Existence and Uniqueness Theorems for Massless Fields on a Class of Spacetimes with Closed Timelike Curves |
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Authors: | J L Friedman M S Morris |
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Institution: | (1) Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA and Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA. E-mail: friedman@thales.phys.uwm.edu , US;(2) Department of Physics & Astronomy, Butler University, 4600 Sunset Ave., Indianapolis, IN 46208, USA. E-mail: msmorris@butler.edu, IN |
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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 ℐ
−
, 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 ℐ
−
. 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 whose meaning is essentially that the Cauchy problem for free fields is well defined in the presence
of CTC’s whenever the problem is well-posed in a 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.
Received: 28 November 1994/Accepted: 20 May 1996 |
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