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3.5 Cylindrically symmetric solutions

Solutions of the Einstein equations with cylindrical symmetry that are asymptotically flat in all directions allowed by the symmetry represent an interesting variation on asymptotic flatness. There are two Killing vectors, one translational (without fixed points) and one rotational (with fixed points on the axis). Since black holes are apparently incompatible with this symmetry, one may hope to prove geodesic completeness of solutions under appropriate assumptions. (It would be interesting to have a theorem making the statement about black holes precise.) A proof of geodesic completeness has been achieved for the Einstein vacuum equations and for the source-free Einstein-Maxwell equations in [51], building on global existence theorems for wave maps [112111]. For a quite different point of view on this question involving integrable systems see [351]. A recent paper of Hauser and Ernst [171] also appears to be related to this question. However, due to the great length of this text and its reliance on many concepts unfamiliar to this author, no further useful comments on the subject can be made here.

Solutions of the Einstein-Vlasov system with cylindrical symmetry have been studied by Fjällborg [140]. He shows global existence provided certain conditions are satisfied near the axis.

Cylindrical symmetry can be generalized by abandoning the rotational Killing vector while maintaining the translational one. This sitation does not seem to have been studied in the literature. It may be that results on solutions with approximate cylindrical symmetry may be obtained using the work of Krieger [218] on wave maps.


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