5.2 Evolution of hyperboloidal data5 Further results5 Further results

5.1 Isotropic singularities 

The existence and uniqueness results discussed in this section are motivated by Penrose's Weyl curvature hypothesis. Penrose suggests that the initial singularity in a cosmological model should be such that the Weyl tensor tends to zero or at least remains bounded. There is some difficulty in capturing this by a geometric condition and it was suggested by [80] that a clearly formulated geometric condition which, on an intuitive level, is closely related to the original condition, is that the conformal structure should remain regular at the singularity. Singularities of this type are known as conformal or isotropic singularities.

Consider now the Einstein equations coupled to a perfect fluid with the radiation equation of state tex2html_wrap_inline847 . Then it has been shown [61, 36] that solutions with an isotropic singularity are determined uniquely by certain free data given at the singularity. The data which can be given is, roughly speaking, half as large as in the case of a regular Cauchy hypersurface. The method of proof is to derive an existence and uniqueness theorem for a suitable class of singular hyperbolic equations. Generalizations of this by Anguige and Tod have been discussed in [79]. Details will be given in Anguige's thesis. Related work was done earlier in a somewhat simpler context by Moncrief[59] who showed the existence of a large class of spacetimes with Cauchy horizons.



5.2 Evolution of hyperboloidal data5 Further results5 Further results

image Local and global existence theorems for the Einstein equations
Alan D. Rendall
http://www.livingreviews.org/lrr-1998-4
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