Search Results (4 total)

Hawking has recently proposed a ‘‘chronology protection conjecture,’’ which states that closed timelike curves cannot form in the real Universe. The most likely mechanism for enforcing this conjecture, if it is correct, is a divergent vacuum polarization at the Cauchy horizon (‘‘chronology horizon’’) where closed timelike curves first try to form. Hawking has proved that, if the chronology horizon is compactly generated, then it contains one or more smoothly closed null geodesics. Because it seems likely that all the horizon’s generators emerge from these closed null geodesics, a sufficiently strongly divergent vacuum polarization at the closed null geodesics is likely to destroy the chronology horizon completely and thereby prevent closed timelike curves from forming. In this paper we compute the details of the divergence near the closed null geodesics, in a generic spacetime with a compactly generated chronology horizon—thereby generalizing earlier computations, in special spacetimes, by Hiscock and Konkowski, Kim and Thorne, and Frolov. We carry out the computation for both a conformal scalar field and a two-component spinor field. We show that, for an observer who will pass through a point on the closed null geodesic after a small interval of proper time δt, the leading-order divergence is always proportional to (δt)^-3 and has the same tensorial structure as the stress-energy of a null fluid moving along the closed null geodesic. We also show that, by contrast with flat spacetime, there is in general no cancellation between the divergent vacuum energies of a combination of fields that, in flat spacetime, would be related by supersymmetry: two conformal scalar fields and one two-component spinor field. We discuss the implications of these results for Hawking’s chronology protection conjecture.

The effects of self-interaction in classical physics, in the presence of closed timelike curves, are probed by means of a simple model problem: The motion and self-collisions of a nonrelativistic, classical billiard ball in a space endowed with a wormhole that takes the ball backward in time. The central question asked is whether the Cauchy problem is well posed for this model problem, in the following sense: We define the multiplicity of an initial trajectory for the ball to be the number of self-consistent solutions of the ball’s equations of motion, which begin with that trajectory. For the Cauchy problem to be well posed, all initial trajectories must have multiplicity one.

A simple analog of the science-fiction scenario of going back in time and killing oneself is an initial trajectory which is dangerous in this sense: When followed assuming no collisions, the trajectory takes the ball through the wormhole and thereby back in time, and then sends the ball into collision with itself. In contrast with one’s naive expectation that dangerous trajectories might have multiplicity zero and thereby make the Cauchy problem ill posed (‘‘no solutions’’), it is shown that all dangerous initial trajectories in a wide class have infinite multiplicity and thereby make the Cauchy problem ill posed in an unexpected way: ‘‘far too many solutions.’’

The wide class of infinite-multiplicity, dangerous trajectories includes all those that are nearly coplanar with the line of centers between the wormhole mouths, and a ball and wormhole restricted by (ball radius)≪(wormhole radius)≪(separation between wormhole mouths). Two of the infinity of solutions are slight perturbations of the self-inconsistent, collision-free motion, and all the others are strongly different from it. Not all initial trajectories have infinite multiplicity: trajectories where the ball is initially at rest far from the wormhole have multiplicity one, as also, probably, do those where it is almost at rest. A search is made for initial trajectories with zero multiplicity, and none are found. The search entails constructing a set of highly nonlinear, coupled, algebraic equations that embody all the ball’s laws of motion, collision, and wormhole traversal, and then constructing perturbation theory and numerical solutions of the equations. A future paper (paper II) will show that, when one takes account of the effects of quantum mechanics, the classically ill-posed Cauchy problem (‘‘too many classical solutions’’) becomes quantum-mechanically well posed in the sense of producing unique probability distributions for the outcomes of all measurements.

This paper initiates a research program to determine whether, and in what situations, quantum field theory enforces averaged energy conditions on the renormalized stress-energy tensors of quantum fields. This program is motivated by the important roles of averaged energy conditions in general-relativistic singularity theorems, and in preventing the existence of classical, traversable wormholes and wormhole-induced closed timelike curves. As a first step in this research program, this paper shows that a quantized, free scalar field in Minkowski space-time has the following properties: The weak-energy condition is satisfied for a wide class of states when averaged along a complete null geodesic, but it can be violated when averaged along a nongeodesic curve. If the curvature coupling constant in the scalar wave equation is restricted to a certain range, which includes conformal coupling, then the strong-energy condition is satisfied for the same wide class of states when averaged along a complete timelike geodesic. It is shown, further, that this enforcement of energy conditions is not universally true in all spacetimes: by closing up Minkowski spacetime in a spatial direction (e.g., by identifying x=0 with x=L), one can produce quantum states of a free scalar field that violate the averaged weak-energy condition.

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 Planck-scale, 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 self-consistency, 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, self-consistency 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, self-consistency 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 self-consistent evolution. A spacetime whose self-consistency 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, 2-dimensional spacetimes the benign ones are not generic. However, it seems likely that every 4-dimensional, asymptotically flat space-time that is stable and has a topology of the form R×(S-one point), where S is a closed 3-manifold, is benign. Wormhole spacetimes are of this type, with S=S¹×S². We suspect that these types of self-consistency 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 hard-sphere 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, self-consistent motions of the ball; other choices produce two different self-consistent motions; and others might (but we are not yet sure) produce no self-consistent motions whatsoever. By contrast, in a path-integral formulation of the nonrelativistic quantum mechanics of such a billiard ball, there appears to be a unique, self-consistent 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.