Title: The Cauchy Problem in Spacetimes with Closed Timelike Curves
by John Goodwin
Abstract: The Cauchy Problem for the scalar wave equation is studied in spacetimes with closed timelike curves. In general such spacetimes contain a chronal and non-chronal region, separated by a chronology horizon. We consider spacetimes that contain compactly generated horizons and those that do not. Initially we consider initial data for the wave equation on partial Cauchy surfaces in the chronal region. In Misner space we can write down an explicit solution to the wave equation in terms of the intial data. It is shown that generic initial data evolves to give a divergent stress-energy scalar at the chronology horizon. The Cauchy problem in the wormhole spacetimes is discussed in the geometric optics limit. We find that initial data can evolve in a finite way through the chronology horizon. Using a covering space we were able to convert the Cauchy problem in Gott space to an equivalent problem in Minkowski space. Modifying results in Hawking and Ellis we show that the Cauchy problem in Gott space is well posed up to and on the chronology horizon. We then considered extending the data for the wave equation beyond the chronology horizon in Gott and Grant space. Performing a change of coordinates so that the isometries of the spacetime are manifest in one periodic coordinate we simplify the wave equation to a reduced wave equation. This is an equation of mixed type changing from elliptic to hyperbolic. Similar things happen when we consider the wave equation in the spinning cosmic string spacetime. The surface of parabolic degeneracy coincides with the "0th-polarised hypersurface". The nature of the 0th-polarised hypersurface is discussed and we give a definition of this surface that is not dependent on the spacetime having isometries. We define a surface called "the essential chronology horizon" which we conjecture coincides with the 0-th polarised hypersurface for spacetimes without compactly generated chronology horizon and the chronology horizon for spacetimes with compactly generated chronology horizon.
Keywords: closed timelike curve, time travel, general relativity, causal structure, causality violation, Cauchy problem
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