Existence and uniqueness theorems for massless fields on a class of spacetimes with closed timelike curves

TL;DR

This paper investigates the existence and uniqueness of solutions for massless scalar fields on certain asymptotically flat spacetimes with closed timelike curves, providing conditions under which solutions exist and are unique, and discussing extensions and conjectures.

Contribution

It establishes existence and restricted uniqueness theorems for massless scalar fields on specific spacetimes with CTCs, and explores extensions to other fields and the well-posedness of the Cauchy problem.

Findings

Existence of smooth solutions with given initial data on certain spacetimes with CTCs.

Restricted uniqueness of solutions in regions outside the CTCs.

Counterexamples showing failure of existence and uniqueness under weaker conditions.

Abstract

We study the massless scalar field on asymptotically flat spacetimes with closed timelike curves (CTC's), in which all future-directed CTC's traverse one end of a handle (wormhole) and emerge from the other end at an earlier time. For a class of static geometries of this type, and for smooth initial data with all derivatives in $L_2$ on ${\cI}^{-}$, we prove existence of smooth solutions which are regular at null and spatial infinity (have finite energy and finite $L_2$-norm) and have the given initial data on $\cI^-$. A restricted uniqueness theorem is obtained, applying to solutions that fall off in time at any fixed spatial position. For a complementary class of spacetimes in which CTC's are confined to a compact region, we show that when solutions exist they are unique in regions exterior to the CTC's. (We believe that more stringent uniqueness theorems hold, and that the present limitations are our own.) An extension of these results to Maxwell fields and massless spinor fields is sketched. Finally, we discuss a conjecture that the Cauchy problem for free fields is well defined in the presence of CTC's whenever the problem is well-posed in the geometric-optics limit. We provide some evidence in support of this conjecture, and we present counterexamples that show that neither existence nor uniqueness is guaranteed under weaker conditions. In particular, both existence and uniqueness can fail in smooth, asymptotically flat spacetimes with a compact nonchronal region.