SPACE AND TIME, DO WE DESCRIBE THEM CORRECTLY? ON THE 4-DIMENSIONAL STRUCTURE OF THE UNIVERSE

SPACE AND TIME, DO WE DESCRIBE THEM CORRECTLY? ON THE 4-DIMENSIONAL STRUCTURE OF THE UNIVERSE PIER SANDRO SCANO [email protected] ABSTRACT The topic is whether the 4-dimensional structure constitutes a proven theory and whether it completely and effectively describes space-time phenomena. Spacetime is almost universally assumed as the basic structure for physical and cosmological thinking. Its meaning lies in the fusion of time and space into a unified entity. In the article, the theoretical foundation and mathematical formalism are analysed. The theory is confirmed by an incontrovertible quantity of experimental verifications. However reasons for reflection emerge, starting from the one-way character of time. The conclusion is that on space and time we know little, in the state, to be able to affirm a definitive and testable theory. Further research is necessary on what is assumed to be indisputable. Finally, it doesn’t seem sustainable that Special Relativity, supplemented by General Relativity, constitutes a complete theory of spatiotemporal relations. 1 INTRODUCTION Space, intuitively represented as three-dimensional until a little over a century ago, was considered to be completely separate from time. Both were regarded independent and absolute. This is what Newton thought, in quoting the most authoritative1. For decades now, spacetime has been assumed as a proven and essential concept consisting of unity; it means that neither space nor time are conceived as independent and absolute. Physicists, astrophysicists, cosmologists, and philosophers reason, almost entirely, in terms of spacetime, that is a four- dimensional structure of the universe. It is recognized as the «…basic structure for physical thought», we quote for all W. Rindler.2 The concept has the first origin in Einstein.3 In pre-relativistic physics, the time coordinate is not modified by the transformations between inertial systems. In the context of special relativity, instead, the transformation from one reference system to another requires the interdependence between space and time coordinates. Space and time are bound from the equations of relativity. Among the most important the difference between the squares of the time component and the sum of the spatial components is invariant under Lorentz transformations:4 ( ∆𝑠)2 = - ∆𝑡 2 + ( ∆𝑥 2 + ∆𝑦 2 + ∆𝑧 2 ). (1) The equation replaces the Euclidean square distance in the description of the phenomena: (∆𝑠 2 ) = (∆𝑥 2 + ∆𝑦 2 + ∆𝑧 2) (2) The spacetime interval (or separation) remains invariant in varying the temporal coordinate and spatial coordinates. In transformations time and space vary, while a specific combination of space and time does not change. This is a quantity which is preserved. A relationship of formal dependence has been created between spatial and temporal coordinates, constituting the system’s dimensions. This approach allows framing the data of the experience, referring them to a continuum consisting of the four-dimensional structure of reality. In this framework, the objective quantities of physics should be expressed as invariants of a four-dimensional structure. Nonetheless, there are a multiplicity of interesting conjectures about space and time, with extra dimensions, multiverses, brane worlds and others extremely stimulating, but according to the thinking of most researchers not yet sufficiently proven with verified predictions. 2 Restricted relativity stimulated the search for a four-dimensional mathematical structure, inclusive of the relations between space and time. It was Hermann Minkowski5 who was the theorizer of spacetime (denoted by the symbols M or M4 or R3.1). We can overlook the anticipators here, such as Menyhert Palagyi6 and others. The solemn lecture delivered on 21-09-1908, during the 80th Scientific Assembly of German Nature and Physicians, about space and time and, besides, the formal mathematical arrangement given to the theory of special relativity (1909) were of paramount importance. Minkowski, and indeed already Poincaré7 in 1906, introduced the uniform treatment of space and time coordinates and, therefore, time as the fourth dimension of a 4-dimensional space-time continuum. Einstein, after an initial puzzlement laced with some irony («...superfluous learning»8), accepted Minkowski's mathematical reformulation.9 In the decades that followed the soon to become boundless literature on the four-dimensionality of reality (among the first, as early as 1921, Wolfang Pauli10 in his famous work on relativity, Relativitätstheorie), with the mathematics of four-vector, algebra and tensorial analysis, consisting of the application of infinitesimal analysis to vectors and tensors, developed by Ricci Curbastro11 and Levi Civita.12 «Tensor calculus has become the ideal mathematical technique for the study of relativity» (W. Rindler).13 Poincaré's group itself, which is a symmetry group of any relativistic field theory and consists of a Lorentz group plus a spacetime translation, is the group of spacetime isometries and is consistent with Minkowski's spacetime geometry. This represents the most powerful formalism for describing special relativity, the basis of general relativity. The mathematical structure is a pseudo-Euclidean space, with zero curvature like Euclidean space, but with a distance not defined as positive. The theoretical construction of spacetime, simple and fascinating, has established itself and has demonstrated, and still does, its power and majestic operation. Einstein notes that «Classical mechanics is also based on three-dimensional continuum of space and time. Only the ‘sections’ corresponding to constant values of time have an absolute reality, that is, independent of the choice of the reference system. The three-dimensional continuum, therefore, splits into a three-dimensional and a one-dimensional (time) continuum, and the four-dimensional point of view does not impose itself as necessary».14 DISCUSSION Restricted relativity has a long theoretical and experimental history, but most researchers assume, as its still viable and valid core, Einstein’s original program, 3 complemented by Minkowski’s 4-dimensional geometric model, which facilitated the construction of general relativity and long time later offered the basis for relativistic field theories. And indeed, it has certain assumptions which, for convenience, should be recalled: 1) The speed of light, c, is not a speed like any other; it is a constant in any reference system and for any observer; 2) Time is treated in the same way as a spatial dimension; 3) One must use c, as a conversion factor, multiplying by c the time coordinate, which, in this way is transformed into a fourth spatial coordinate. The crux of the spacetime theory is that space and time have no existence in themselves, «…condemned to dissolve into nothing more than shadows» (Minkowski, in the recalled lecture) and that what is real is their fusion into a unified entity, in which they intermingle and exchange. From this point of view, the distinction of spacetime into space and time is unnatural as claimed, e.g., F. Wilczek15. In summary, space and time are unified into a single entity, and the time coordinate, t, is joined to the position vector x = (x1, x2, x3) (3) of three-dimensional space to obtain the 4-vector, (t,x) = (𝑥 0 , 𝑥 1 ,𝑥 2 , 𝑥 3 ). (4) They are different dimensions of the same continuum. Similarly we proceed for momentum and energy, which are unified in the 4-vector energy-quantity of motion. The four-vector in relativity is the formalism that generalizes the three- dimensional vectors of classical mechanics ( position, velocity, acceleration, momentum…) by transforming them into the corresponding four-vectors. There is a second assertion, which is indeed the necessary presupposition and a cornerstone of special relativity: the speed of light, invariant in the vacuum, is a universal value, independent of source and reference system. A third assertion, according to most scientists, completes the scenario: the speed of light is the speed of all bodies and is a fixed quantity, valid for every reality, a kind of plenum that is divided to varying degrees between spatial and temporal components. The sum of the two parts always gives c, for anybody. In short, c would be the speed with which everything moves in the universe across the spacetime continuum. We note, too, that for special relativity the norm of the four-velocity ( the norm of a four-vector is a Lorentz invariant) of an object at rest, thus with the time axis as the direction of the 4 vector, is equal to c. It seems fair to observe that without this third step, the first, namely the merging of space and time into Minkowski's 'independent reality,' might appear problematic and lacking in the physical meaning. How space can mutate into time and time into space remains, however, a puzzle to be solved, beyond the fruitful operation of multiplying the duration by c and by i = √−1 or choosing the type of scalar product. The tetra-dimensional spacetime has four coordinates (ct, x, y, z) to be considered equivalent, each component of the four-vector. The vector components are the variation of the spatial and temporal coordinates with respect to the proper time. To have, in fact, a time coordinate invariant under Lorentz transformations, it is necessary to take the 'proper time', τ. Parameterization takes place as a function of proper time, precisely because it is invariant, unlike measurements of transformed time. At first, the invariance of relativistic separation is restricted to inertial systems framework like the space components, rotating in the same way in spacetime. In this view, Lorentz transformations are nothing more than a coordinate system rotation in three-dimensional space. It generally assumes however the difference in sign, between space coordinates and time coordinates,16 by the mathematical formalism adopted and because time is not exactly symmetrical with respect to space, as, moreover, the common human experience constantly reminds us, as well as experimental data, such as those on the decay of K mesons. Many authors point out that this is not a simple 4-dimensional generalization of the ordinary Euclidean space, in which time is nothing but a fourth dimension, and that, because of the signature, the time coordinate is not on the same plane as the three spatial ones, and, finally, that the universe, a 4-dimensional Riemannian17 space, has anisotropic properties in a way that is quite dissimilar to Euclidean space. The sign difference arises from the introduction, with Minkowski, of the ict coordinate, in that by multiplying t by c we obtain a space, i.e., homogeneous coordinates, and by multiplying by i =√−1, we obtain the indispensable sign difference. There are other formalisms, characterized mainly by the scalar product with which they are endowed, for defining the concepts of distance, length, and angle. In the familiar Euclidean space, the scalar product is positively definite, while in the pseudo- Euclidean space, it is not positively definite (so distance is not necessarily ≥ 0); in the pseudo-Riemannian variety, the requirement of positive determinacy is relaxed and it is sufficient for the tensor to be non degenerate. In addition, moving from 𝑅3 to 𝑅4 , the vector and the associated convector no longer have the same 5 components and the spatial components of the convector have opposite sign to those of the corresponding vector. The difference in sign follows, since the two different signatures (3, 1) and (1, 3) are used in the different approaches, namely : [ - , + , + ,+ ] [ + , - , - , - ] (5) from the solutions adopted for the space type, scalar product and metric. If we take the four-vector, the contravariance and covariance components are always the same, except for the temporal contravariance and covariance components, which have, however, opposite sign: 𝐴0 = −𝐴0 ; 𝐴𝑚 = 𝐴𝑚 (6) Clearly, as we move from 𝑅4 to 𝑅𝑛 , more powerful mathematical tools, such as algebra and tensor analysis, which describe the invariant-spatial changes and transformations in the various references, become indispensable. The traditional treatment galilean-newtonian of the fundamental concepts of physics (space, time, mass, matter, energy, motion) has been revolutionized, as is well known, by the breakthrough of relativity, first Special Relativity (SR) then General Relativity (GR), and quantum mechanics. Separate concepts were brought back to unity. First, space and time with SR, which blends their coordinates; mass and energy, as early as 1905, with the foundations of relativistic dynamics, then with GR, which similarly merges their quantities and substitutes energy for mass as a measure of inertia. But it also radically changes the relationship between space and time on the one hand and mass and energy on the other. Einstein, in the quoted text18: «...the totality of physical ‘events’ is thus thought of as being embedded in a four-dimensional continuous manifold». A mighty conceptual unification. J.A. Wheeler19 systematically expressed the new vision in a celebrated formidable synthesis: mass tells spacetime how to bend and spacetime tells mass how to move. Perhaps we can no longer think of something remaining itself, either moving in space but not in time, moving in time but not in space, varying mass but not energy, varying energy but not mass, changing spacetime but not mass-energy or changing mass-energy but not position in spacetime. The revolution is spectacular but maybe still incomplete. If one were to stop here, the merging of space and time might appear as a mathematical sagacity, in which even the change of sign between coordinates might seem gratuitous and whose physical meaning would not be understood. 6 Another line of reasoning begins to develop, not only for these reasons. Space-time would perhaps be nothing more than mass-energy; that is, it could be, too, a form of mass-energy. The intuition passes, on the one hand, through field theories (field include motions and alterations, moving electric charges radiating electromagnetic waves, moving masses radiating gravitational waves, and whatnot) and, on the other, through the paths of quantum theory, for which «The gravitational field, like all physical things, must have quantum properties», as C. Rovelli writes.20 Thus, again, F. Wilczek: «Space-time is also a form of matter».21 With this last passage, we may have closed the circle. The fundamental concepts of physics, namely mass, matter, space, time, and motion, would be brought back to a single fundamental concept: energy. It would open a way, among other things, to try to understand how space, in moving from one reference system to another, can translate into time and time into space. Both would be nothing but energy. Whatever the case may be, in the absence of a coherent theory (the one just mentioned is a conjecture), the four-dimensions of reality, the total interchangeability between space and time, and all the processing based on the four-vector (or single 3-1-dimensional field, with 3 spatial dimensions and 1 temporal) would not be easily explained. However, they describe well the spatial and temporal relationships. The conjecture, but intuition should give rise to experimentally verified predictions, would have another relevant theoretical consequence. It is a widespread assessment that attempts to quantify space and time, the node of quantum gravity, have not yet been successfully verified so far. The fact is that space, which we could see as the coexistence of distinct events (i.e., reality is not point-like), and time, which we could see as the succession of specific events (i.e., reality does not remain identical), is possible that have a relational, non-substantial, nature. This would make quantization implausible, but research is underway and is open to different outcomes. In the hypothesis that quantum gravity turned out to be an unpracticable road, the theory would have to use a different physical-mathematical arsenal, i.e. the metric tensor and other formalisms of the physics of relativity and the continuum. If, on the other hand, spacetime were attributable to a form of energy, the scenario would change radically. The search for a unitary idea, which allows us to describe the world well, would have a great new chance. But, of course, even a theory that the world can be traced back to energy alone has its own 7 challenge to test its effectiveness. There are different theoretical hypotheses competing. The grand complication of the theories of four-dimensionality, four-vector, and ultimately spacetime is the uniqueness of the direction of time. This is a point ''where the shoe hurts,'' to use a typical Einstein’s expression. It is the question of irreversibility, which is much harder than a change in the sign of the components. It seems problematic, although the state cannot be ruled out, that the mathematics of the four-vector can encompass and absorb the irreversibility of time. Such irreversibility, moreover, appears to be in harmony with the one-way dispersion of energy, in accordance with the second principle of thermodynamics, that is, the inexorable process from statistically improbable situations to situations with a more probable arrangement. That is, time is not exclusively change, it is also toward change and inexorability of finite duration. The arrow of time means that the duration of time, and therefore of the universe, can only have an end. It would take, perhaps, an inconceivably long but finite time. There are other aspects to reason about, and the first is the concept of space. The space in which we live appears to us, intuitively, with the property of 3 dimensions. Our brains can maximally picture space and objects in 3 dimensions. Starting from this basis, in the historical development of mathematics, it came gradually at the mathematical model of space and, therefore, its dimensional variability. Following Gauss,22 Riemann,23 the concept of variety, which generalizes that of space, the developments in differential geometry and so on, today, the differentiable manifold is one of the most studied mathematical objects, both in geometry and in topology. Physics has embraced and assimilated the mathematical model of space/variety. In common use, for example, are 1-dimensional and 2-dimensional space, even to simplify and translate concepts into visualizable patterns. The 4-dimensional model takes on a special importance beginning with special and general relativity, because time is framed as one of the 4 dimensions of spatiotemporal reality. Spacetime is described as a curved 4-dimensional variety. Central role assumes the metric tensor, a scalar product defined on the spacetime of each point (curvature of the variety), which allows concepts such as distance, length of a curve, angle, geodesic, curvature, etc. to be defined. With the development of mathematical physics, modelling, going beyond the so-called 'low dimension' (variety of dimensions up to 4-d), fully unfolds with n-dimensional theories and techniques (Lagrangian 8 mechanics, relativity, QFT, string theories, quantum gravity, cosmological models etc.). The question of dimensions in mathematics (real and complex analysis, topology, algebra, differential geometry…) and physics should always be kept in mind. Mathematics has, in all its branches, exclusively internal constraints of coherence and completeness. The possibilities are limited only by its rules. So, for example, we use geometry with infinite dimensions without being surprised at all. Physics has external constraints, and reality, whatever it is, limits the spectrum of possibilities. For the sake of brevity, we can restrict ourselves to Feynman: «We cannot make a mathematics of the world ».24 We have to deal with the objects of reality unless we are among those who think that reality consists essentially of relations between numbers. This is not at all forbidden, although most researchers think the world is not reducible to numbers or only to numbers. It is not only legitimate, necessary and useful to calculate in any number of dimensions, but it is a question of verifying what is being modelled, whether it is compatible with certain properties, and whether it can translate into a physical meaning. If we wanted to determine a point in space + a time coordinate + any scalar property (e.g. temperature or mass), we would need 5 numbers, so if we found it useful, we could use a 5-dimensional model. It would not mean that there is a world with 5 dimensions underneath. Any system can be defined as an n-space, considered correct and effective for the given purposes, correlated, including the spatial and temporal coordinates, to the n-number of objects considered. Is it the reality that has n-dimensions, or is it the observer who chooses, depending on the purposes and situations, the n- dimensional functional degree? The number of dimensions is probably our choice, depending on the objective pursued. Does this also work for the 4-dimension time? The choices will have different degrees of effectiveness and functionality; sometimes, they will also be a model of something, sometimes not. The 4-space variety is extremely effective for the theory of spatial and temporal relations, but this is not the same as deducing that it necessarily reflects reality with 4 dimensions. The model describes the system; it does not take the place of the system itself. Each event can be indicated by 4 coordinates, and the choice of coordinates seems to be arbitrary, e.g. three spatial at will and any measure of time. By changing the reference system, the same object will be represented by a different sequence of numerical components, following precise transformation laws. As Stephen Hawking observes «... there is no real distinction between spatial and temporal 9 coordinates».25 An event happens in a precise place, in a precise instant. Many other variables (temperature, mass, shape, density ...) are associated with the event. «The time variable is one of the many variables that describe the world»26 If we adopt a 4-space, keeping the 3 spatial dimensions still, there would be many variables, starting with time, among which to choose 4. It seems to be an arbitrary choice of the observer, more or less effective. The effectiveness varies for different cases and different observers. A single dimension would be enough for an elevator, I don't know how many for a flock of birds or a complex electronic device. As for us, in the traditional common sense, we move in a 3-space, while scientific thought tells us in a 4-space, adding the time coordinate. A complex general theme was thus recalled, but the subject of this work is much more circumscribed. Now, let’s get to the point: in the 4-dimensional model, which describes spacetime, one of the four dimensions, the temporal one, seems to be heterogeneous with respect to the three spatial dimensions. The four-vector requires a time component. In theory, it is assumed that by multiplying the time coordinate by c, we obtain (with ct, for use and convenience) four spatial coordinates are considered consequently homogeneous. Once time has been traced back to the distance travelled by light, in that given time interval, everything seems to go smoothly. The time interval, it should be emphasized, is inevitably seen and measured by a reference inertial system different from the primary one, as widely argued in the article ‘On the velocities addition relativistic equation'.27 But it is precisely the equivalence between the temporal component and the spatial components that we are discussing, not in the sense of the legitimacy of this operation, which appears to be completely correct. The question is if this exhausts the time entity and constitutes a complete description. Spatial transformations can occur in any direction; the temporal transformation is one-way. It is absolutely unidirectional, although it can be conjugated with any spatial direction. Heterogeneity seems given in facts, by the unidirectionality and irreversibility of time. To arguments of this kind, Einstein objected that «…irreversibility is based only on considerations related to probabilities»,28 but it should be noted that the univocal data of experience is added to the reasoning. It seems to most that time constitutes one of the essential preconditions of our existence and the world. The universality and irrevocability of the irreversibility of time are perhaps the strongest data of the experience of humankind. It’s not at all simple to consider time an equivalent and homogeneous component to the three spatial dimensions. 10 There is reason to think that the direction of time poses a real problem. There are very different and conflicting opinions on this point. In Y. Z. Zhang's interesting book 'Special relativity and its experimental foundations' we read: «It is obvious that the four-dimensional Minkowski space M4 is analogous to the ordinary three-Euclidean space : the ‘interval’ and Minkowski metric are analogous, respectively, to the ordinary space interval and Euclidean metric».29 It is not difficult to argue, and it was done with reasoning of various kinds, including geometric processes, how this just mentioned is by no means a logically correct procedure. We can make calculations, as Hilbert30 used to say, with tables, chairs, and mugs of beer, but keeping the definitions and the rules of the procedure firm. However, the equation X0 = ct guarantees that the components have the same unit of measurement and the treatment is completely satisfactory from a mathematical point of view, but it does not give us any information on the nature of time. Measuring time in meters is certainly legitimate, given that we have a constant speed in a vacuum c and very high-precision clocks and it is also very effective. Just remember that it is the distance travelled by light in a certain time interval. From Einstein onward, and even to this day, we employ conceptually the speed of light for synchronization. Measuring it as a length doesn't mean we know what time is. Certainly, being able to connect the ‘where’ and the ‘when’ gives us a very powerful tool. Other formalisms describe the electromagnetic radiation, also by relating t (time) and 𝜈 (frequency) and other related quantities (wavelength λ, angular frequency ω ...). These are formalisms, which contain time coordinates, quantities which sometimes present themselves as dimensionally non-homogeneous with respect to spatial components. However things stand, at the centre of special relativity is an equality between a time and a length. Reducing it to a trivial example, we could combine the speed of light with the cycle of a sand hourglass: we would have the distance travelled by light in the interval marked by the hourglass, but we will not obtain any information on the nature of the sand. The reasoning does not change if we use, instead of an hourglass, a state- of-the-art ‘optical’ atomic clock. The point is what we are measuring, when we measure time, whatever clock is used (light type, hourglass, or atomic). A clock is but a periodic phenomenon, of which we measure the frequency of oscillations. The question can, of course, have a disarming answer: time is what is measured by a clock. A lot of things can be imagined about time; we have seen that the most 11 advantageous choice is to make it a dimension of space, but the equivalence of time and space, defined with SR, remains an open problem. In the formalism of general relativity, the mathematical object tensor is used to describe spacetime, a 4-dimensional curved variety. It allows the generalization of all structures of linear algebra. The tensor and tensor field describe invariances in the most general way. The tensor, which defines the differential invariant, is the four-tensor 𝑔𝑖𝑘 , the so-called fundamental tensor (which allows the operations of index shifting and covariant derivation), with the squared interval as the metric of the space. Thus, D𝑠 2 = 𝑔𝑖𝑘 𝑑𝑥 𝑖 𝑑𝑥 𝑘 (7) From wich ds = √𝑔𝑖𝑘 𝑑𝑥 𝑖 𝑑𝑥 𝑘 (8) This metric determines a Riemannian space, or pseudo-Riemannian if provided with a relaxed vector product on the positive definition requirement, and is more general than the n-dimensional Euclidean space metric. Precisely, 𝑔𝑖𝑘 is the so-called metric tensor. In tensor analysis, the covariant derivative, when all Christoffel symbols are zero, in other words, when the metric tensor is constant, coincides with the ordinary derivative. SR, which forms the basis of GR and spacetime, requires no more powerful mathematics to be described than is usually used, starting with Einstein himself. However, the four-dimensional representation, which is of great elegance, leads to new techniques and a deep theoretical arrangement. Also, the following procedure has to be kept in mind is: it is assumed that a valid equation in SR, when it has, or can be expressed in, tensorial form, i.e. when it is independent of the choice of a basis, remains valid, by applying the tensorial transformation rules, even in curved spacetime, i.e. in GR and any system. What is more, «…tensor equations have the property of being true in all systems if they are true in any system».31Therefore «…a four-tensor equation subject to a Lorentz transformation transform into the same four-tensor equation in the new system.32 Tensors, in fact, distinguish intrinsic geometric and physical properties from coordinate-dependent ones. The invariant is a useful quantity for distinguishing objects, and tensor algebra and analysis are the strongest invariant theory. It is clear, however, that a discussion involving the original equation (the input) cannot be resolved by the sequence of tensor transformations and remains (in the output). Therefore, the use of the tools of tensor mathematics, in our 12 opinion, offers no reason to change the setting and outcomes of the present work in its precise and circumscribed object. The properties of the time component do not question the use of the four- dimensional formalism to model, for example, elapsed time and distance travelled or even more complex and abstract relationships. Still, quite another issue would be the claim to have demonstrated, once and for all, that space and time are different manifestations of the same entity and, therefore, that space can turn into time and time into space. The claim that the sum of the time and space components must necessarily equal c for anybody also presents some problems. If so, there should probably be two extreme states: a) the state in which everything is a spatial component, with the temporal component equal to zero; b) the opposite state, in which everything is a temporal component, with the spatial component equal to zero. In the theoretical framework of SR we would have : a) is the state of massless particles, including photons; b) in a system at rest we have the velocity vector = 0 and the Lorentz factor = 1, from which it follows immediately that the norm of the four-velocity of an object at rest is ∥X∥ = c (9) Now, non-massive particles have physical reality, but does the norm of four- velocity, in a system at rest with v = 0 and 𝛾 = 1, equal c, possess a clear physical relevance? We do not know, for that matter, whether it is conceivable to move in time while standing still in space (x=0, t≠0) or to move in space while standing still in time (x≠0, t=0), and in any case, the absence of motion, by the principle of relativity, at the base of SR, is neither observable nor assertable. The reality, in any case, is not a purely logical matter. The energy-momentum four-vector is constructed in analogy to the four-velocity, replacing, since a four-vector must always have a time component, the time coordinate with E/c, and the spatial coordinates x, y, z with 𝑝𝑥 , 𝑝𝑦, 𝑝𝑧 . Thus, the transformation gives us three spatial parts, the components of motion, and a fourth time-like component, energy. Without going into specific analysis here, it seems clear that this is a relativistic four-vector that possesses all the characteristic properties of a four-vector. The velocity four-vector, which, when multiplied by the rest mass gives the momentum four-vector, has square norm -𝑐 2 . The negative square norm implies that the velocity and impulse four-vectors are both time-type and by changing sign in the sequence ( +, -, -, - ) (10) 13 the square norm chances sign. A simple examination of the equations found in all texts dealing with special relativity can strengthen the questions. Let us take two important equations : a) t=𝛾𝜏 (11) Which clearly is just a different way of writing 𝜏 t= (12) √1−𝛽 2 and b) 𝜏 = ct (13) in which 𝜏 is the proper time and t is the time interval transformed according to the Lorentz equations. Combining the two expressions gives t=𝛾𝑐𝑡 (14), 𝑐𝑡 which we can write as 𝑡= (15) √1−𝛽 2 It can be seen immediately that in the three cases, with v<c, with v=c, and with v>c, there will always be some results, devoid of mathematical or physical meaning, which could signal the existence of some problems that are still open in the description of phenomena with the formalism considered.33 The fact is that 𝜏 is the time measured by a single clock integral with one's inertial reference system and t is that same time measured, applying Lorentz transformations, by another inertial reference frame in relative motion with respect to the first. In other words, 𝜏, seen in the primary system, is transformed into time t, seen in the new system, and measured by another ( or two, as the case, maybe) stationary clock. The proper time 𝜏 thus becomes the distance travelled by light in an interval of time measured by another clock connected with another inertial reference system. One proceeds correctly with Lorentz transformations, but surprisingly not everything fits. In summary, the whole edifice is based on assuming the time component in a form considered equivalent and homogeneous to the spatial components. And this is precisely the most challenging point of the argument. The choice of a signature, whether one adopts it by multiplication by √−1 or by the scalar product type or by other mathematically correct options, offers patterns of interpretation of spatiotemporal phenomena, but it does not constitute proof of any theory, among those currently in the field, space and time and, therefore, of the workings of nature. One may wonder, moreover, whether time is reducible to the measurement 14 of time alone. Is time nothing but a clock? In this context, it is not superfluous to mention that there is no operator to observe time in standard quantum mechanics. A recent article by A. Ananthaswamy quotes what W.Pauli wrote on the subject in 1933: «We, therefore, conclude that the introduction of a temporal operator… must be abandoned fundamentally»,34 which is equivalent to saying that quantum mechanics cannot accommodate an operator for time. En passant, the cited article, referring to ongoing research being carried out at LMU Munich by Dürr, Aristarhov, Das and F. Schmidt-Kaler's group35 at Gutenberg Mainz, consists of a promising experimental study of quantum-mechanical data of the distribution of particle arrival times, using D. Bohm36 mechanics to test the fundamentals of quantum theory. This research is of great interest to those concerned with the relationships between time measurement and physical theories. Other lines of research, in addition to the unification of space and time in the same entity, consist in investigating their non-fundamental character (they could both emerge from more fundamental components of reality), hypothesizing a universe with x dimensions, thinking that space, and perhaps also time, are constituted by quantum entanglement, in hypothesizing an AdS (anti-de Sitter space)/CFT37, conformal field theory, correspondence (for this theory the universe could be a hologram, furthermore space and time not needed to describe reality) and several other very interesting theoretical hypotheses, mostly not yet tested or rather difficult to test. The philosopher of physics, Eleanor Knox, of King's College London, thinks that «'Where are things?’ and ‘Where do they live?’ Are not the right questions to ask».38 E. Knox may be right, but, it can be respectfully observed, so we could also get rid of the questions for which we have not yet answered. Of great importance is the position expressed by C. Rovelli ( and also from other authoritative physicists) in various of his works, which proposes the hypothesis of the timeless world and the plausibility of intending for time nothing more than to happen. «The fundamental theory of the world… does not need a time variable».39 In this framework there may be no space or time as distinct variables and everything would be resolved in the relational structure of the observed events and variables. Rovelli is also very clear in stating that the world «…does not even form a four- dimensional geometry».40 Of the long theoretical and experimental history of SR what still remains today, as mentioned above, is essentially Einstein’s original program, supplemented by Minkowski’s mathematical reformulation. This framework still remains the 15 reference, and among the underlying issues is the four-dimensional formalism of mechanics and electrodynamics. Summing up, the question of spacetime calls into question the interpretation of relativity: whether SR constitutes, generalized and developed in GR, a complete and definitive theory of spatial and temporal relations or whether, as Einstein41 himself thought, “…for all man-made theories” not only for the 'disturbing' quantum theory of Bohr42 and Heisenberg,43 the question of incompleteness and temporariness is raised. In this perspective even valid theories, as the special and general relativity have shown up to now in an undeniable way, wouldn’t be the last word. In fact, «…physics is an attempt to grasp reality conceptually, as it is conceived independently of the fact of being observed. In this sense, we speak of ‘physical reality».44 Special relativity is a theory of inertial reference systems, that is, in simple terms, of the relationship existing between the points of view of different inertial observers and is a theory of invariants, fixed in SR limited to inertial reference system, and generalized in GR for any arbitrary observer. Can it be defined as a theory of spacetime reality "regardless of whether it is observed" or should it be defined as a theory that does not disregard the observer, on the contrary considers it fundamental? Relativistic effects (contraction of lengths, dilation of durations, relativity of simultaneity) would be, in the absence of reference to inertial and, with GR, arbitrary observers, not only not falsifiable but meaningless concepts. Furthermore, in the theory of relativity, as well as in the formalism of the four- velocity, the irreversibility of time does not find an expression. These aspects and those analyzed by quite a few scientists, and also in the article On the velocities addition relativistic equation45 seem to tell us that maybe something is missing. CONCLUSIONS In summary, it can be argued that we do not yet know what space and time really are. We know that the world is not point-like ( and therefore there must be something we call space). We know that the world is not always the same and that it changes ( and therefore there must be something we call time). We know, with SR, that spatial distances and temporal durations, taken separately, do not account for phenomena and that interdependence between coordinates describes observation much better. We also know that space has fewer restrictions than time. In the former we can move with numerous options, schematically in three directions, which can be travelled ‘two-way’, each of which has an infinite number of variables; while in the latter we can do only one thing, move forward. We are capable of 16 measuring both and the measure of time is, indeed, the most precise one we can have. This, moreover, can be spatialized, that is, translated into length, or it can be split from any spatial reference, as with an atomic clock assumed to be at rest, thus emphasizing its scalar character. We also know that the mass and distance, like other properties, vary with time, and that the rate of time, in turn, varies with overall mass and velocity. So far we do not fully understand, finally, how much of this is intrinsic and how much depends on the relationship to the observer. I hope I helped make it clearer that SR, supplemented by GR, could be plausibly a not yet complete and not definitive theory of spatial and temporal relations. Furthermore, I express the conviction, for which I cannot adduce, beyond the arguments set out, other than (resorting to an ironic quote from Einstein: «…my only witness is the pricking of my little finger… an authority that cannot inspire the slightest respect outside the sphere of my hand»)46 that we know little about space and time at the moment to be able to say a lot solidly founded and tested. At this stage, even the idea that time and space are simply labels, coordinates or schemes, for ordering events cannot a priori be ruled out. One might think that we have just emerged from a somewhat unfinished phase of the scientific understanding of space and time. There is still a lot to understand about space and time and the first thing to know is precisely what must be questioned about what passes for indisputable. 1 NEWTON I., Philosophiae Naturalis Principia Mathematica, first pub. 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