The 1990 analysis of the Cauchy problem in spacetimes with closed timelike curves (Friedman, Morris, Novikov, Echeverria, Klinkhammer, Thorne, and Yurtsever) established that classical and first-quantized evolution in wormhole time-machine spacetimes can be globally self-consistent, and posed the questions it could not answer: whether quantum-field back reaction prevents the formation of closed timelike curves (later sharpened by Hawking into the chronology protection conjecture); whether a self-consistent, second-quantized field theory exists on such spacetimes; whether 'benign' self-consistency behavior is generic among four-dimensional time-machine spacetimes; and whether open sets of classical initial data admit no self-consistent evolution at all. More than three decades of subsequent literature have rigorously established that the renormalized stress-energy tensor is necessarily singular or ill-defined on compactly generated chronology horizons (Kay–Radzikowski–Wald), but has also identified loopholes — non-compactly generated horizons, states with vanishing leading divergence, and the breakdown of the semiclassical approximation near the horizon — so that chronology protection, the quantum-field-theoretic formulation, the genericity of benignness, and the classical completeness question all remain open.

Background

General relativity admits spacetime geometries containing closed timelike curves (CTCs) — timelike worldlines that return to their own past. A concrete classical mechanism was exhibited by Morris, Thorne, and Yurtsever (1988): a traversable wormhole, a short 'handle' connecting two distant regions of space whose throat is held open only by matter violating the averaged null energy condition (the requirement that energy density averaged along light rays be nonnegative; quantum field theory permits bounded violations), can be converted into a time machine by the 'twin paradox' maneuver — carrying one wormhole mouth away and back — so that the induced time shift between the mouths produces CTCs to the future of a Cauchy horizon. A Cauchy horizon is the boundary beyond which initial data posed on an earlier hypersurface no longer determine the fields; the initial region is, by definition, globally hyperbolic, meaning the field equations posed there have a unique, predictable evolution. The Cauchy problem is the question of whether, and from what data, the future can be predicted at all.

Friedman, Morris, Novikov, Echeverria, Klinkhammer, Thorne, and Yurtsever (1990) asked whether the Cauchy problem makes sense in time-machine spacetimes. They proposed the principle of self-consistency: a local solution of the physical laws can occur in the real universe only if it extends to a global solution that is single-valued and well defined on the nonsingular region of the classical spacetime — events reached along several causal paths must agree with themselves. In explicit wormhole models built from flat spacetime with two identified mouths, they showed for a classical massless scalar field that standard radiative data on past null infinity determine a unique self-consistent solution, and that self-consistency constrains initial data only mildly: in some neighborhood of every event one is free to specify the data arbitrarily, with the data elsewhere adjusted to guarantee a self-consistent evolution. A spacetime whose self-consistency constraints have this property is called benign; in the geometric-optics (short-wavelength) limit the constraints retreat to a measure-zero set of events. In flat two-dimensional spacetimes with compact space and time identifications, benign spacetimes are not generic; the paper conjectured, by contrast, that every stable four-dimensional asymptotically flat spacetime with topology R×(S−{point})R\times(S-\{\text{point}\}) (wormhole spacetimes correspond to S=S1×S2S=S^1\times S^2) is benign for noninteracting, linearly superposing fields. For a classical billiard ball able to meet its younger self, some initial data give a unique self-consistent trajectory, others give two or more, and — the authors were explicitly unsure — some might give none; a path integral restricted to self-consistent histories formally gave a unique probability distribution for the first-quantized ball. The paper left open the following questions: whether the renormalized expectation value ⟨Tμν⟩\langle T_{\mu\nu}\rangle of the quantum stress-energy tensor (the vacuum-polarization source of back reaction in semiclassical gravity, where the metric is classical but matter is quantized) generically diverges at the Cauchy horizons of time machines, and whether such divergences always destroy the would-be CTCs; whether the self-consistent sum over histories can be promoted to a convergent second-quantized field theory with well-defined probabilities and global conservation laws; whether benignness is genuinely generic in four dimensions; whether open sets of classical initial data without any self-consistent evolution exist; and whether quantum gravity ultimately forbids macroscopic time machines.

Hawking (1992) sharpened the first question into the chronology protection conjecture: the laws of physics prevent the formation of CTCs. For the standard class of time machines whose Cauchy horizon is compactly generated (its null generators, followed into the past, all enter and remain in a compact set), Kay, Radzikowski, and Wald (1997) proved a rigorous theorem: no quantum state that is Hadamard on the initial globally hyperbolic region (Hadamard being the condition singling out physically admissible states, i.e. those locally resembling the Minkowski vacuum at short distances) can remain Hadamard across the full spacetime, so the renormalized ⟨Tμν⟩\langle T_{\mu\nu}\rangle is necessarily singular or ill-defined on the horizon. This is the strongest known result toward protection, and the point of departure for everything that follows.

Problem Statement

Determine whether spacetimes with closed timelike curves produced by time-machine constructions are physically admissible, within semiclassical physics (classical general relativity plus quantum matter fields). The single common objective — the viability and predictability of time machines — decomposes into four linked questions raised and left open by the 1990 analysis:

(a) Chronology protection. Does the back reaction of quantum fields always prevent the formation of closed timelike curves? It is now known (Kay, Radzikowski, Wald) that the renormalized stress-energy is necessarily singular or ill-defined on every compactly generated Cauchy horizon (one whose null generators, followed into the past, all enter and remain in a compact set) for Hadamard states (states with the standard short-distance regularity of the Minkowski vacuum); the open question is whether this, together with the semiclassical Einstein equations sourced by ⟨Tμν⟩\langle T_{\mu\nu}\rangle, always destroys the would-be time machine — in particular for the known loophole classes: non-compactly generated horizons, states with vanishing leading divergence, and spacetimes where semiclassical gravity ceases to be a valid approximation before the horizon is reached.

(b) Self-consistent quantum field theory. Does a well-defined quantum theory of (interacting) fields exist on a spacetime with CTCs — in particular, does the sum over self-consistent histories converge for second-quantized fields and yield well-defined probabilities and the usual global conservation laws — or is some modified consistency condition required, such as Deutsch's fixed-point condition, which demands that the density matrix on the CTC region be a fixed point of the evolution?

(c) Genericity of benignness. Is every stable, four-dimensional, asymptotically flat spacetime with topology R×(S−{point})R\times(S-\{\text{point}\}) benign for noninteracting classical fields (benign: self-consistency constrains initial data only in the mild sense that near every event the data can be specified freely, with data elsewhere adjusted), as conjectured? Here 'stable' is the qualifier the source attaches to the spacetimes the conjecture covers — stable against small perturbations — and 'genericity' is meant in the sense of that universal conjecture over the stated class, not a measure-theoretic genericity.

(d) Completeness of classical evolution. Do there exist open sets of initial data for which no self-consistent classical evolution exists in an interacting system, and if so, what physical principle resolves the resulting indeterminacy? The question is existential over all interacting systems — it asks whether some interacting system admits such an open set, so its negation is the general statement 'for every interacting system, no such open set exists'; colliding billiard balls in a wormhole time machine are only an illustrative, non-exhaustive case — the subsequent billiard-ball literature (refs 11-13) finds self-consistent solutions to exist there, the pathology being multiplicity rather than non-existence.

The broader quantum-gravity question — whether a complete theory of quantum gravity enforces energy conditions or forbids traversable wormholes and hence macroscopic time machines — is the acknowledged context of facet (a).

References

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