9 ROTATING CYLINDERS AND THE POSSIBILITY OF GLOBAL... In addition to the already mentioned static nature of the field, there is the fact that it is not even asymptotically Minkowskian (especially when aR >!). Still, the gravitational potential of the cylinder's Newtonian analog also diverges at radial infinity, yet this potential is a good approximation near the surface in the middle of a long but finite cylinder, and if we shrink the rotating cylinder down to a "ring" singularity, we end up with the Kerr field, which also has CTL. These facts suggest that there is a region near the surface of a jinite cylinder where g, becomes negative, implying causality violation. 16 Since H0 for r0, there are no event horizons around the infinite cylinder. By analogy with the static case, I expect this to be true for a finite cylinder; if so, then a timelike line from any event in the universe could enter the region where A is negative and return to any other event."7 In short, general relativity suggests that if we construct a sufficiently large rotating cylinder, we create a time machine. I would like to thank Dr. D. Schmidt for helpful discussions, and Professor D. R. Brill for reading the manuscript. 2205 (A4) Suppose first that w0. Then a little manipulation yields u = A (lnr) +B, k = A2(lnr) + C, a=D, where A, B, C, D are constants. By the transformation tap, 4 =4', 2 = 2', r=r', we discover that except for global topology this solution is just the Weyl solution (3a). Suppose now that w0. It is at this point that Davies and Caplan err; their "general" solution in fact places implicit restrictions on the value of their constant A. The complete general solution is obtained via the following procedure. Let v=eTM*, p = (wr), so that u = -in(v), and d/dr =2w'r(d/dp), which gives d'a r dr d'u 1 du da + + 2,5 dr = 0, (A1) (A2) dr dk r dr dr dr 2-2(d) + 2(d)-0. We have three coupled equations for three functions: second order in u, second order in a, first order in k. Thus we expect five arbitrary constants. A general physical solution to the above system will be defined to be a set of functions a, u, k in which the five constants are allowed to assume all real values from to. I will show that this general solution is given by Eqs. (3a)(3c). Equation (A2) can be written Let = ln(p), d/dp = (1/p)d/dt. (A6) becomes (Q/w)2 = 4 w ± A', which can be written & which is identical to Eq. (2.3) of Davies and Caplan (in Ref. 14). The computation proceeds as they outline to obtain k and a. Frehland has shown that this solution is the same as the Weyl solution (3a). Suppose now that A-0. We get Ru-r(1-Ew Inw'r 'D'), B1 = wr In(w3y 2D2), 8. = -rE(2 +Ew Inw3r 3D2), 10 (A9) where E, D, F are constants. These solutions are identical to (3b), with a suitable choice of constants. Suppose now that the integration constant is -A'. We obtain 2wr cos(In(w^r^)+C), A *Work supported in part by the National Science Foundation under Grant No. GP-25548. 18. W. Hawking and R. Penrose, Proc. R. Soc. Lond. A 314, 629 (1970). IC. W. Misner, in Relativity Theory and Astrophysics: Relativity and Cosmology, edited by J. Ehlers (American Mathematical Society, Providence, R.I., 1967), Vol. 8, p. 160. 'R. Geroch, Ann. Phys. (N.Y.) 48, 526 (1968). 'B. Carter, Phys. Lett. 21, 423 (1966). B. Carter, Phys. Rev. 174, 1559 (1968). Carter's causality theorem can be stated as follows: A necessary and sufficient condition for nontrivial causality violation in a connected, time-oriented spacetime with a timewise orthogonally transitive Abelian isometry group is the nonexistence of a covariant vector in the Lie algebra such that the corresponding differential form in the surface of transitivity is everywhere well behaved and everywhere timelike. If the above criterion is satisfied, then there exist both future- and pastdirected timelike lines between any two points of the spacetime. For the van Stockum metrics (2) and (3a)-(3b), the group generated by the Killing vectors (8/ə z ̧ à3⁄4 ̧ a/a) is timewise orthogonally transitive and Abelian. It is easily checked that for >1/a in (2) and aR> in (3), there is no linear combination +By+ Cz (where A, B, C are constants) such that the form de is everywhere timelike. At &u= &r = Fy−(1 + ^®) /2 ̧ 8,, = r { sin( In(w^r^) +C] + D cos( In(w^r^)+C}}. g is determined by the relation FL + M3 =y', where A, C, D, F are constants. (A10) Thus the general exterior field is given by (3). 'K. Gödel, Rev. Mod. Phys. 21, 447 (1949). region where g changes sign (see Ref. 7). Penrose situations. 1oC. W. Misner, in Astrophysics and General Relativity, 12J. Ehlers and W. Kundt, in Gravitation: An Introduc - 13H. Levy and W. J. Robinson, Proc. Camb. Philos. I. Davies and T. A. Caplan, Proc. Camb. Philos. Soc. 15. S. Thorne, Comments Astrophys. Space Phys. W. Israel, Nature 216, 149 (1967), 216, 312 (1967). "Assuming, of course, that the cylinder has existed for all time. If it is create, then this statement will have to be qualified somewhat, but observable causality violation will still occur. To perhaps oversimplify one aspect of the question of future space programs, I would like to present three so-called "alternative futures", not just for the United States, but for the world. I think which future becomes reality may depend to a surprisingly significant extent on what is decided in the near future by the relatively few members of this committee (compared to over 4,000 million members of the human species). It is perhaps appropriate rather than with the Senate, the President, or even the United Nations. One possible future is for NASA to continue at its present more or less subdued pace. In light of the testimony space offers to industry, employment, and the human condition this approach is clearly irresponsible and will not be discussed further herein. The other approach, then, is for NASA to quicken its pace perhaps, for example, taking a realistic and flexible 2 step-by-step approach such as that previously detailed in these hearings by Gerard K. O'Neill or G. Harry Utine. The real question in my mind therefore becomes whether we will treat the so-called "high frontier" program as a space program or a people program. By "treating it as a space program", I mean looking at it from a narrow technological short-run point of view which no doubt will aid industry, increase employment, and probably improve the human condition and devoid of any overarching long-term ethical (rather than technical) goal. I propose a more comprehensive human long-term "high frontier" program with an explicitly stated overall ethical goal and time table to strive for. Does it really make much difference whether we take a "space program" or "people program" approach? I suggest It is not often we get a chance to eliminate all human poverty and end all war. Specifically, for the first time in human history we now have via space and space science the practical means to acheive what the social scientists have only speculated about. We may not get another chance in centuries, if ever. There are realistic reasons why we can stop spreading poverty and war beyond our tiny planet when we have been unable to end it aboard ship. As urban planners have noted it would be easier to build a new 3 city rather than have to work with a decaying old one. Re "fighting" the population problem and giving "utopia" (democracy) a fair chance, the settling of America (not merely the slow evolution of Europe) was the road (unconsciously) taken. The difficult job of settling the American frontier seemed to a lot of people at the time to be a silly, nutty idea. The United States has been called the first new nation. And because we were new, we could idealistically but perhaps for the first time realistically give our ideals a fair ("neutral") chance. At the time, the miraculous results greatly surprised a large number of supposedly intelligent level-headed people. But we have come a long way in 200 years. The question today is not whether a non-democratic government is a necessary evil. Democracy was very idealistic - but perhaps it needed a new land, a new frontier, to also prove itself realistic. Today the question is whether we will end all human poverty and war. With only a few exceptions, space scientists admit that technologically we theoritically can end all human poverty via "the high frontier", possibly by the mid-21st century. There are no doubt various space programs (past, present, future) which are exceedingly justified. These programs are not discussed herein. The program herein under discussion should more properly be viewed as a people program, not |
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