Search Results (16 total)

We investigate the effect of quantum metric fluctuations on qubits that are gravitationally coupled to a background spacetime. In our first example, we study the propagation of a qubit in flat spacetime whose metric is subject to flat quantum fluctuations with a Gaussian spectrum. We find that these fluctuations cause two changes in the state of the qubit: they lead to a phase drift, as well as the expected exponential suppression (decoherence) of the off-diagonal terms in the density matrix. Although in principle observable, the current state of technology prohibits the experimental demonstration of the phase drift. Second, we calculate the decoherence of a qubit in a circular orbit around a Schwarzschild black hole. The no-hair theorems suggest a quantum state for the metric in which the black hole’s mass fluctuates with a thermal spectrum at the Hawking temperature. Again, we find that the orbiting qubit undergoes decoherence and a phase drift that both depend on the temperature of the black hole. Third, we study the interaction of coherent and squeezed gravitational waves with a qubit in uniform motion. Finally, we investigate the decoherence of an accelerating qubit in Minkowski spacetime due to the Unruh effect. In this case decoherence is not due to fluctuations in the metric, but instead is caused by coupling (which we model with a standard Hamiltonian) between the qubit and the thermal cloud of Unruh particles bathing it. When the accelerating qubit is entangled with a stationary partner, the decoherence should induce a corresponding loss in teleportation fidelity.

I show how the holographic entropy bound can be derived from elementary flat-spacetime quantum field theory when the total energy of Fock states is constrained gravitationally. This energy constraint makes the Fock space dimension (whose logarithm is the maximum entropy) finite for both bosons and fermions. Despite the elementary nature of my analysis, it results in an upper limit on entropy in remarkable agreement with the holographic bound, and also provides a microscopic deviation of a more general entropy bound recently introduced by Gour.

Recent work has raised the possibility that quantum-information-theory techniques can be used to synchronize atomic clocks nonlocally. One of the proposed algorithms for quantum clock synchronization (QCS) requires distribution of entangled pure singlets to the synchronizing parties [R. Jozsa et al., Phys. Rev. Lett. 85 2010 (2000)]. Such remote entanglement distribution normally creates a relative phase error in the distributed singlet state, which then needs to be purified asynchronously. We present a relativistic analysis of the QCS protocol that shows that asynchronous entanglement purification is not possible, and, therefore, the proposed QCS scheme remains incomplete. We discuss possible directions of research in quantum-information theory, which may lead to a complete, working QCS protocol.

The now-famous Majumdar-Papapetrou exact solution of the Einstein-Maxwell equations describes, in general, N static, maximally charged black holes balanced under mutual gravitational and electrostatic interaction. When N=2, this solution defines the two-black-hole spacetime, and the relativistic two-center problem is the problem of geodesic motion on this static background. Contopoulos and a number of other workers have recently discovered through numerial experiments that, in contrast with the Newtonian two-center problem, where the dynamics is completely integrable, relativistic null-geodesic motion on the two-black-hole spacetime exhibits chaotic behavior Here I identify the geometric sources of this chaotic dynamics by first reducing the problem to that of geodesic motion on a negatively curved (Riemannian) surface.

I propose a simple generalization of the ANEC (averaged null energy condition), in which the right-hand side of the ANEC inequality is replaced by a finite (but in general negative) state-independent lower bound. It is plausible that unlike the original ANEC this version might hold generally in four-dimensional curved spacetime. I study some of the implications of the generalized ANEC, and show, in particular, that if it holds in static traversable wormhole spacetimes (which is likely but remains to be shown), then the generalized ANEC rules out macroscopic (but not necessarily microscopic, Planck-size) static wormholes.

For a large class of quantum states, all local (pointwise) energy conditions widely used in relativity are violated by the renormalized stress-energy tensor of a quantum field. In contrast, certain nonlocal positivity constraints on the quantum stress-energy tensor might hold quite generally, and this possibility has received considerable attention in recent years. In particular, it is now known that the averaged null energy condition, the condition that the null-null component of the stress-energy tensor integrated along a complete null geodesic is non-negative for all states, holds quite generally in a wide class of spacetimes for a minimally coupled scalar field. Apart from the specific class of spacetimes considered (mainly two-dimensional spacetimes and four-dimensional Minkowski space), the most significant restriction on this result is that the null geodesic over which the average is taken must be achronal. Recently, Ford and Roman have explored this restriction in two-dimensional flat spacetime, and discovered that in a flat cylindrical space, although the stress energy tensor itself fails to satisfy the averaged null energy condition (ANEC) along the (nonachronal) null geodesics, when the ‘‘Casimir-vacuum’’ contribution is subtracted from the stress-energy the resulting tensor does satisfy the ANEC inequality. Ford and Roman name this class of constraints on the quantum stress-energy tensor ‘‘difference inequalities.’’ Here I give a proof of the difference inequality for a minimally coupled massless scalar field in an arbitrary (globally hyperbolic) two-dimensional spacetime, using the same techniques as those we relied on to prove the ANEC in an earlier paper with Wald. I begin with an overview of averaged energy conditions in quantum field theory.

It is by now well known that the standard local (i.e., pointwise) energy conditions always can be violated in quantum field theory in curved (and flat) spacetime, even when these energy conditions hold for the corresponding classical field. Nevertheless, some global constraints on the stress-energy tensor may exist. Indeed recent work has shown that the averaged null energy condition (ANEC), which requires the positivity of energy suitably averaged along null geodesics, holds for a wide class of states of a minimally coupled scalar field on Minkowski spacetime, and also (in the massless case) on a wide class of states in curved two-dimensional spacetimes satisfying certain asymptotic regularity properties. In this paper, we strengthen these results by proving that, for the massless scalar field in an arbitrary globally hyperbolic two-dimensional spacetime, the ANEC holds for all Hadamard states along any complete, achronal null geodesic. In our analysis, the general, algebraic notion of ‘‘state’’ is used, so, in particular, it is not even assumed that our state belongs to any Fock representation. Our proof shows that the ANEC is a direct consequence of the general positivity condition which must hold for the two-point function of any state. Our results also can be extended (with a restriction on states) to the massive scalar field in two-dimensional Minkowski spacetime and (with an additional restriction on states) to the (massless or massive) minimally coupled scalar field on four-dimensional Minkowski spacetime. In the case of a (curved) four-dimensional spacetime with a bifurcate Killing horizon, our proof also extends to establish the ANEC for the null geodesic generators of the horizon (provided that there exists a stationary Hadamard state of the field). This latter result implies that the ANEC must hold for the massive Klein-Gordon field in de Sitter spacetime.

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.

Although linear quantum field theory on the background of a single gravitational plane wave is trivial, colliding plane-wave solutions are likely to feature interesting quantum effects; these might yield insight into similar effects in other more general inhomogeneous and dynamical spacetimes. This paper presents the initial results of an investigation into the behavior of quantum fields in colliding plane-wave backgrounds. Throughout the paper, we restrict our attention to the analysis of quantum field theory on the Khan-Penrose spacetime. Since spacetime is flat before the arrival of either plane wave, there exists a unique, well-defined set of in modes and a corresponding in vacuum state. We introduce a physically plausible prescription for constructing a unique canonical set of out-mode solutions, and we evaluate the Bogolubov transformation between the in and out modes explicitly for a massless scalar field propagating on the Khan-Penrose spacetime. We then use these results to approximately compute the spectrum of created particles in the out region. Next, we study the quantity 〈T_μν〉≡〈0, in|T_μν|0, in〉, the renormalized in-vacuum expectation value of the stress-energy tensor for a massless, conformally coupled (ξ=1 / 6) scalar field. In a colliding plane-wave spacetime, 〈T_μν〉 vanishes everywhere except in the interaction region. To compute 〈T_μν〉 in the interaction region, we make a number of assumptions about its general form; these assumptions are entirely reasonable for the specific geometry of the Khan-Penrose spacetime, but they may not hold for a general colliding plane-wave solution. Combined with the conservation property of 〈T_μν〉 and the choice of ξ=1 / 6, our assumptions reduce the determination of 〈T_μν〉 throughout the interaction region to the solution of a coupled system of first-order partial differential equations for two functions. These equations cannot be solved exactly; but they can be used to obtain crucial information on the behavior of 〈T_μν〉 near the singularity of the Khan-Penrose spacetime. Although our method of computing 〈T_μν〉 is unlikely to be adequate for other colliding plane-wave solutions we use the information obtained through our calculations to speculate about 〈T_μν〉 in more general colliding gravitational-wave spacetimes. We argue that these speculations have important consequences for cosmology, but they must be verified by further calculations.

It is well known that when gravitational plane waves propagating and colliding in an otherwise flat background interact they produce singularities. In this paper we explore the structure of the singularities produced in the collisions of arbitrarily polarized gravitational plane waves and we consider the problem of whether (or under what conditions) singularities can be produced in the collisions of almost-plane gravitational waves with finite but very large transverse sizes. First we analyze the asymptotic structure of a general arbitrarily polarized colliding plane-wave spacetime near its singularity. We show that the metric is asymptotic to a generalized inhomogeneous-Kasner solution as the singularity is approached. In general, the asymptotic Kasner axes as well as the asymptotic Kasner exponents along the singularity are functions of the spatial coordinate that runs tangentially to the singularity in the non-plane-symmetric direction.

It becomes clear that for specific values of these asymptotic Kasner exponents and axes the curvature singularity created by the colliding waves degenerates to a coordinate singularity, and that a nonsingular Killing-Cauchy horizon is thereby obtained. Our analysis proves that these horizons are unstable in the full nonlinear theory against small but generic plane-symmetric perturbations of the initial data, and that in a very precise and rigorous sense, ‘‘generic’’ initial data for colliding arbitrarily polarized plane waves always produce all-embracing, spacelike curvature singularities without Killing-Cauchy horizons. Next we turn to the problem of colliding almost-plane gravitational waves, and by combining the results that we obtain in this paper and in other previous papers with the Hawking-Penrose singularity theorem and the Cauchy stability theorem, we prove that if the initial data for two colliding almost-plane waves are sufficiently close to being exactly plane symmetric across a sufficiently large but bounded region of the initial surface, then their collision must produce spacetime singularities.

Although our analysis proves the existence of these singularities rigorously, it does not give any information about either their global structure (e.g., whether they are hidden behind an event horizon) or their local asymptotic behavior (e.g., whether they are of Belinsky-Khalatnikov-Lifshitz generic-mixmaster type).

It is argued that, if the laws of physics permit an advanced civilization to create and maintain a wormhole in space for interstellar travel, then that wormhole can be converted into a time machine with which causality might be violatable. Whether wormholes can be created and maintained entails deep, ill-understood issues about cosmic censorship, quantum gravity, and quantum field theory, including the question of whether field theory enforces an averaged version of the weak energy condition.

It is well known that when gravitational plane waves propagating on an otherwise flat background collide, they produce spacetime singularities. In this paper we consider the problem of whether (or under what conditions) singularities can be produced by the collision of gravitational waves with finite but very large transverse sizes. On the basis of (nonrigorous) order-of-magnitude considerations, we discuss the outcome of the collision in two fundamentally different regimes for the parameters of the colliding waves; these parameters are the transverse sizes (L_T)_i, typical amplitudes h_i, typical reduced wavelengths λ/_i≡λ_i/2π, thickneses a_i, and focal lengths f_i∼λ/_i²/a_ih_i² (i=1,2) of the waves 1 and 2.

For the first parameter regime where (L_T)₁≫(L_T)₂ and h₁≫h₂, we conjecture the following. (i) If (L_T)₂≪√λ/₂f₁ (h₁/h₂)^1/4, the almost-plane wave 2 will be focused by will be focused by wave 1 down to a finite, minimum size, then diffract and disperse [Fig. 1(a)]. (ii) If (L_T)₂≫√λ/₂f₁(h₁/h₂)^1/4 (and if wave 1 is sufficiently anastigmatic), wave 2 will be focused by wave 1 so strongly that it forms a singularity surrounded by a horizon, and the end result is a black hole flying away from wave 1 [Fig. 1(b)]. For the second parameter regime where (L_T)₁∼(L_T)₂≡L_T and h₁∼h₂, we conjecture that if L_T≫√f₁f₂≡f, a horizon forms around the two colliding waves shortly before their collision, and the collision produces a black hole that is at rest with respect to the reference frame in which f₁∼f₂∼f (Fig. 2). As a first step in proving this conjecture, we give a rigorous analysis of the second regime in the case L_T≫f, for the special situation of colliding parallel-polarized (almost-plane) gravitational waves which are exactly plane-symmetric across a region of transverse size ≫f, but which fall off in an arbitrary way at larger transverse distances. Our rigorous analysis shows that this collision is guaranteed to produce a spacetime singularity with the same local structure as in an exact plane-wave collision, but it does not prove that the singularity is surrounded by a horizon.

When gravitational plane waves propagating and colliding in an otherwise flat background interact, they produce spacetime singularities. If the colliding waves have parallel (linear) polarizations, the mathematical analysis of the field equations in the interaction region is especially simple. Using the formulation of these field equations previously given by Szekeres, we analyze the asymptotic structure of a general colliding parallel-polarized plane-wave solution near the singularity. We show that the metric is asymptotic to an inhomogeneous Kasner solution as the singularity is approached. We give explicit expressions which relate the asymptotic Kasner exponents along the singularity to the initial data posed along the wave fronts of the incoming, colliding plane waves. It becomes clear from these expressions that for specific choices of initial data the curvature singularity created by the colliding waves degenerates to a coordinate singularity, and that a nonsingular Killing-Cauchy horizon is thereby obtained. Our equations prove that these horizons are unstable in the full nonlinear theory against small but generic perturbations of the initial data, and that in a very precise sense, ‘‘generic’’ initial data always produce all-embracing, spacelike curvature singularities without Killing-Cauchy horizons. We give several examples of exact solutions which illustrate some of the asymptotic singularity structures that are discussed in the paper. In particular, we construct a new family of exact colliding parallel-polarized plane-wave solutions, which create Killing-Cauchy horizons instead of a spacelike curvature singularity. The maximal analytic extension of one of these solutions across its Killing-Cauchy horizon results in a colliding plane-wave spacetime, in which a Schwarzschild black hole is created out of the collision between two plane-symmetric sandwich waves propagating in a cylindrical universe.

It is well known that when two precisely plane-symmetric gravitational waves propagating in an otherwise flat background collide, they focus each other so strongly as to produce a curvature singularity. This paper is the first of several devoted to almost-plane gravitational waves and their collisions. Such waves are more realistic than plane waves in having a finite but very large transverse size. In this paper we review some crucial features of the well-known exact solutions for colliding plane waves and we argue that one of these features, the breakdown of ‘‘local inextendibility’’ can be regarded as nongeneric. We then introduce a new framework for analyzing general colliding plane-wave spacetimes; we give an alternative proof of a theorem due to Tipler implying the existence of singularities in all generic colliding plane-wave solutions; and we discuss the fact that the recently constructed Chandrasekhar-Xanthopoulos colliding plane-wave solutions are not strictly plane symmetric and thus do not satisfy the conditions and the conclusion of Tipler’s theorem.

Our alternative proof of Tipler’s theorem emphasizes the role and the necessity of strict plane symmetry in establishing the existence of singularities in colliding plane-wave spacetimes. However, we argue on the basis of previous work that the breakdown of strict plane symmetry as exhibited in the Chandrasekhar-Xanthopoulos solutions is a nongeneric phenomenon. We then propose a definition of general gravitational-wave spacetimes, of which almost-plane waves are a special case; and we develop some mathematical tools for studying them. An old result of Dautcourt implies that the only gravitational-wave spacetimes with a Killing propagation direction are plane fronted waves with parallel rays (PP waves); and we prove a new, related result, that only the gravitational-wave spacetimes with a precisely sandwiched curvature distribution are PP waves. These properties imply that almost-plane waves cannot propagate without diffraction, and that as opposed to the case for precisely plane waves, the curvature in an almost-plane-wave spacetime cannot be precisely sandwiched between two null surfaces (i.e., the wave must have tails). We also prove a ‘‘peeling theorem’’ for components of the Weyl curvature in general gravitational-wave spacetimes.

We construct an infinite-parameter family of exact solutions to the vacuum Einstein field equations describing colliding gravitational plane waves with parallel polarizations. The interaction regions of the solutions in this family are locally isometric to the interiors of those static axisymmetric (Weyl) black-hole solutions which admit both a nonsingular horizon, and an analytic extension of the exterior metric to the interior of the horizon. As a member of this family of solutions we also obtain, for the first time, a colliding plane-wave solution where both of the two incoming plane waves are purely anastigmatic, i.e., where both incoming waves have equal focal lengths.

It is well known that when plane-symmetric gravitational waves collide, they produce singularities. Presently known exact solutions representing such collisions fall into two classes: those in which the singularities are spacelike, and those in which timelike singularities appear preceded by a Killing-Cauchy horizon. This paper shows that Killing-Cauchy horizons in plane-symmetric spacetimes are unstable against plane-symmetric perturbations and thence argues that generic spacetimes representing colliding plane waves are likely to have spacelike singularities without Killing-Cauchy horizons. More specifically, this paper gives an explicit definition of Killing-Cauchy horizons in plane-symmetric spacetimes and classifies these horizons into two types: those which are smooth surfaces, called ‘‘type I,’’ and those which are singular, called ‘‘type II.’’ It is then shown that type-I horizons are unstable with respect to any generic, plane-symmetric perturbation data posed on a suitable initial null boundary and evolved with arbitrarily nonlinear field equations satisfying some very general requirements; linearized gravitational perturbations constitute a special case of this instability. Horizons of type II are shown to be unstable with respect to generic, plane-symmetric perturbations satisfying linear evolution equations; a special case again is linearized gravitational perturbations.