Selfsimilar symmetry in Tolman models
Abstract
The conditions under which relativistic spherical distribution of dust possesses selfsimilar symmetry are studied. It is found that there are homothetic, partially homothetic, and conformal solutions depending on how the mass and energy are distributed through the matter. It is shown that the properties of such solutions are quite different from the properties of selfsimilar Newtonian dust. In particular, the 'big bang' hypersurface is entirely determined by the symmetry in the relativistic case, while it is entirely free in the Newtonian case. The similarity variable is determined uniquely by a simple differential equation. In all cases the space has similarity in the usual geometrical sense. The solutions are regular everywhere, except the hyperbolic and elliptic homothetic solutions which are singular at the center of symmetry. The results are used to gain insight into the nature of selfsimilarity. The generality of the results is discussed.
 Publication:

Monthly Notices of the Royal Astronomical Society
 Pub Date:
 May 1991
 DOI:
 10.1093/mnras/250.1.69
 Bibcode:
 1991MNRAS.250...69P
 Keywords:

 Einstein Equations;
 Interstellar Matter;
 Relativistic Theory;
 Similarity Theorem;
 SpaceTime Functions;
 Spatial Distribution;
 Big Bang Cosmology;
 Energy Distribution;
 Mass Distribution;
 Symmetry;
 Astrophysics