Cauchy problem in spacetimes with closed timelike curves
Abstract
The laws of physics might permit the existence, in the real Universe, of closed timelike curves (CTC's). Macroscopic CTC's might be a semiclassical consequence of Planckscale, quantum gravitational, Lorentzian foam, if such foam exists. If CTC's are permitted, then the semiclassical laws of physics (the laws with gravity classical and other fields quantized or classical) should be augmented by a principle of selfconsistency, which states that a local solution to the equations of physics can occur in the real Universe only if it can be extended to be part of a global solution, one which is well defined throughout the (nonsingular regions of) classical spacetime. The consequences of this principle are explored for the Cauchy problem of the evolution of a classical, massless scalar field Φ (satisfying □Φ=0) in several model spacetimes with CTC's. In general, selfconsistency constrains the initial data for the field Φ. For a family of spacetimes with traversible wormholes, which initially possess no CTC's and then evolve them to the future of a stable Cauchy horizon scrH, selfconsistency seems to place no constraints on initial data for Φ that are posed on past null infinity, and none on data posed on spacelike slices which precede scrH. By contrast, initial data posed in the future of scrH, where the CTC's reside, are constrained; but the constraints appear to be mild in the sense that in some neighborhood of every event one is free to specify initial data arbitrarily, with the initial data elsewhere being adjusted to guarantee selfconsistent evolution. A spacetime whose selfconsistency constraints have this property is defined to be ``benign with respect to the scalar field Φ.'' The question is posed as to whether benign spacetimes in some sense form a generic subset of all spacetimes with CTC's. It is shown that in the set of flat, spatially and temporally closed, 2dimensional spacetimes the benign ones are not generic. However, it seems likely that every 4dimensional, asymptotically flat spacetime that is stable and has a topology of the form R×(Sone point), where S is a closed 3manifold, is benign. Wormhole spacetimes are of this type, with S=S^{1}×S^{2}. We suspect that these types of selfconsistency behavior of the scalar field Φ are typical for noninteracting (linearly superposing), classical fields. However, interacting classical systems can behave quite differently, as is demonstrated by a study of the motion of a hardsphere billiard ball in a wormhole spacetime with closed timelike curves: If the ball is classical, then some choices of initial data (some values of the ball's initial position and velocity) give rise to unique, selfconsistent motions of the ball; other choices produce two different selfconsistent motions; and others might (but we are not yet sure) produce no selfconsistent motions whatsoever. By contrast, in a pathintegral formulation of the nonrelativistic quantum mechanics of such a billiard ball, there appears to be a unique, selfconsistent set of probabilities for the outcomes of all measurements. This paper's conclusion, that CTC's may not be as nasty as people have assumed, is reinforced by the fact that they do not affect Gauss's theorem and thus do not affect the derivation of global conservation laws from differential ones. The standard conservation laws remain valid globally, and in asymptotically flat, wormhole spacetimes they retain a natural, quasilocal interpretation.
 Publication:

Physical Review D
 Pub Date:
 September 1990
 DOI:
 10.1103/PhysRevD.42.1915
 Bibcode:
 1990PhRvD..42.1915F
 Keywords:

 04.60.+n;
 03.70.+k;
 04.20.Cv;
 Theory of quantized fields;
 Fundamental problems and general formalism