The Need for a New Extension of Principle of Relativity

Prior to setting the principle of relativity, there is the definition of coordinate systems and of the possible transformations between these systems. Indeed this principle is a statement about the universality of the laws of physics, whatever the system of coordinates in which they are expressed. So let us try to analyse further what we mean by a system of coordinates. Physics is, above all, a science based on measurements. Its laws apply not to objects by themselves, but rather to the numerical results of measurements which have been or may be performed upon these objects. So the definition of coordinate systems should include all the relevant information which is necessary to describe these results and to relate them in terms of physical laws.

It is an experimental fact that four numbers are necessary and sufficient to locate an event (i.e., a position and an instant): space-time is of topological dimension 4. This operation of location of an event is found to have the following properties:

(i) It cannot be made in an absolute way. This means that an event can be located only with respect to another event, never to some absolute position or instant. What are measured are always space intervals and time intervals. This relativity of events implies that coordinate systems must be firstly characterized by the setting of an origin, O.

(ii) Then one needs to define the axes of the coordinate system. They may be rectilinear, but more generally curvilinear. This means that space-time is covered by a continuous grid or lattice of lines (i.e., of topological dimension 1). In present physics, this curvilinear coordinate system is also assumed to be differentiable.

(iii) We want to characterize by numerical values the position and instant of a second event with respect to O. However length and time intervals are themselves relative quantities: there is indeed no absolute scale in Nature. This second relativity, which we shall call "relativity of scales", is currently translated by the need to use some units for measuring length and time intervals. But we shall argue in the following that its consequences for laws of physics may be far more profound.

(iv) A last property of space-time coordinate measurements (and of any measurement) is that they are always made with some finite resolution. We claim that resolution should be included in the definition of coordinate systems.^32,33 Being itself a length or time interval, it is subjected to the relativity of scales. This resolution corresponds to the minimal unit which may be used when characterizing the length or time interval by a final number: e.g., if the resolution of a rod is 1 mm, it would have no physical meaning to express a result in Å. This resolution sometimes corresponds to the precision of the measuring apparatus: it may then eventually be improved, this corresponding to an improved precision of the result in classical physics. But it may also correspond to a physical limitation. For example it is probable that the distance from the Sun to the Earth would never be measured with a precision of 1 Å: this would have no physical meaning. And last but not least, resolution of the measurement apparatus plays in quantum physics a completely new role with respect to the classical, since the results of measurements become dependent on it, as a consequence of Heisenberg's relations.

It is well-known that any set of physical data takes its complete sense only when it is accompanied by the measurement errors or uncertainties, and more generally by the resolution characterizing the system under consideration. Complete information about position and time measurement results is obtained when not only space-time coordinates (t, x, y, z), but also resolutions (Dt, Dx, Dy, Dz) are given. Though this analysis already plays a central part in the theory of measurement and in the interpretation of quantum mechanics, one may remark however that its consequences for the nature of space-time itself have still not been drawn: we suggest that this comes from the fact that, up to now, resolutions have never appeared explicitly in the definition of coordinate systems, while, as shown hereabove, they are explicitly related to the information which is relevant for our understanding of the meaning of actual measurements.

Once the properties of coordinate systems defined, the next task is to describe the possible transformations that are acceptable between these systems. These transformations change the various quantities which define the state of coordinate systems, i.e., following the above analysis: origin, axes, units and resolution.

First consider changes of origin. The invariance of physical laws under static changes of the origin of coordinates systems is translated in terms of homogeneity of space and uniformity of time (more generally, homogeneity of space-time). This is the basis, under Noether's theorem, for the conservation of momentum and energy. So the very existence of energy-momentum as a fundamental conservative quantity (which is itself identified as the charge for gravitation in general relativity) relies on the first relativity, that of position and instants. Velocity-dependent changes of origin may also be considered in space: this leads to Galileo's relativity of inertial motion. But they may also be subsequently included into static rotations in space-time, so this leads us to the second transformation, that of axes.

Axes transformations first include changes of orientation. The invariance of physical laws under rotations in space corresponds to the isotropy of space and yields the conservation of angular momentum. Including rotations in space-time in the transformations considered allows one to describe the relativity of motion as a relativity of orientations in space-time. This yields the Lorentz transform. Finally Einstein's special theory of relativity accounts, in terms of the Poincaré invariance group, for the full relativity of positions, instants and axis orientation.

Then including any continuous and differentiable transformation yields Einstein's general relativity: the change to curvilinear coordinate systems introduces not only non-inertial motion but also curvature of space-time that manifests itself as gravitation.

On this road, it is clear that if one still wanted to generalize the class of acceptable transformations, one should give up differentiability, and then, as a last possibility, continuity (recall that non-differentiability does not imply discontinuity). Let us call "extended covariance" the covariance of the equation of physics under general continuous transforms, including non-differentiable ones. It is also clear that the achievement of such an extended covariance would imply a profound change in the physico-mathematical tool, since the whole mathematical physics is currently founded on integro-differentiation. In this book, we shall try to convince the reader that such an achievement is indeed necessary for our understanding of the foundation of the laws of Nature, in particular in the microphysical domain.

Let us indeed point out what may be considered as remaining defects of the present state reached by physics, which indicate that the principle of relativity of motion itself needs to be extended. Although it became definitively clear after Mach's and Einstein's analysis that the concept of an absolute space-time was to be given up and superseded by a space-time depending on its material and energetic content, the present quantum theory of microphysics still assumes space-time to be Minkowskian, i.e. absolute. This is at variance with the radically new quantum properties of matter and energy, as compared to the classical ones on which the special and general theories of relativity were founded. In other words, the Minkowskian and Riemannian nature of space-time was deduced from the classical properties of objects. In the quantum domain, we know that all objects have quantum properties (all are subjected to Heisenberg's inequalities). We also know that the structure of space-time must depend on its material and energetic content: how, under these conditions, can a space-time whose content is universally quantal be Minkowskian, i.e. flat and absolute?

An additional remark may be made. The goal of a completely general relativity cannot presently be considered as reached, since it is clear that the methods of the present theory of general relativity do not apply to reference frames which would be swept along in the quantum motion, which is continuous but non-differentiable, as discovered by Feynman.^34,35 This non-differentiability of virtual and real quantum paths is one of the key points to our own approach. We shall at length come back on it, showing in particular that it can be described in terms of Brownian motion-like fractal properties.^36,33

Basing ourselves on these considerations, we have suggested that the principle of relativity still needs to be extended.^32,33 Our concept of space-time has evolved from the Galilean independent space and time, to the Minkowskian absolute space-time, then to the Riemannian relative space-time of Einstein's theory. If one wants to include the non-differentiable fractal quantum motion into those described by a theory of relativity, a radically new geometrical structure of space-time must be introduced. Our suggestion is that the quantum space-time is relative and fractal,³³ i.e., divergent with decreasing scale (we shall adopt this definition of the word "fractal" here; see Mandelbrot^37,38 for other definitions). We shall indeed demonstrate (see [57] Sec. 3.10) that continuity and non-differentiability implies scale divergence. The same conclusion concerning the fractal stucture of space-time will be reached in the next Section, by basing ourselves on the relativity of all scales in Nature.

In our approach, throughout the present essay, we assume space-time to remain a continuum, even if it is no longer assumed to be differentiable. An ultimate choice for physics would be to give up the hypothesis of continuity itself. Some attempts to introduce discontinuous space-times have been made.^39,27-29 In this respect let us quote one of these attempts, Moulin's "arithmetic relators", which are defined using purely integer numbers.⁴⁰ Arithmetic relators are quadratic cellular automata which include internal variables and environment variables. They have proved to be efficient for providing structures, in particular biological ones.⁴¹ Such an ability of making structures emerge from very few conditions is reminiscent of "mappings" often used in the study of dynamical chaos. In particular, arithmetic relators yield a hierarchisation, i.e., the various structures appear at different levels of imbrication.

We shall adopt here a more conservative point of view, by keeping the space-time continuum hypothesis and by including the scale dependence in the fundamental principle themselves.
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Quoted from: Nottale L., 1993, "Fractal Space-Time and Microphysics: Toward of Theory of Scale Relativity", Chap. 2 (© World Scientific, 1993).