromagnetic fields. As is well known, Lorentz Electrodynamics,inspite some initial success, failed in correctly describe such kind of interaction. This problemwill be solved just with the advent of Quantum Electrodynamics [55]. From a conceptualpoint of view, Einstein attributed the weakness of Lorentz theory to the fact that it tried todetermine the interaction phenomena by a combination of partial differential equations andtotal differential equations, a procedure that, in his opinion, is obviously not natural.In 1888, the English Mathematician Oliver Heaviside (1850-1925) showed that the electricfield ( ~E ) of a moving electric charge (with velocity v ) differ from that ( ~E ◦ ) of a stationarycharge as indicated below [56]: ~E ◦ = kqr ˆ r ⇒ ~E = kqr γ (cid:20) − β − β sin θ (cid:21) / ˆ r where β = v/c and γ = (1 − β ) − / . So, we can see that, in the direction of motion ( θ = 0),the electric field behaves like ~E k = 1 γ kqr ˆ r Therefore, this result was interpreted by Heaviside as a contraction of the electrostatic field.This result was published in 1889 and it was discussed by Heaviside, the British physicistOliver Lodge (1851-1940) and the Irish physicist George FitzGerald (1851-1901) [57]. Inspi-red on this result, FitzGerald proposed that the objects contract along their line of flight. The model according to which the physical world would be composed of ponderable matter, electricallycharged mobile particles and ether, such that electromagnetic and optical phenomena would be based on theposition and movement of these particles. γ factor.Pre-Minkowskian applications of non-Euclidean geometry in Physics weren’t many andthey were reviewed in Ref. [59].Now, we would like to stress that, although Lorentz demonstrated, in 1904, that timeis related to tri-dimensional space through the relations known as Lorentz Transformations(LT) [60], it was only the Russian-German mathematician Hermann Minkowski (1864-1909)who understood [61] that the LT represent a kind of rotation in a 4-dimensional flat spacehaving coordinates ( x , x , x , x ), with a metric (measurement of the distance between twopoints in this space) defined by:d s = X µ,ν g µν d x µ d x ν = d x + d x + d x + d x where g µν = δ µν is the four-dimensional Kronecker delta, x = x , x = y , x = z , x = ict ,and i = √− √−
1, the mathematiciansdefined a signature for g µν , such that the indices µ and ν can assume the values 1, 2, 3, 4(+ , + , + , − ) with x = ct , or 0, 1, 2, 3 (+ , − , − , − ) with x = ct , where ± means ± i.e. , assuming it does not depend on the velocity of the movingbody [62, 63].For Lorentz, the local time ( t ′ ) introduced in the coordinate transformations betweeninertial references, would be just an auxiliary parameter necessary to maintain the invarianceof the laws of Electromagnetism, as stated at the end of the second edition of his Theory ofElectrons [64]: “If I had to write the last chapter now, I should certainly have given a more prominentplace to Einstein’s theory of relativity (...) by which the theory of electromagnetic pheno-mena in moving systems gains a simplicity that I had not been able to attain. The chiefcause of my failure was my clinging to the idea that the variable t alone can be consideredas the true time and that my local time t ′ must be regarded as no more than an auxiliarymathematical quantity. In Einstein’s theory, on the contrary, t ′ plays the same part as t ; ifwe want to describe phenomena in terms of x ′ , y ′ , z ′ , t ′ we must work with these variablesexactly as we could do with x, y, z, t .” On the other hand, the conception and interpretation of Lorentz’s transformations as ageometric transformation in a pseudo-Euclidean space of dimension 4, for Minkowskii, wasonly possible thanks to Einstein’s assertion, as quoted in a meeting of scientists in 1908, inCologne [65]: 10
But the credit of first recognizing clearly that the time of the one electron is just as goodas the time of the other, that t and t ′ are to be treated identically, belongs to A. Einstein.” However, another point made clear by Einstein is that for him the introduction of time asa fourth explicit coordinate in the transformations of inertial reference systems derives fromthe principle of relativity. In his words [66]: “It is a widespread error that special theory of relativity is supposed to have, to a certainextent, first discovered, or at any rate, newly introduced, the four-dimensionality of thephysical continuum. This, of course, is not the case. Classical mechanics, too, is based onthe four-dimensional continuum of space and time. But in the four-dimensional continuumof classical physics the subspaces with constant time value have an absolute reality, inde-pendent of the choice of the reference system. Because of this [fact], the four-dimensionalcontinuum falls naturally into a three-dimensional and a one-dimensional (time), so thatthe four-dimensional point of view does not force itself upon one as necessary . The specialtheory of relativity, on the other hand, creates a formal dependence between the way inwhich the spatial coordinates, on the other hand, and the temporal coordinates, on theother, have to enter into natural laws.”
In this paper we have reviewed how mathematicians, physicists and philosophers havepositioned themselves on whether or not a fourth dimension does exist. It was shown thatthe development of non-Euclidean geometries opened a new possibility to describe Physics.In addition to the classical example of Special Relativity, the possibility that extra spatialdimensions can play an important role in Physics is not new. It can be traced back to thepioneer works of Kaluza [67] and Klein [68], in which a fifth dimension was considered, tryingto unificate Electromagnetism and Gravitation. Following the general unification idea ofKaluza-Klein [69], several higher-dimensional theories were developed, like String Theory andSupersymmetry [70, 71], based on theoretical ideas that go beyond the Standard Model ofParticle Physics and show promise for unifying all forces. In all these examples, the extradimension is always space-like.Indeed, the introduction of extra dimensions in Fundamental Interactions Physics hasbeen enabling a remarkable progress in two major contemporary programs: the quantizationof gravity and the unification of the force fields of Nature, for which the mechanisms ofreduction and dimensional compacting are of utmost importance [72].It is interesting to point out that difficulties concerning the search for a Unified Theoryof Elementary Physical Interactions (electromagnetic, strong, weak and gravitational) bringphysicists to develop the
M Theory , which is a unifying theory in an eleven dimensional space(with just one temporal). Seven of those spatial dimensions are curled out and compactifiedin a Calabi-Yau space having dimensions equivalents to Planck’s length ( ≈ − cm) and tothem are attributed other properties, like mass and electric charge [36].As a last remark, we can refer to the possibility of developing field theories with morethan one coordinate time, in the course of 20th century, as reviewed in Ref. [30].11 eferˆencias bibliogr´aficas [1] Barnes, J. (Ed.): The Complete Works of Aristotle . The revised Oxford Translation,Princeton: Princeton University Press, volume 1, p. 447 (1984).[2] Pauli, W.: The influence of Archetypal Ideas on the Scientific Theories of Kepler. In C.G. Jung & W. Pauli:
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