Marie-Noëlle Célérier

Quantum-classical transition in Scale Relativity

The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schr¨ odinger and Klein–Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using the mathematical tool of Hamilton’s bi-quaternions, leads us to a derivation of the Dirac equation within the scale-relativity paradigm. The complex form of the wavefunction in the Schr¨ odinger and Klein–Gordon equations follows from the non-differentiability of the geometry, since it involves a breaking of the invariance under the reflection symmetry on the (proper) time differential element (ds ↔− ds). This mechanism is generalized for obtaining the bi-quaternionic nature of the Dirac spinor by adding a further symmetry breaking due to non-differentiability, namely the differential coordinate reflection symmetry (dx µ ↔− dx µ ) and by requiring invariance under the parity and time inversion. The Pauli equation is recovered as a non-motion-relativistic approximation of the Dirac equation.

Timelike and null focusing singularities in spherical symmetry: A solution to the cosmological horizon problem and a challenge to the cosmic censorship hypothesis

Extending the study of spherically symmetric metrics satisfying the dominant energy condition and exhibiting singularities of power-law type initiated in SI93, we identify two classes of peculiar interest: focusing timelike singularity solutions with the stress-energy tensor of a radiative perfect fluid (equation of state:

The Pauli equation in scale relativity

p={1\over 3} \rho

Effects of inhomogeneities on the expansion of the Universe: A challenge to dark energy?

) and a set of null singularity classes verifying identical properties. We consider two important applications of these results: to cosmology, as regards the possibility of solving the horizon problem with no need to resort to any inflationary scenario, and to the Strong Cosmic Censorship Hypothesis to which we propose a class of physically consistent counter-examples.

A SOLUTION TO THE HORIZON PROBLEM : A DELAYED BIG BANG SINGULARITY

In standard quantum mechanics, it is not possible to directly extend the Schrodinger equation to spinors, so the Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. In the scale relativity framework, the Schrodinger, Klein–Gordon and Dirac equations have been derived from first principles as geodesics equations of a non-differentiable and continuous spacetime. Since such a generalized geometry implies the occurrence of new discrete symmetry breakings, this has led us to write Dirac bi-spinors in the form of bi-quaternions (complex quaternions). In the present work, we show that, in scale relativity also, the correct Pauli equation can only be obtained from a non-relativistic limit of the relativistic geodesics equation (which, after integration, becomes the Dirac equation) and not from the non-relativistic formalism (that involves symmetry breakings in a fractal 3-space). The same degeneracy procedure, when it is applied to the bi-quaternionic 4-velocity used to derive the Dirac equation, naturally yields a Pauli-type quaternionic 3-velocity. It therefore corroborates the relevance of the scale relativity approach for the building from first principles of the quantum postulates and the quantum tools. This also reinforces the relativistic and fundamentally quantum nature of spin, which we attribute in scale relativity to the non-differentiability of the quantum spacetime geometry (and not only of the quantum space). We conclude by performing numerical simulations of spinor geodesics, that allow one to gain a physical geometric picture of the nature of spin.

Redshift drift as a test for discriminating between different cosmological models

The current standard model of cosmology, the ΛCDM model, is based on the homogeneous FLRW solutions of the Einsteins equations to which some perturbations are added to account for the CMB features and structure formation at large scales. This model fits rather well to the observations, provided 95% of the energy density budget of the Universe should be of an unknown physical nature, i.e., dark matter and dark energy. Now, the aim of a cosmological model is not merely to reproduce the observations, but also to give a physical understanding of the Universe we live in. Moreover, even if the assumption of homogeneity seems to be more or less valid at large scales, it appears to be in contradiction with observations at intermediate scales (between the scale of non-linear structure formation and that where structures virialize). This is the reason why, during the last decade, a community of researchers has been formed, whose aim has been to look for the best way to take into account the influence of the inhomogeneities seen in the Universe, and to construct accurate cosmological models, which could possibly get rid of the dark components. This task, which is still in its infancy, is currently progressing towards promising results. Two types of methods can be found in the literature: Spatial averaging of scalar quantities, and use of exact inhomogeneous solutions of general relativity. We have given, here, a brief report of the second one.

Inhomogeneities in the Universe with exact solutions of General Relativity

Abstract One of the main drawbacks of standard cosmology, known as the horizon problem, was until now thought to be only solvable in an inflationary scenario. A delayed Big Bang in an inhomogeneous universe is shown to solve this problem while leaving unimpaired the main successful features of the standard model.

ELECTROMAGNETIC KLEIN–GORDON AND DIRAC EQUATIONS IN SCALE RELATIVITY

Models of inhomogeneous universes constructed with exact solutions of Einsteins General Relativity have been proposed in the literature with the aim of reproducing the cosmological data without any need for a dark energy component. Besides large scale inhomogeneity models spherically symmetric around the observer, Swiss-cheese models have also been studied. Among them, Swiss-cheeses where the inhomogeneous patches are modeled by different particular Szekeres solutions have been used for reproducing the apparent dimming of the type Ia supernovae (SNIa). However, the problem of fitting such models to the SNIa data is completely degenerate and we need other constraints to fully characterize them. One of the tests which is known to be able to discriminate between different cosmological models is the redshift-drift. This drift has already been calculated by different authors for Lema\^itre-Tolman-Bondi (LTB) models. We compute it here for one particular axially symmetric quasi-spherical Szekeres (QSS) Swiss-cheese which has previously been shown to reproduce to a good accuracy the SNIa data, and we compare the results to the drift in the

Relativistic corrections to fractal analyses of the galaxy distribution

\Lambda

REDSHIFT-DRIFT AS A TEST FOR DISCRIMINATING BETWEEN DECELERATING INHOMOGENEOUS AND ACCELERATING UNIVERSE MODELS

CDM model and in some LTB models that can be found in the literature. We show that it is a good discriminator between them. Then, we discuss our models remaining degrees of freedom and propose a recipe to fully constrain them.