On the ‘‘averaged weak energy condition’’ and Penrose’s singularity theorem

In a previous paper, we used techniques developed by Tipler to prove that Penrose's singularity theorem will still hold if the weak energy condition is replaced by a weaker nonlocal energy condition and if the null generic condition is satisfied. The purpose of the present paper is to strengthen our original result somewhat by slightly weakening the nonlocal energy condition and by showing that the null generic condition is unnecessary. We also comment briefly on whether recent criticisms of Tipler's techniques affect the validity of our results. In addition, we take the opportunity to rewrite some of our previous analysis in what we feel is a more direct mathematical language and, in so doing, we provide the details of a proof which were not included in our earlier paper.