Principe cosmologique,principe variationnel et theorie des groupes

The Friedmann universes are built on the cosmological principle only. The Robertson-Walker metric is common to all the theories based on a homogeneous, isotropic and irrotational universe. In the present work we examine ways of constructing a metric conformal with that of Robertson and Walker, by means of a variational principle which takes into account the cosmological principle as stated by Weinberg (1972), and based on the existence of orbits generated by a one-parameter group of diffeomorphisms of physical space. The application of the cosmological principle to variational methods allows the determination of first integrals which can characterize the physical properties of the Universe. To this end, we show that the Lagrangian of the Universe, considered as a mechanical system, can be chosen from the germs of functions, and that the form variations δq ⁱ are tangent vectors of the group orbits in a Riemannian manifold. Thus the variation of the action vanishes automatically. There appears a first integral of the Euler equations, which is δq ⁱ (?L/?q1 ⁱ ) = C ^te , and also the condition ?L/?t=0, which means the uniformity of time in a Lagrangian conservative system, and which is a direct application of the cosmological principle. These conditions allow the effective determination of a form invariant Lagrangian in the case of isometries. These conditions can be generalized to the case in which the group trajectories are a partition of physical space. Thus, it is possible to define a time from the group trajectories inV ₃: a second of the group time is a lengthm measured along any orbit θ _p of the group. Any pointp of the manifold can then be considered as the starting point of a bundle of orbits, along which the tangent vectors δq ⁱ could be calculated. From this group time, we can build a metric ds ² conformal to the initial ds ² and for which the orbits, which are geodesic, are orthogonal to the transitivity surfaces of the group in the manifold. This implies new statements of the cosmological principle:

1. At any point of space-time it is possible to construct a metric ds ² from the trajectories generated by a one-parameter group of diffeomorphisms ofV ₄.
2. Any two points of space-time can always be joined by means of trajectories of group.

The variational implications of these two principles are the appearance of spectral line shifts such as 1+z=F(p, t _p)/F(q, t_q), wherep andq are arbitrary points of the manifold, andF the transformation function which allows passage from one metric to another. The identification of group trajectories with physical trajectories depends on these two principles. The photon trajectories inV ₃ is an example of this identification. The trajectories of charged particles inV ₄ are another. Principle (b) stated an entropy condition; its application allows a new expression of action variation, this one leading to a general formulation of the shift of spectral lines by a variational method. If we choose the parabolic Friedmann universe as a realistic model, it is the expansion itself which is the generator of the diffeomorphisms allowing the establishment of a group structure in the manifold. The photons are carried away by expansion and do not resist it. The massive particles moderate this expansion locally, and their trajectories inV ₃ are the result of the reaction. In this scheme there is no theoretical difference between the treatment of particles of vanishing proper mass and massive particles. The Robertson-Walker metric fork=0 corresponds to a picture of the Universe which can be drawn by study of the movement of photons in physical space. Only the study of particles can allow the generalization of this scheme and, from this, make a real Universe which is not just a reflection of the physical properties of the photons alone.