26 PAUL E. EHRLICH AND KEVIN L. EASLEY where the gravitational plane waves fit in "the ladder of causality." Thus, the first author deduced that this figure had apparently taken on the stature of the canon against which things were to be tested! This impression was only further confirmed when the first author and his wife were dining the Friday evening before the Beemfest opened with Professors Beem, Dostoglou, and most of the other conference participants. Suddenly, to his amaze- ment, two of the participants started talking animatedly about "the ladder of causality" and wondering whether "causally simple" was on the table, trying to recall the precise definition of this concept. The first author had already shown his two transparencies on which this section is based to his wife, explaining that he thought he could end on a humorous note. "See what I mean," he said to her, "that is exactly what I showed you several days ago." But he did not enlighten the other two conference participants as the the meaning of his remarks, not wanting to give away the conclusion of his lecture. As a final example, during the academic year 2002-2003 the first author had somewhat trying circumstances during which the air conditioning and heating of his building were renovated at the University of Florida, while he and his colleagues shifted around amongst the building from semester to semester sharing offices as various portions were worked on. Thus, the first author found himself during the Fall semester, 2002, relocating across the building and sharing an office with Pro- fessor Paul Robinson. The first author had long felt guilty since he had not given Paul a copy of the Second Edition when we received our complimentary copies, since Robinson's area of research was symplectic geometry, and other members of the Florida group worked on topics which were more closely allied to the materials treated in this monograph. But finally an extra copy of this expensive book had come into the first author's possession. He told Paul Robinson that to reward him for being such a fine office mate, he would present him with an autographed copy of the book in December, 2002, just as we were preparing to vacate that part of Little Hall and return to our old offices for the Spring semester. Paul gladly accepted this present and started reading away. The first author was pleased to find that Robin- son was quite taken with some new material presented on pages 26-28 regarding the well-known folk result in General Relativity that the null vectors determine the metric up to a conformal factor. By early January, Robinson had produced a preprint "Null cones and conformality" giving three more alternative proofs with a greater degree of abstraction and conceptualization. References 1. D. E. Allison, Lorentzian Warped Products and Static Space-Times, Ph.D. Thesis, University of Missouri-Columbia, 1985. 2. D. E. Allison, Geodesic completeness in static space-times, Geom. Dedicata 26 (1988), 85-97. 3. D. E. Allison, Energy conditions in standard static space-times, Gen. Rei. Grav. 20 (1988), 115-122. 4. D. E. Allison, Pseudo-convexity in Lorentzian doubly warped products, Geom. Dedicata 39 (1991), 223-227. 5. D. E. Allison, Lorentzian Claimut submersions, Geom. Dedicata 63 (1996), 309-319. 6. D. E. Allison and B. Una!, Geodesic structure of standard static space-times, J. Geom. Phys. 46 (2003), 193-200. 7. T. Aubin, Metriques Riemannienes et Courbure, J. Diff. Geom. 4 (1970), 383-424. 8. R. Bartnik, Existence of maximal surfaces in asymptotically flat space-times, Commun. Math. Phys. 94 (1984), 155-175.

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