Superluminal censorship

Transcription

1 Superluminal censorship Matt Visser, Bruce Bassett, and Stefano Liberati, Physics Department, Washington University, Saint Louis, Missouri , USA, International School for Advanced Studies (SISSA), Via Beirut 2 4, Trieste, Italy Istituto Nazionale di Fisica Nucleare (INFN), sezione di Trieste (8 October 1998; L A TEX-ed October 9, 1998) We investigate the relationship between superluminal travel and the null energy condition (NEC). First, by working perturbatively around Minkowski space we use linearized Einstein gravity to show that stable causality is perturbatively protected, because the NEC forces the light cones to contract (narrow). Given the NEC the spacetime geometry of any weak gravitational field always inherits the property of stable causality from the underlying Minkowski space. The NEC is also responsible for guaranteeing that the Shapiro time delay is always a delay relative to the Minkowski background, and never an advance. Secondly, we provide a non-perturbative definition of superluminal travel in asymptotically flat spacetimes, and prove a rigorous superluminal censorship theorem: Superluminal travel requires violations of the averaged null energy condition (ANEC). Introduction: The relationship between the causal aspects of spacetime and the stress-energy of the matter that generates the geometry is a deep and subtle one. In this Letter we shall start by perturbatively investigating the connection between the null energy condition (NEC) and stable causality. We shall demonstrate that in linearized gravity the NEC always forces the light cones to contract (narrow): thus the fact that in weak gravitational fields the Shapiro time delay is always a delay rather than an advance is related to the validity of the NEC for ordinary matter. This simple observation also has implication for the physics of effective faster-than-light (FTL) travel via warp drive. It is well established, via a number of rigorous theorems, that any possibility of effective FTL travel via traversable wormholes necessarily involves NEC violations [?,?,?,?,?,?,?]. On the other hand, for effective FTL travel via warp drive (say via the Alcubierre warp bubble [?] or the Krasnikov FTL hyper-tube [?]) NEC violations are observed in specific examples but there is as yet no really general theorem guaranteeing that FTL travel implies NEC violations. Part of the problem arises in even defining what we mean by FTL, and recent progress in this regard is reported in [?,?]. In the first part of this Letter we shall restrict attention to weak gravitational fields and work perturbatively around flat spacetime. One advantage of doing so is that the background Minkowski space provides an unambiguous definition of FTL travel. A second advantage is that the linearized Einstein equations are simply (if formally) solved via the gravitational Liénard Wiechertpotentials. The resulting expression for the metric perturbation provides information about the manner in which light cones are perturbed. In the second part of this Letter we provide a general non-perturbative definition of effective FTL travel in asymptotically flat spacetimes, using ideas of global analysis. By a modification of the argument leading to Penrose Sorkin Woolgar positive mass theorem [?] we prove a superluminal censorship theorem. (Minor variants of the argument simultaneously provide a censorship theorem for FTL travel via warp drives or traversable wormholes, an alternate proof of the topological censorship theorem [?], and show that negative asymptotic mass implies FTL). Linearized gravity: For a weak gravitational field we can write the metric as [?,?,?] g µν = η µν + h µν, (1) with h µν 1. Then adopting the Hilbert Lorentz gauge (aka Einstein gauge, harmonic gauge, de Donder gauge, Fock gauge) ν [h µν 1 ] 2 ηµν h =0, (2) the linearized Einstein equations are [?,?,?] [ h µν = 16πG T µν 1 ] 2 η µνt. (3) This has the formal solution [?,?,?] [ h µν ( x, t) =16πG d 3 Tµν (y, t) 1 2 y η µνt (y, t) ], (4) x y where t is the retarded time t = t x y. (5) These are the gravitational analog of the Liénard Wiechert potentials of ordinary electromagnetism, and the integral has support on the unperturbed backward light cone from the point x. Now consider a vector k µ whichwetaketobeanull vector of the unperturbed Minkowski geometry η µν k µ k ν =0. (6) 1

2 In terms of the full perturbed geometry this vector has a norm k 2 g µν k µ k ν (7) = h µν k µ k ν (8) =16πG d 3 y T µν(y, t) k µ k ν. (9) x y Now assume the NEC T µν k µ k ν 0, (10) and note that the kernel x y 1 is positive definite. Using the fact that the integral of a everywhere positive integrand is also positive, we deduce g µν k µ k ν 0. (11) Barring degenerate cases (e.g., a completely empty spacetime) the integrand will generically be positive so that g µν k µ k ν > 0. (12) That is, a vector that is null in the Minkowski metric will be spacelike in the full perturbed metric. Thus the null cone of the perturbed metric must everywhere lie inside the null cone of the unperturbed Minkowski metric. (Colloquially: Gravity sucks light cones contract.) Therefore the full perturbed metric, g µν, possesses a widening, η µν, that satisfies the chronology condition. This is one of the definitions of stable causality [?,?,?], and what we have shown is that weak gravitational fields whose sources satisfy the NEC are automatically stably causal. Furthermore, because the light cones contract, the coordinate speed of light must everywhere decrease (not the physical speed of light as measured by local observers, as always in General Relativity, that is of course a constant). This does however mean that the time required for a light ray to get from one spatial point to another must always increase compared to the time required in flat Minkowski space. This is the well-known Shapiro time delay, and we see two important points: (1) to even define the delay (delay with respect to what?) we need to use the flat Minkowski metric as a background, (2) the fact that in the solar system it is always a delay, never an advance, is due to the fact that everyday bulk matter satisfies the NEC. While subtle quantum-based violations of the NEC are known to occur [?,?,?,?,?], they are always small and are in fact tightly constrained by the Ford Roman quantum inequalities [?,?,?]. There are also classical NEC violations that arise from non-minimally coupled scalar fields [?], but these NEC violations require rather unphysical Planck-scale expectation values for the scalar field. NEC violations are never appreciable in a solar system setting. From the point of view of warp drive physics, this analysis is complementary to that of [?,?]. Though the analysis (so far) is perturbative around Minkowski space, it has the advantage of establishing a direct and immediate connection between FTL travel and NEC violations. Generalizing this result beyond the weak field perturbative regime is somewhat tricky. To even define effective FTL one will need to compare two metrics (just to be able to ask the question FTL with respect to what? ). If we work perturbatively around a general metric, instead of perturbatively around the Minkowski metric, the complications are immense: (1) the Laplacian in the linearized gravitational equations must be replaced by the Lichnerowicz operator; (2) the Green function for the Lichnerowicz operator need no longer be concentrated on the past light cone [physically, there can be backscattering from the background gravitational field, and so the Green function can have additional support from within the backward light cone]; and (3) the Green function need no longer be positive definite. Even for perturbations of a Friedman Robertson Walker (FRW) cosmology the analysis is not easy [?]. Because linearized gravity is not conformally coupled to the background the full history of the spacetime back to the Big Bang must be specified to derive the Green function. From the astrophysical literature concerning gravitational lensing it is known that voids (as opposed to over-densities) can sometimes lead to a Shapiro time advance [?,?,?]. This is not in conflict with the present analysis and is not evidence for astrophysical NEC violations. Rather, because those calculations compare a inhomogeneous universe with a void to a homogeneous FRW universe, the existence of a time advance is related to a suppression of the density below that of the homogeneous FRW cosmology, i.e. avoid[?]. Non-perturbative FTL: Turning to non-perturbative analyses, a pair of highly specialized results have already been announced. In [?] the warp drive field is assumed to be localized in space and time with the initial and final points of the timelike curve in the flat Minkowski space outside the warp field. There is then an implicit assumption that the entire spacetime has the same topology as that of ordinary Minkowski space (otherwise you would not be able to tell the difference between a warp drive and a traversable wormhole), and the flat Minkowski metric is used to establish the definition of FTL travel. On the other hand, in [?] a general theorem is proposed that guarantees NEC violations under certain specific geometrical conditions, but it is far from clear to us that these these conditions accurately capture the notion of FTL. We start by giving a general definition of FTL travel in any asymptotically flat spacetime. By definition, an asymptotically flat spacetime M has a boundary M = i I i o I + i +, (13) that is both isomorphic and isometric to the boundary of Minkowski space. Now pick any null generator k of past null infinity I,anypointuon this generator, and any null generator k + of future null infinity I +. Then consider any null curve in the spacetime M that 2

3 starts from (k,u) and terminates somewhere on k +.We can compare this with the fastest null geodesic 0, in Minkowski space M 0, that starts from from (k,u)and terminates on k +. (The only reason this works is because M and M 0 have the same boundary, both topologically and geometrically.) If reaches the generator k + in I + before the fastest Minkowski geodesic 0 then the spacetime M exhibits (effective) superluminal travel. This will be our definition of FTL. (The notion of fastest null geodesic from past null infinity to future null infinity is explored in great detail in proving the Penrose Sorkin Woolgar positive mass theorem [?].) k k FIG. 1. Asymptotic Penrose diagram for an asymptotically flat spacetime containing faster-than-light (FTL) causal curves. The dashed lines denote null curves in Minkowski space, while the solid lines are null curves traversing the interior of the FTL spacetime. Both spacetimes share the same boundary. Near the boundary, null curves in the FTL spacetime must asymptote to 45 degrees (because of the assumed asymptotic flatness). Deep in the interior there are warp fields (or traversable wormholes) that allow the local light cones to widen (or tip over) with respect to the asymptotic light cones. If some null curve in the full spacetime crosses from past null infinity to future null infinity and arrives before the null curve 0 in Minkowski space that starts at the same point, then the spacetime exhibits FTL travel. u This definition should now be augmented in such a way that we can tell the difference between FTL via warp drive and FTL via traversable wormhole. The relevant concepts are developed in proving the Friedman Schleich Witt topological censorship theorem [?]. The FTL travel takes place via traversable wormhole if the FTL curve is not homotopic to the trivial null curve through M. If the FTL curve is homotopic to the trivial curve, then the FTL travel takes place via warp drive (e.g., Alcubierre warp drive or Krasnikov FTL hypertube). Theorem (Superluminal censorship): If an asymptotically flat spacetime, whose domain of outer communication is globally hyperbolic, possesses a FTL null curve from past null infinity to future null infinity, then there exists a fastest null geodesic from past null infinity to future null infinity that is in the same homotopy class and does not have any conjugate points (and therefore the ANEC is violated along this fastest null geodesic). To prove this theorem we invoke the machinery of the Penrose Sorkin Woolgar positive mass theorem [?]. Lemma (fastest null geodesics): In any asymptotically flat spacetime (with globally hyperbolic domain of outer communication), given a point u on any generator k of past null infinity, and any generator k + of future null infinity, there will be a fastest null geodesic from (k,u) to k +. This fastest null geodesic will either lie entirely in the boundary M or will traverse the interior of M, in which case it cannot have conjugate points. (Proof: See [?]. There is also a brief discussion in [?].) Of course, once one has a complete null geodesic without conjugate points, it follows that the hypotheses of the focussing theorems must be violated, and in particular the ANEC must be violated along this null geodesic. (See [?,?].) Now suppose we have an asymptotically flat spacetime that possesses a FTL null curve : (1) If this null curve is not homotopic to the trivial null curve through M then there will be a fastest null curve in the same homotopy class of that by construction does not lie entirely in the boundary M. Fromthe above stated Lemma, this null geodesic has no conjugate points and must then violate the ANEC. This is essentially the alternate proof of the topological censorship theorem briefly discussed in [?]. (2) If the null FTL curve is homotopic to the trivial null curve we must use a different argument: By the definition of FTL the null curve is faster than any null geodesic in Minkowski space. Thus the fastest null geodesic in M is faster than any null geodesic in M 0 that connects the same generators. This fastest null geodesic cannot lie entirely in the boundary M, because then it would be a null geodesic of Minkowski space M 0, violating the definition that it is a FTL curve. (Therefore traverses the interior of M, possesses no conjugate points, and violates the ANEC.) We mention in passing that if an asymptotically flat spacetime has a negative asymptotic mass then there will be null curves in the asymptotic region that exhibit a Shapiro time advance [?]. Thus negative mass implies that there are some curves traversing the interior of M that are faster than any null curve in Minkowski space. (That is, by our definition above, an asymptotically flat spacetime with negative mass also exhibits FTL travel). This observation then leads to the Penrose Sorkin Woolgar positive mass theorem [?]. 3

4 Thus the argument of this Letter provides a unified censorship theorem that simultaneously deals with FTL via warp drives (warp drive censorship), FTL via traversable wormholes (topological censorship), and FTL via negative asymptotic mass (positive mass theorem). Discussion: This Letter completes the program of proving that any form of FTL travel requires violations of the ANEC. The perturbative analysis presented in the first portion is very useful in that it demonstrates that it is already extremely difficult to get even started: Any perturbation of flat space that exhibits even the slightest amount of FTL (defined as widening of the light cones) must violate the NEC. The perturbative analysis also serves to focus attention on the Shapiro time delay as a diagnostic for FTL, and it is this feature of the perturbative analysis that can be extended to the nonperturbative regime to provide a non-perturbative definition of FTL: Any asymptotically flat spacetime that exhibits a Shapiro time advance, (a null curve which is advanced relative to the flat Minkowski space you can smoothly attach to the conformal boundary) is said to exhibit FTL. The proof presented here also unifies the ideas of superluminal censorship, topological censorship, and the positive mass theorem in that the same technology (with extremely minor variations) serves to prove all three theorems. The proof is also in agreement with, and an extension of, the results of Olum [?,?]. Finally, there are also some related comments by Coule [?], regarding energy condition violations and opening out the light cones, that can be placed in a proper non-perturbative context by the analysis presented here. Acknowledgments: This research was supported in part by the US Department of Energy (MV) and by the Italian Ministry of Science (BB, and SL). MV wishes to thank SISSA (Trieste, Italy) for hospitality during early phases of this research. MV also wishes to thank LAEFF (Laboratorio de Astrofísica Espacial y Física Fundamental; Madrid, Spain), and Victoria University (Te Whare Wananga o te Upoko o te Ika a Maui; Wellington, New Zealand) for hospitality during final stages of this work. Electronic mail: visser@kiwi.wustl.edu Electronic mail: bruce@stardust.sissa.it. Address after 1 October 1998, Dept. of Theoretical Physics, University of Oxford, UK. Electronic mail: liberati@sissa.it [1] M.S. Morris and K.S. Thorne, Am. J. Phys. 56, 395 (1988). [2] M.S. Morris, K.S. 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5 k + 0 u - k