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In the first part, after showing that the most natural approach to define anorder on sets of conformal classes fails, we define a nontrivial order $\leq_2$on the set of conformal classes of compact Cauchy slabs with fixed pastboundary. The second part, using the conformal ideas of the first part, provesBartnik's splitting conjecture for a conformally invariant set $F$ ofspacetimes larger than the one considered by Bartnik in \cite{rB2}, and $F$ isstable under $\leq_2$. In the third part we show that if we replace the strongenergy condition with the null energy condition, then in any dimension greateror equal to $3$ the conclusion of Bartnik's splitting conjecture would bewrong, more precisely: On a manifold of dimension $\geq 3$, {\em every}globally hyperbolic spatially compact conformal class contains future completemetrics satisfying the null energy condition. In the spatially noncompact case,the same is true in the future of any Cauchy surface.

Orders on sets of conformal classes applied to Bartnik's conjecture