### 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 [112, 111]. 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.