On the structure of conformal singularities in classical general relativity

Abstract

Consideration is given to Big Bang singularities which can be conformally transformed to a spacelike hypersurface, with the conformal factor and conformal metric both smooth on the extended manifold. A precise definition of such ‘conformal singularities’ is proposed by analogy with the established definition of conformal infinity. The energy tensor of the physical space-time is assumed to have a form appropriate to an isentropic perfect fluid. The smoothness condition implies that the adiabatic index of this fluid must tend, at the singularity, to the value 4/3 appropriate to a highly relativistic initial state. The smoothness condition also facilitates a natural choice of conformal factor in terms of the Synge index of the fluid. With this choice, the four-dimensional curvature is uniquely determined at the initial hypersurface by the induced 3-metric, independently of the equation of state, subject to a condition on the rate at which the adiabatic index approaches its limiting value. The study refines, in the smooth case, previous work of Goode & Wainwright (1985) concerning ‘isotropic singularities’. (Justification is given for the new terminology.) The work provides the foundation for a subsequent investigation of the Cauchy problem for space-times which originate from a Big Bang of conformal type.