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 jinile cylinder where g, becomes negative, implying causality violation.

Since H0 for 0, 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 g is negative and return to any other event." 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 w=0. Then a little manipulation yields

u = A (lnr) +B, k = A'(lnr) +C, a=D,

where A, B, C, D are constants.

By the transformation =t' - ap, 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 = e` p = (wr), so that u=-in(v), and d/dr = 2w3r(d/dp), which gives

*Work supported in part by the National Science Foundation under Grant No. GP-25548.

19. W. Hawking and R. Penrose, Proc. R. Soc. Lond. A 314, 529 (1970).

C. 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/ǝz, /, /ay) is timewise orthogonally transitive and Abelian. It is easily checked that for >1/4 in (2) and a R> in (3), there is no linear combination & At + B + Cz (where A, B, C are constants) such that the form de is everywhere timelike.

K. Gödel, Rev. Mod. Phys. 21, 447 (1949). 'R. M. Wald, Phys. Rev. Lett. 26, 1653 (1971). C. W. Misner, Phys. Rev. Lett. 28, 994 (1972). *For a2 + e2 >m2 there are no event horizons and so causality violation is global, but it is not clear that a star with such high values of angular momentum and/or charge would collapse sufficiently far to uncover the

region where g changes sign (see Ref. 7). Penrose has argued (in Proceedings of the Sixth Texas Symposium on Relativistic Astrophysics, 1972 (unpublished)] that a naked Kerr singularity would be a good model for a rapidly rotating star which has collapsed into a disk. CTL would be expected when e 0, but one might contend that these occur so close to the singularity (and hence in regions where we expect general relativity to break down anyway) that they are without physical significance. van Stockum's work shows, however, that CTL are not necessarily associated with extreme curvature in physically significant situations.

1°C. W. Misner, in Astrophysics and General Relativity, edited by M. Chrétien, S. Deser, and J. Goldstein (Gordon and Breach, New York, 1969), Vol. 1. W. J. van Stockum, Proc. R. Soc. Edinb. 57, 135 (1837). Other authors, such as S. C. Maitra (J. Math. Phys. 7, 1025 (1966)) have noted that the van Stockum interior solution possesses CTL.

12J. Ehlers and W. Kundt, in Gravitation: An Introduc • tion to Current Research, edited by L. Witten (Wiley, New York, 1962), p. 84.

13H. Levy and W. J. Robinson, Proc. Camb. Philos. Soc. 60, 279 (1964).

II. Davies and T. A. Caplan, Proc. Camb. Philos. Soc. 69, 325 (1971), E. Frehland, Commun. Math. Phys. 23, 127 (1971).

K. S. Thorne, Comments Astrophys. Space Phys. 191 (1970).

W. Israel, Nature 216, 149 (1967); 216, 312 (1967). Assuming, of course, that the cylinder has existed for all time. If it is created, then this statement will have to be qualified somewhat, but observable causality violation will still occur.