Quantized Fields and Chronology Protection

Several recent possible counterexamples to the Chronology Protection Conjecture are critically examined. The “adapted” Rindler vacuum state constructed by Li and Gott for a conformal scalar field in Misner space is extended to nonconformally coupled and self-interacting scalar fields. For these fields, the vacuum stress-energy always diverges on the chronology horizons. The divergence of the vacuum stress-energy on Misner space chronology horizons cannot be generally avoided by choosing a Rindler-type vacuum state. PACS number(s): 04.62.+v Typeset using REVTEX 1 Over the last decade, there has been substantial interest in whether it is possible, within the known body of physical law, to create closed timelike curves (CTC) in a spacetime which is initially free of such objects [1–3]. In colloquial terms, the question is whether it is possible in principle to construct a “time machine”. The main impediment found to such a construction is the divergence of the vacuum stress-energy of quantized fields in such spacetimes [4,5]. This divergence takes place on the chronology horizon, the null surface beyond which CTCs first form. It is believed (but not proven) that the gravitational backreaction to such a diverging stress-energy would alter the spacetime in such a way as to prevent the formation of CTCs. The generic notion that nature will not allow the formation of CTCs is embodied in Hawking’s Chronology Protection Conjecture (CPC): The laws of physics do not allow the appearance of closed timelike curves [6]. Recently, a number of examples have been found of spacetimes containing CTCs in which the vacuum stress-energy tensor of a particular quantum field does not diverge on the chronology horizon [7–11]. These cases have been interpreted by some [10,11] as possible counterexamples to the Chronology Protection Conjecture, at least in the form where the vacuum stress-energy of quantized matter fields is the agent which protects chronology. However, to be taken seriously, proposed counterexamples to the CPC should have to satisfy the same sort of criteria that are used to evaluate potential counterexamples to the Cosmic Censorship Hypothesis [12]. To begin, let us assume that the divergence of 〈T ν μ 〉 on chronology horizons is in fact the mechanism of chronology protection. Then, if one finds a combination of quantized field(s), vacuum state, and spacetime such that 〈T ν μ 〉 does not diverge on the chronology horizon, that combination can only be considered a valid counterexample to the CPC if: (1) the non-divergence of 〈T ν μ 〉 holds on an open set of spacetime metrics. Counterexamples must not depend on “fine-tuning” of metric parameters or topological identification scales. (2) the vacuum stress-energy does not diverge for a collection of interacting realistic fields. Counterexamples must not depend on special field properties (e.g., being conformally 2 invariant, massless, or free). Condition (1) is general; condition (2) applies only to the extent that the divergence of quantized field’s vacuum stress-energy is considered to be the mechanism of chronology protection. The recent examples in which the vacuum stress-energy is regular on the chronology horizon violate one or both of these conditions. For example, Boulware [8] and Tanaka and Hiscock [9] showed that a massive scalar field will have regular vacuum stress-energy on the chronology horizon of Grant space [13], provided the field mass is sufficiently large. In this case, one might argue that the first condition above is at least partially satisfied, since no fine-tuning of the Grant space parameters is required to render the vacuum stress-energy finite. However, the second condition is obviously violated, since not all quantized fields in the real world are massive. As a second example, Sushkov [7] demonstrated that a complex automorphic massless scalar field would have a nondivergent vacuum stress-energy on the chronology horizon of Misner space [14] if the automorphic parameter (the angle by which the complex field is rotated upon topological identification) has a special value. This violates the first condition above, since the automorphic parameter must be fine-tuned to eliminate the divergence in the vacuum stress-energy. In addition, one could not expect all fields in nature to be free massless automorphic fields; as Sushkov points out, the addition of any interaction terms will likely restore the divergence in the vacuum stress-energy. On the other hand, Cassidy [10] and Li and Gott [11,15] have shown that there exists a quantum state in Misner space, an “adapted” Rindler vacuum state, for which the vacuum stress-energy of conformally invariant fields is finite, in fact precisely zero, provided the Misner space identification scale is chosen to have a unique special value. They have indicated that they believe this may serve as a counterexample to the CPC, or at least to the idea that the divergence of the vacuum energy of quantized fields can protect chronology. This “counterexample” violates the first condition above, since the vacuum stress-energy is nondivergent only for a single value of the Misner identification scale, a set of measure zero. 3 In this Letter, I demonstrate that the adapted Rindler vacuum also violates the second condition. I show that the (Rindler) vacuum stress-energy of a nonconformally coupled scalar field, or a conformally coupled massless field with a λφ self-interaction will diverge on the chronology horizon for all values of the Misner identification scale. In addition, the vacuum polarization of the field, 〈φ〉, diverges in all cases, even for the conformally invariant case examined by Li and Gott. Hence, the regular behaviour found by Cassidy and Li and Gott holds only for a conformally invariant, non-interacting field, and only for the stress-energy tensor. While some fields in nature (e.g., the electromagnetic field, before interactions are added) are conformally invariant, others – notably gravity itself – are not; and interactions are the rule, not the exception. All calculations are performed in the Lorentzian signature spacetime, avoiding any conceivable ambiguity associated with regularization in the Euclidean sector [15]. Misner space is constructed from Minkowski space by identifying the points: (t, x, y, z) ↔ (t coshnb+ x sinhnb, x cosh nb+ t sinh nb, y, z) (1) where (t, x, y, z) are the usual Minkowski Cartesian coordinates, b is an arbitrary positive constant, and n is an integer. The identifications take on a simpler form in Misner (equivalently, flat Kasner, or Rindler) coordinates, (η, ζ, y, z), where t = ζ cosh η, x = ζ sinh η . (2) The metric in these coordinates takes the form ds = − dζ + ζ dη + dy + dz , (3) and the points which are identified are now simply (ζ, η, y, z) ↔ (ζ, η + nb, y, z) . (4) . The Misner space coordinates cover only the past (P , with t > |x|) and future (F , with t < |x|) quadrants of Minkowski space. There are two null surfaces ζ = 0 (corresponding 4 to t = x and t = −x) which are Cauchy and chronology horizons. The metric may be analytically extended across the boundaries at ζ = 0; the maximal extension of Misner space is obtained by performing both extensions, although at the cost of obtaining a nonHausdorff spacetime with a quasiregular singularity at the origin t = x = 0 [16]. Due to the topological identifications of Eq.(1), the extended spacetime now contains regions of closed timelike curves, namely the right (R, x > |t|) and left (L, x < |t|) quadrants of Minkowski space. In these regions, the η and ζ coordinates reverse roles, with ζ becoming a timelike coordinate and η spacelike, ds = − ζdη + dζ + dy + dz , (5) with the relation to Minkowski coordinates now being t = ζ sinh η, x = ζ cosh η in R and L. Consider now the vacuum stress-energy of a non-conformally coupled quantized massless scalar field in Misner space. The vacuum stress-energy tensor may be written in terms of the renormalized Hadamard function as: 〈Tμν〉ren = 1 2 lim X→X [ (1− 2ξ)∇μ∇ν′ + (2ξ − 1 2 )gμν∇α∇ α − 2ξ∇μ∇ν ] G ren , (6) where G ren is the renormalized Hadamard function. For the adapted Rindler vacuum state considered by Li and Gott, the renormalized Hadamard function is obtained by taking the sum over Misner identifications of the image sources for the Rindler Hadamard function [17],