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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 twodimensional 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.

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 planesymmetric sandwich waves propagating in a cylindrical universe.

The now-famous Majumdar-Papapetrou exact solution of the EinsteinMaxwell equations describes, in general, N static, maximally charged black holes balanced under mutual gravitational and electrostatic interaction. When N = 2, this solution denes 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 numerical 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 rst reducing the problem to that of geodesic motion on a negatively curved (Riemannian) surface.

The maximum entropy that can be stored in a bounded region of space is in dispute: it goes as volume, implies (nongravitational) microphysics; it goes as the surface area, asserts the "holographic principle." Here I show how the holographic 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.PACS numbers: 04.70.-s, 04.62.+v, 04.60.-m, 03.67.-a An outstanding recent puzzle in gravitational physics is to find a local, microscopic explanation for the "holographic principle" [1], which asserts that the maximum entropy that can be stored inside a bounded region R in 3-space must be proportional to the surface area A(R) [as opposed to the volume V (R)] of the region:where k B is Boltzmann's constant, and l p = G/c 3 is the Planck length. The most compelling conceptual evidence for the holographic bound comes from black-hole physics and thermodynamics. If there were a physical system enclosed in R whose entropy exceeded S max , it would be possible to violate the second law in the following way: First, one could dump as much energy into R as necessary to bring it to the threshold of gravitational collapse. This process can only increase the entropy contained in R, making it exceed S max even further. One could then tip the system into full gravitational collapse, leaving nothing but a black hole inside R. The resulting event horizon, being contained in R, necessarily has surface area no larger than A(R). But according to the Bekenstein formula [2], the entropy of this black hole, given by the right hand side of Eq. (1) with A(R) replaced by the horizon area, cannot exceed S max . Thus gravitational collapse would appear to cause a sudden decrease in entropy, violating the second law of thermodynamics.The holographic principle presents a puzzle since derivations based on standard (non-gravitational) microphysics yield an entropy bound proportional to the volume V (R) instead of the surface area. To discuss this in the simplest microscopic model, let me choose R to be a standard three-dimensional spacelike cube of size L in Minkowski space, and consider a real, massless (linear) scalar field φ confined in R. The Fock space is built out of the modes of the field φ, which are the positive frequency solutions of the scalar wave equation φ = 0 that vanish on ∂R. These modes are given (up to normalization) by the solutions sin( k · x − ω k t), where ω k = c| k|, and the admissible wave vectors k are labelled by non-negative integers m x , m y , m z : (k x , k y , k z ) = (π/L)(m x , m y , m z ). I will often use single-letter labels i, j etc. to denote a composite multi-index like (m x , m y , m z ). Mode counting and summing various quantities over the modes (and all my...

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. Secondly, 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. Thirdly, 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.

The mathematical formalism for linear quantum field theory on curved spacetime depends in an essential way on the assumption of global hyperbolicity. Physically, what lie at the foundation of any formalism for quantization in curved spacetime are the canonical commutation relations, imposed on the field operators evaluated at a global Cauchy surface. In the algebraic formulation of linear quantum field theory, the canonical commutation relations are restated in terms of a welldefined symplectic structure on the space of smooth solutions, and the local field algebra is constructed as the Weyl algebra associated to this symplectic vector space. When spacetime is not globally hyperbolic, e.g. when it contains naked singularities or closed timelike curves,