THE DUALITY OF SPACE AND TIME AND THE THEORY OF RELATIVITY B. Guy Ecole Nationale Supérieure des Mines de Saint-Etienne, 158, Cours Fauriel, 42023 Saint-Etienne Cedex 2, France e-mail: [email protected] Abstract Space and time concepts cannot be defined independently of each other; thus, there is a mathematical link between the corresponding parameters describing our physical world. An expression of this link is met in that, to-day, assessments of space and time standards are both derived from one single phenomenon, light movement, whose velocity is decided at an arbitrary mathematical value. A way to describe the space-time-associated reality is done by means of a phase-space frame where the position of the entities are represented with respect to each other without the need of a special time parameter. The “ extraction ” of a single time parameter from this frame is done by favoring, among all the physical entities, one that is assumed to be moving by itself (e.g. the photon), the position of which defines the time. Because time is related to position, one may give a pre- temporal significance to a three-dimensional spatial parameter. Three- dimensional pre-time, to which is associated three-dimensional space parameter through a tensorial duality, allows a straightforward expression of the consequences of space-time duality, particularly for frame transformations, with space and time parameters on an equal footing. If one wants to have a suitable connection between the physical descriptions of phenomena both in a mobile frame and in a rest one, the same ratio of space to time standards must be taken in both frames. As a result, effects similar to Lorentz transformation are obtained; they may be understood as picturing a non-Euclidean metrics for the mobile frame as represented in the rest frame. They may also be conceived as uncertainty relations connecting the values of time and space standards to that of light velocity. 1 Introduction In order to build a quantitative representation of the physical world, it is necessary to define a frame where the position and the time of the events are measured. By doing so, space and time co-ordinates are viewed as four independent parameters, setting up a four-dimensional mathematical frame. This frame is considered before any other frame, having a relative motion with respect to it, is examined; so it has been named “ rest frame ” since A. Einstein (1). The well-known associated changes for space and time co- ordinates in Special Theory of Relativity, only occur when a new moving frame with respect to the rest frame is considered. Inside the starting rest frame, space and time co-ordinates are supposed to be measured separately by use of rules and clocks; time measurements have a meaning in the whole frame after all the clocks, located at the different space points, have been synchronized. These operations and the assumptions on which they lie (rigidity of rules, punctuality of clocks etc.) have been abundantly discussed since Einstein. Paradoxically however, it happens today that space and time in the rest frame are not defined (nor measured) independently. Both are determined together by the same non-rest phenomenon: the displacement of an electromagnetic signal (2), (3). The velocity of this signal is inferred, or, rather, decided, to be constant; its value is chosen at an arbitrary mathematical value. In this context, space is no longer measured by rigid rods but by the time of light travel. Practically, space cannot be defined without time because measurement operations of space take time; more fundamentally, in order space has a physical meaning, its different parts must be connected by the time- consuming travel of a signal. For a located observer, true space and the distant objects that constitute it are always discovered through a time mediation. Conversely, time is correlated to displacement of light in space. It may be defined as the position of a photon. This situation takes us back to Aristotle who defined time by movement, time not being intrinsic but defined by the relative position of the objects. In a way, time and space thus appear as two points of view on the same reality. Only the relative positions and relative displacements of physical entities (material and light entities will be treated equivalently in our discussion) matter to build the physical 2 appearance of our world and thus to define space and time. The velocity of material entities can be divided by an arbitrary factor, or any velocity may be subtracted from all the velocities and no apparent change will result. Equivalently, any quantity may be subtracted from all the positions or times, with no change. On the whole, there is an arbitrary both on origin and scale, for positions, times and velocities of physical entities, and, again, there is something too much in counting space and time independently. At last, according to the above modus operandi, light is a physical phenomenon to be described within our representation of the world, and at the same time the tool to measure space and time: therefore it is no longer possible to define separately the velocity of light and the value of space and time units; we are in a circle. In short, time and space are computed as independent parameters within a four- (and not three-) dimensional frame, and, in the same time, are linked at the basic definition level. In the present paper, we propose a first quantitative assessment of this paradox and examine the question whether something may be discussed about space-time dimension and about time and space parameter independence. The paper will remain at a cinematic level; dynamical and electrodynamical questions will only be alluded to. Our theory does not bear on observational data. We think it will help discuss, and possibly solve, several internal inconsistencies in Relativity Theory that have been raised by many authors, and also give a new content to the formulae of Relativity Theory, that will be partly modified. Outline of the paper In first step (Sect. 1), we start with a 3-dimensional spatial frame, called E3, wherein the position of each entity of the universe is merely defined by the distance from the origin of co-ordinates. Each spatial coordinate may be duplicated by the light travel time from the origin. This is a first way to express the fundamental link between space and time co-ordinates and space E3 thus already has a temporal meaning. On this basis, a phase-space may be constructed, called Q3N, with the 3N co-ordinates of the N entities composing the universe. Velocities need not be represented in this phase- space for reasons that will appear clear in the sequel. 3 The second step (Sect. 2) will consist in favouring, among all the entities of the preceding universe, and thanks to postulates that will be indicated, one specific entity that is moving by itself (e.g. a photon), in order to define a time for the whole frame. This is another way to say that time was already embodied in spaces E3 and Q3N; so, a phase-space with an additional time parameter of the type Q3N+1 might actually be replaced by a timeless phase- space Q3N, and, by way of consequence, one dimension of standard space- time E3+1 may be seen as superfluous with respect to the three dimensions of timeless space E3. In other words, if a space such as E3+1 is compulsorily needed to draw the trajectories of physical entities from a practical point of view, one must know that this space is in part redundant, because the spatial axes already have a temporal meaning and the time axis already a spatial meaning. In phase space Q3N the trajectories of the entities are simply defined with respect to the other entities. Provisionally, the three spatial coordinates of the above photon allow to define a three-dimensional pre-time (that might equivalently be defined by three photons moving along the three co-ordinate axes). Pre-time coordinates are merely position coordinates that are computed separately from other spatial parameters and that serve as a basis to choose the ordinary time parameter. Together with the three spatial coordinates of the entities, the three temporal co-ordinates of the photon define an intermediate 6-dimensional frame, E6, that proves to be useful, in particular, to discuss frame changes. In third step (Sect. 3), we will examine more fully the case when starting three-dimensional space E3 is non-Euclidean; in that case, one is naturally led to give two associated tensorial characters (covariant and contravariant respectively) for the three pre-time and the three space co-ordinates of the photon considered at previous step. In fourth step (Sect. 4), we will be ready to consider the case of one moving frame with respect to another, either in the E3 (or E6) spatio-temporal representation, or in the Q3N phase-space representation; because of the fundamental link between time and space parameters, such frame movement may equivalently be viewed as an intrinsic unprecision on the space and time measurements within one single frame. The following sections (5, 6 and 7) will give a physical, Lorentz- transformation type, interpretation to the preceding results and discuss 4 corresponding equations for the three-dimensional time case. One important contribution of the three dimensional pre-time is to help connect moving frames when the relative velocity vector is arbitrary in three dimensional space (and not parallel to one axis), a situation that causes some difficulties in the classical Relativity Theory. In next section (8), we will sum up how to get back to a four-dimensional space (transition from E6 to E4), with a time with one dimension instead of three.