A BEEMIAN SAMPLER: 1966-2002 23 On a very different theme, the many useful warped product curvature and connection formulae were exploited by Beem, Ehrlich and Powell [26] to derive an interesting result on the null geodesic incompleteness of Robertson-Walker space- times. For a Lorentzian warped product of the form (M = (a, b) x f H, -dt2 EB fh), where (H, h) is Riemannian, a given future directed null geodesic "Yo in M =(a, b) x JH may be reparametrized in the form "Y(t) = (t, c(t)), where /' is a smooth null pregeodesic in ( M, g). The expression of the connection '\7 on M in terms of the corresponding objects on the base and fiber leads directly to the equation - I 1 I [ ] I V--n It= 2f(t) I' (t), relating the warping function f to the scaling factor necessary for reparametrizing the pregeodesic /' into a null geodesic. Defining the monotone increasing (hence invertible) function p(t) =it VJ[ ) ds, wo the authors invoked a result from the classical differential geometry of curves to conclude that the curve f'Op- 1 (t) = (p- 1 (t),cop-1(t)) is a null geodesic. Defining A= lim p(t), and B = lim p(t), t-+a+ t-+b- one has a bijection p : (a, b) --- (A, B) and the reparametrized null geodesic /' o p- 1 : (A, B) --- M. Finiteness of A (respectively, B) thus ensures that M is past (respectively, future) null geodesically incomplete. The authors stated the result as follows in [26], and in pp. 109-110 of [24]: THEOREM 7. 5. Let M = (a, b) x f H be a warped product with Lorentzian metric g = -dt2 EB fh where -oo ::=: a b ::=: +oo, (H, h) is an arbitrary Riemannian manifold, and f : (a,b)--- (O,oo). Set S(t) = /TfJ5. Then if lim jwo S(s)ds t-+a+ t [respectively, limt_,.b- J~ 0 S(s) ds] is finite, every future directed null geodesic in (M, g) is past {respectively, future} incomplete. Recall that the Robertson-Walker solutions model the stress-energy content of the universe as a perfect fluid (U = Bt, p, p) with energy density p and pressure p. One relationship between the geometry (warping function f) and the physical content of the model (p and p) is expressed through the equation ( cf. [78], p. 346]) S" ( t) (7.1) 3 S(t) = -41r(p + 3p), where S(t) = /TfJ5, as above. Taken together, Equation (7.1) and Theorem 7.5 provide a direct connection between null geodesic completeness (and incom- pleteness) and the physical parameters which characterize the Robertson-Walker models. During his time at the University of Missouri, John Beem guided the research of six doctoral students in mathematics we conclude this section with a brief dis- cussion of their work. The inclusion of this material in this section is particularly appropriate since five of the six students worked on research topics involving warped
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