Global Causal Structure Research Articles

Weakly regular $T^2$-symmetric spacetimes. The global geometry of future Cauchy developments

We provide a geometric well-posedness theory for the Einstein equations within the class of weakly regular vacuum spacetimes with T^2 -symmetry, as defined in the present paper, and we investigate their global causal structure. Our assumptions allow us to give a meaning to the Einstein equations under weak regularity as well as to solve the initial value problem under the assumed symmetry. First, introducing a frame adapted to the symmetry and identifying certain cancellation properties taking place in the standard expressions of the connection and the curvature, we formulate the initial value problem for the Einstein field equations under the proposed weak regularity assumptions. Second, considering the Cauchy development of any weakly regular initial data set and denoting by R the area of the orbits of symmetry, we establish the existence of a global foliation by the level sets of R such that R grows to infinity in the future direction. Our weak regularity assumptions only require that R is Lipschitz continuous while the metric coefficients describing the initial geometry of the symmetry orbits are in the Sobolev space H^1 and the remaining coefficients have even weaker regularity.

Quantum correlations with no causal order

The idea that events obey a definite causal order is deeply rooted in our understanding of the world and at the basis of the very notion of time. But where does causal order come from, and is it a necessary property of nature? Here, we address these questions from the standpoint of quantum mechanics in a new framework for multipartite correlations that does not assume a pre-defined global causal structure but only the validity of quantum mechanics locally. All known situations that respect causal order, including space-like and time-like separated experiments, are captured by this framework in a unified way. Surprisingly, we find correlations that cannot be understood in terms of definite causal order. These correlations violate a 'causal inequality' that is satisfied by all space-like and time-like correlations. We further show that in a classical limit causal order always arises, which suggests that space-time may emerge from a more fundamental structure in a quantum-to-classical transition.

The characteristic initial value problem for plane symmetric spacetimes with weak regularity

We investigate the existence and the global causal structure of plane symmetric spacetimes with weak regularity when the matter consists of an irrotational perfect fluid with pressure equal to its mass-energy density. Our theory encompasses the class of W1, 2 regular spacetimes whose metric coefficients have square-integrable first-order derivatives and whose curvature must be understood in the sense of distributions. We formulate the characteristic initial value problem with data posed on two null hypersurfaces intersecting along a two-plane. Relying on Newman–Penrose's formalism and expressing our weak regularity conditions in terms of the Newman–Penrose scalars, we arrive at a fully geometrical formulation in which, along each initial hypersurface, two scalar fields describing the incoming radiation must be prescribed in L1 and W−1, 2, respectively. To analyze the future boundary of such a spacetime and identify its global causal structure, we introduce a gauge that reduces the Einstein equations to a coupled system of wave equations and ordinary differential equations for well-chosen unknowns. We prove that, within the weak regularity class under consideration and for generic initial data, a true spacetime singularity forms in finite proper time. Our formulation is robust enough so that propagating discontinuities in the curvature or in the matter variables do not prevent us from constructing a spacetime whose curvature generically blows up on the future boundary. Earlier work on the problem studied here was restricted to sufficiently regular and vacuum spacetimes.

Causal structure and algebraic classification of non-dissipative linear optical media

In crystal optics and quantum electrodynamics in gravitational vacua, the propagation of light is not described by a metric, but an area metric geometry. In this article, this prompts us to study conditions for linear electrodynamics on area metric manifolds to be well-posed. This includes an identification of the timelike future cones and their duals associated to an area metric geometry, and thus paves the ground for a discussion of the related local and global causal structures in standard fashion. In order to provide simple algebraic criteria for an area metric manifold to present a consistent spacetime structure, we develop a complete algebraic classification of area metric tensors up to general transformations of frame. This classification, valuable in its own right, is then employed to prove a theorem excluding the majority of algebraic classes of area metrics as viable spacetimes. Physically, these results classify and drastically restrict the viable constitutive tensors of non-dissipative linear optical media.

SPHERICALLY SYMMETRIC TRAPPING HORIZONS, THE MISNER–SHARP MASS AND BLACK HOLE EVAPORATION

We discuss some of the issues relating to information loss and black hole thermodynamics in the light of recent work on local black hole horizons. Understood in terms of pure states evolving into mixed states, the possibility of information loss in black holes is closely related to the global causal structure of space–time, as is the existence of event horizons. However, black holes need not be defined by event horizons, and in fact we argue that in order to have a fully unitary evolution for black holes, they should be defined in terms of something else, such as a trapping horizon. The Misner–Sharp mass in spherical symmetry shows very simply how trapping horizons can give rise to black hole thermodynamics, Hawking radiation and singularities. We show how the Misner–Sharp mass can also be used to give insights into the process of collapse and evaporation of locally defined black holes.

Simple space-time symmetries: Generalizing conformal field theory

We study simple space-time symmetry groups G which act on a space-time manifold M=G∕H which admits a G-invariant global causal structure. We classify pairs (G,M) which share the following additional properties of conformal field theory. (1) The stability subgroup H of o∊M is the identity component of a parabolic subgroup of G, implying factorization H=MAN−, where M generalizes Lorentz transformations, A dilatations, and N− special conformal transformations. (2) Special conformal transformations ξ∊N− act trivially on tangent vectors v∊ToM. The allowed simple Lie groups G are the universal coverings of SU(m,m),SO(2,D),Sp(l,R),SO*(4n), and E7(−25) and H are particular maximal parabolic subgroups. They coincide with the groups of fractional linear transformations of Euclidean Jordan algebras whose use as generalizations of Minkowski space-time was advocated by Günaydin [Mod. Phys. Lett. A 8, 1407 (1993)]. All these groups G admit positive energy representations. It will also be shown that the classical conformal groups SO(2,D) are the only allowed groups which possess an automorphism with properties appropriate for a time reflection.

Interpreting the C-metric

The basic properties of the C-metric are well known. It describes a pair of causally separated black holes which accelerate in opposite directions under the action of forces represented by conical singularities. However, these properties can be demonstrated much more transparently by making use of recently developed coordinate systems for which the metric functions have a simple factor structure. These enable us to obtain explicit Kruskal–Szekeres-type extensions through the horizons and construct two-dimensional conformal Penrose diagrams. We then combine these into a three-dimensional picture which illustrates the global causal structure of the spacetime outside the black hole horizons. Using both the weak field limit and some invariant quantities, we give a direct physical interpretation of the parameters which appear in the new form of the metric. For completeness, relations to other familiar coordinate systems are also discussed.

SCALAR FIELD IN A MINIMALLY COUPLED BRANE WORLD: NO-HAIR AND OTHER NO-GO THEOREMS

In the brane-world framework, we consider static, spherically symmetric configurations of a scalar field with the Lagrangian (∂ϕ)2/2-V(ϕ), confined on the brane. We use the 4D Einstein equations on the brane obtained by Shiromizu et al., containing the conventional stress tensor [Formula: see text], the tensor [Formula: see text] which is quadratic in [Formula: see text], and [Formula: see text] describing interaction with the bulk. For models under study, the tensor [Formula: see text] has zero divergence, allowing one to consider [Formula: see text]. Such a brane, whose 4D gravity is decoupled from the bulk geometry, may be called minimally coupled. Assuming [Formula: see text], we try to extend to brane worlds some theorems valid for scalar fields in general relativity (GR). Thus, the list of possible global causal structures in all models under consideration is shown to be the same as is known for vacuum with a cosmological constant in GR: Minkowski, Schwarzschild, (anti-) de Sitter and Schwarzschild–(anti-)de Sitter. A no-hair theorem, saying that, given a potential V≥0, asymptotically flat black holes cannot have nontrivial external scalar fields, is proved under certain restrictions. Some objects, forbidden in GR, are allowed on the brane, e.g, traversable wormholes supported by a scalar field, but only at the expense of enormous matter densities in the strong field region.

Scalar Fields in Multidimensional Gravity. No-Hair and Other No-Go Theorems

Global properties of static, spherically symmetric configurations of scalar fields of sigma-model type with arbitrary potentials are studied in $D$ dimensions, including space-times containing multiple internal factor spaces. The latter are assumed to be Einstein spaces, not necessarily Ricci-flat, and the potential $V$ includes contributions from their curvatures. The following results generalize those known in four dimensions: (A) a no-hair theorem: in case $V\geq 0$, an asymptotically flat black hole cannot have varying scalar fields or moduli fields outside the event horizon; (B) nonexistence of particlelike solutions in models with $V\geq 0$; (C) nonexistence of wormholes under very general conditions; (D) a restriction on possible global causal structures (represented by Carter-Penrose diagrams). The list of structures in all models under consideration is the same as is known for vacuum with a cosmological constant in general relativity: Minkowski (or AdS), Schwarzschild, de Sitter and Schwarzschild--de Sitter, and horizons which bound a static region are always simple. The results are applicable to a wide range of Kaluza-Klein, supergravity and stringy models with multiple dilaton and moduli fields.

Scalar-tensor gravity and conformal continuations

Global properties of vacuum static, spherically symmetric configurations are studied in a general class of scalar-tensor theories (STTs) of gravity in various dimensions. The conformal mapping between the Jordan and Einstein frames is used as a tool. Necessary and sufficient conditions are found for the existence of solutions admitting a conformal continuation (CC). The latter means that a singularity in the Einstein-frame manifold maps to a regular surface Strans in the Jordan frame, and the solution is then continued beyond this surface. Strans can be an ordinary regular sphere or a horizon. In the second case, Strans connect two epochs of a Kantowski-Sachs type cosmology. It is shown that the list of possible types of global causal structure of vacuum space–times in any STT, with any potential function U(φ), is the same as in general relativity with a cosmological constant. This is even true for conformally continued solutions. A traversable wormhole is shown to be one of the generic structures created as a result of CC. Two explicit examples are presented: the known solution for a conformal field in general relativity, illustrating the emergence of singularities and wormholes due to CC, and a nonsingular three-dimensional model with an infinite sequence of CCs.

New black holes in the brane world?

It is known that the Einstein field equations in five dimensions admit more general spherically symmetric black holes on the brane than four-dimensional general relativity. We propose two families of analytic solutions (with g_tt\not=-1/g_rr), parameterized by the ADM mass and the PPN parameter beta, which reduce to Schwarzschild for beta=1. Agreement with observations requires |\beta-1| |\eta|<<1. The sign of eta plays a key role in the global causal structure, separating metrics which behave like Schwarzschild (eta<0) from those similar to Reissner-Nordstroem (eta>0). In the latter case, we find a family of black hole space-times completely regular.

Analytical treatment of critical collapse in (2+1)-dimensionalAdS spacetime: a toy model

We present an exact collapsing solution to 2 + 1 gravity with a negativecosmological constant minimally coupled to a massless scalar field,which exhibits physical properties making it a candidate critical solution.We discuss its global causal structure and its symmetries in relationto those of the corresponding continuouslyself-similar solution derived in the Λ = 0 case. Linear perturbations onthis background lead to approximate blackhole solutions. The critical exponent isfound to be γ = (2/5).

DUAL DILATION DYONS

We find four-dimensional black hole solutions of the tree-level string effective action, that carry both electric and magnetic charge. The solutions involve non-trivial axion and dilaton fields, and are related to one another by a form of duality transformation. These black holes have drastically different global properties from the usual charged holes without axions and dilatons. The global causal structures of the effective metrics seen by different types of particles (for the same solution) can differ.

Spherically symmetric solutions admitting a spacelike self-similar motion

Global properties and causal structure are considered for spherically symmetric, perfect fluid solutions admiting a self-similar motion orthogonal to the four-velocity. The fluid admits a stiff equation of state. The momentum-energy tensor is equivalent to that of a free massless scalar field. All solutions have center singularities, some of which are timelike. In the latter cases, there are regions with negative density covered by Cauchy horizons. Regular boundaries at infinity display an asymptotically Minkowski behavior. Models of ‘‘vacuum bubbles’’ arise by performing a C1 matching with a section of Minkowski space-time along the Cauchy horizon. Generalizations associated with a nonzero cosmological constant are briefly examined.

Instability of black hole inner horizons

Linear perturbations of black hole models by a variety of fields are considered. Perturbing fields include the zero rest mass scalar field in the case of Reissner-Nordstrom, and gravitational, electromagnetic and zero rest mass scalar perturbation in the case of the Kerr model. The analysis deals with the Ψ 0 components (in the Newman-Penrose (1962) formalism) of non-zero spin fields. The symmetry properties of the models are used to derive the crucial condition th at the field be singular on the inner horizon. This condition is independent of the field propagation equation. Initial data are then given in terms of incoming radiation from f - is shown that there exist wellbehaved initial data sets for which the resultant fields are singular on the inner horizon. It is emphasized that this instability result is dependent only on the global symmetries and causal structure of the models considered, and is independent of the precise nature of the perturbing field.