Geodesic
Cauchy problem on non-globally hyperbolic space-times
Teoretičeskaâ i matematičeskaâ fizika, Tome 157 (2008) no. 3, pp. 334-344

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We consider solutions of the Cauchy problem for hyperbolic equations on non-globally hyperbolic space-times containing closed timelike curves (time machines). We prove that for the wave equation on such space-times, there exists a solution of the Cauchy problem that is discontinuous and in some sense unique for arbitrary initial conditions given on a hypersurface at a time preceding the formation of closed timelike curves. If the hypersurface of initial conditions intersects the region containing closed timelike curves, then the solution of the Cauchy problem exists only for initial conditions satisfying a certain self-consistency requirement.
Keywords: cauchy problem, non-globally hyperbolic space-time, closed timelike curve. 
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     author = {I. Ya. Aref'eva and I. V. Volovich and T. Ishiwatari},
     title = {Cauchy problem on non-globally hyperbolic space-times},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {334--344},
     publisher = {mathdoc},
     volume = {157},
     number = {3},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2008_157_3_a1/}
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I. Ya. Aref'eva; I. V. Volovich; T. Ishiwatari. Cauchy problem on non-globally hyperbolic space-times. Teoretičeskaâ i matematičeskaâ fizika, Tome 157 (2008) no. 3, pp. 334-344. http://geodesic.mathdoc.fr/item/TMF_2008_157_3_a1/