Self-similar Groups Research Articles

A systematic approach to self-similarity in Newtonian space-time

After an introductory mathematical review of the general concept of self-similarity with respect to a given rescaling algebra, attention is focused on the case of Newtonian systems in Galilean space-time, whose self-similarity transformations will form subgroups of the maximal self-similarity group of the Galilean space-time structure itself. As a prerequisite for a systematic general investigation of self-similarity in such Newtonian systems, it is shown how an appropriately adapted similarity transporting coordinate system can be constructed explicitly for an arbitrary generator of the 12 parameter Galilean self-similarity group. A concrete application is provided by the case of the Keplerian disk, which is kinematically self-similar under the action of a three parameter subgroup. It is shown in the explicit example of a Eulerian fluid system how the generic self-similarity problem can be formulated as an effective stationarity problem.

Exact power law metrics in cosmology

Spatially homogeneous and non-exceptional spatially self-similar spacetime metrics which are 'exact power law metrics' are defined, explicitly parametrised and shown to have fixed conformal 3-geometry in the natural slicing of the spacetime by the orbits of the symmetry group and to admit a homothetic Killing vector field not tangent to that slicing. In fact the exact power metrics are exactly those spatially homogeneous and spatially self-similar metrics which admit a homothetic Killing vector field not tangent to the spacelike orbits of the homogeneity or self-similarity group. Such metrics arise as 'singular point solutions' of gravitational field equations when formulated as a certain system of first order differential equations. These special exact solutions play an important role in the qualitative behaviour of the general solution of a given set of field equations and sources with these symmetries.

Self-similar cosmological models

The kinematics and dynamics of self-similar cosmological models are discussed. The degrees of freedom of the solutions of Einstein's equations for different types of models are listed. The relation between kinematic quantities and the classifications of the self-similarity group is examined. All dust local rotational symmetry models have been found.