Causal Boundary Research Articles

On the Poisson boundary of the relativistic Brownian motion

Dans cet article, nous determinons la frontiere de Poisson du mouvement brownien relativiste dans deux classes de varietes lorentziennes, les espaces modeles de courbure constante et les espaces de Robertson–Walker qui constituent une vaste famille d’espace-temps courbes. Notre objectif est double : d’une part, il s’agit de comprendre les relations entre la geometrie a l’infini de ces varietes et le comportement asymptotique des trajectoires browniennes, avec comme objectif de comparer la compactification stochastique formee par les points de sortie du processus aux frontieres purement geometriques, conformes ou causales. D’autre part, nous souhaitons illustrer la pertinence de la methode de devissage introduite par les auteurs (Seminaire de Probabilites XLVIII (2016) 199–229 Springer), methode qui s’avere particulierement bien adaptee aux differents contextes geometriques consideres.

L-extensions and L-boundary of conformal spacetimes

The notion of L-boundary, a new causal boundary proposed by R. Low based on constructing a `sky at infinity' for any light ray, is discussed in detail. The analysis of the notion of L-boundary will be done in the 3-dimensional situation for the ease of presentation. The proposed notion of causal boundary is intrinsically conformal and, as it will be proved in the paper, under natural conditions provides a natural extension $\bar{M}$ of the given spacetime $M$ with smooth boundary $\partial M = \bar{M} \backslash M$. The extensions $\bar{M}$ of any conformal manifold $M$ constructed in this way are characterised exclusively in terms of local properties at the boundary points. Such extensions are called L-extensions and it is proved that, if they exist, they are essentially unique. Finally it is shown that in the 3-dimensional case, any L-extension is equivalent to the canonical extension obtained by using the L-boundary of the manifold.

Gravitational collapse and formation of universal horizons in Einstein-æther theory

We numerically study the gravitational collapse of a massless scalar field with spherical symmetry in Einstein-aether theory, and show that apparent, spin-0 and dynamical universal horizons (dUHs) can be all formed. The spacetime and the aether field are well-behaved and regular, including regions nearby these horizons (but away from the center of spherical symmetry). The spacetime outside the apparent and spin-0 horizons settles down to a static configuration, and some of such resulting static black holes were already found numerically in the literature. On the other hand, the proper distance of the outermost dUH from the apparent (or spin-0) horizon keeps increasing on $\mbox{\ae}$ther-orthogonal time slices. This indicates that the outermost dUH is evolving into the causal boundary, even for excitations with large speeds of propagation.

A novel notion of null infinity for c-boundaries and generalized black holes

We give new definitions of null infinity and black hole in terms of causal boundaries, applicable to any strongly causal spacetime (M, g). These are meant to extend the standard ones given in terms of conformal boundaries, and use the new definitions to prove a classic result in black hole theory for this more general context: if the null infinity is regular (i.e. well behaved in a suitable sense) and (M, g) obeys the null convergence condition, then any closed trapped surface in (M, g) has to be inside the black hole region. As an illustration of this general construction, we apply it to the class of generalized plane waves, where the conformal null infinity is not always well-defined. In particular, it is shown that (generalized) black hole regions do not exist in a large family of these spacetimes.

Indexically Structured Ecological Communities

Ecological communities are seldom, if ever, biological individuals. They lack causal boundaries as the populations that constitute communities are not congruent and rarely have persistent functional roles regulating the communities’ higher-level properties. Instead we should represent ecological communities indexically, by identifying ecological communities via the network of weak causal interactions between populations that unfurl from a starting set of populations. This precisification of ecological communities helps identify how community properties remain invariant, and why they have robust characteristics. This respects the diversity and aggregational nature of these complex systems while still vindicating them as units worthy of investigation.

Causality and c-completion of multiwarped spacetimes

In this paper, a systematic study of the causal structure and global causality properties of multiwarped spacetimes is developed. This analysis is used to provide a detailed description of the causal boundary of these spacetimes. Some applications of our results in examples of physical interest, for instance, in the context of Maldacena’s conjecture, are considered.

Hausdorff Closed Limits and Rigidity in Lorentzian Geometry

We begin with a basic exploration of the (point-set topological) notion of Hausdorff closed limits in the spacetime setting. Specifically, we show that this notion of limit is well suited to sequences of achronal sets, and use this to generalize the `achronal limits' introduced in [12]. This, in turn, allows for a broad generalization of the notion of Lorentzian horosphere introduced in [12]. We prove a new rigidity result for such horospheres, which in a sense encodes various spacetime splitting results, including the basic Lorentzian splitting theorem. We use this to give a partial proof of the Bartnik splitting conjecture, under a new condition involving past and future Cauchy horospheres, which is weaker than those considered in [10] and [12]. We close with some observations on spacetimes with spacelike causal boundary, including a rigidity result in the positive cosmological constant case.

The Bayesian stance: Equations for ‘as-if’ sensorimotor agency

The verb ‘to do’ plays a vital part in our understanding of the world, and it goes hand-in-hand with words such as active, action and agent. But the physical sciences describe only mechanical happenings, not acts. Their theoretical language is, in essence, a strict mathematical formalism applied to the description of variables (usually quantitative ones) that can – at least in principle – be measured by mechanical instruments. In such a language, what is the definition of an agent? Of an act? In contrast to previous approaches, which attempt to discriminate between agent and non-agent systems, we pursue a more Dennettian approach that attempts only to characterise the explanatory logic of intentional (agentive) interpretations of a physical system; we wish to do so purely in terms of the formal relations that hold between variables in a dynamical system or stochastic process. Our approach is straightforward: we use Pearl’s causal formalism to identify physical variables at the causal boundary between ‘agent’ and ‘environment’, and identify these with variables in Bayesian decision theory; this provides a rigorous bridge between mathematical models of physics and mathematical models of rational decision-making.

Erratum to: Complete affine connection in the causal boundary: static, spherically symmetric spacetimes

Erratum to: Complete affine connection in the causal boundary: static, spherically symmetric spacetimes

A conformal boundary for space-times based on light-like geodesics: The 3-dimensional case

A new causal boundary, which we will term the l-boundary, inspired by the geometry of the space of light rays and invariant by conformal diffeomorphisms for space-times of any dimension m≥3, proposed by one of the authors [R. J. Low, The Space of Null Geodesics (and a New Causal Boundary), Lecture Notes in Physics 692 (Springer, 2006), pp. 35–50] is analyzed in detail for space-times of dimension 3. Under some natural assumptions, it is shown that the completed space-time becomes a smooth manifold with boundary and its relation with Geroch-Kronheimer-Penrose causal boundary is discussed. A number of examples illustrating the properties of this new causal boundary as well as a discussion on the obtained results will be provided.

Complete affine connection in the causal boundary: static, spherically symmetric spacetimes

The boundary at $$\mathcal {I}^+$$ , future null infinity, for a standard static, spherically symmetric spactime is examined for possible linear connections. Two independent methods are employed, one for treating $$\mathcal {I}^+$$ as the future causal boundary, and one for treating it as a conformal boundary (the latter is subsumed in the former, which is of greater generality). Both methods provide the same result: a constellation of various possible connections, depending on an arbitrary choice of a certain function, a sort of gauge freedom in obtaining a natural connection on $$\mathcal {I}^+$$ ; choosing that function to be constant (for instance) results in a complete connection. Treating $$\mathcal {I}^+$$ as part of the future causal boundary, the method is to impute affine connections on null hypersurfaces going out to $$\mathcal {I}^+$$ , in terms of a transverse vector field on each null hypersurface (there is much gauge freedom on choice of the transverse vector fields). Treating $$\mathcal {I}^+$$ as part of a conformal boundary, the method is to make a choice of conformal factor that makes the boundary totally geodesic in the enveloping manifold (there is much gauge freedom in choice of that conformal factor). Similar examination is made of other boundaries, such as timelike infinity and timelike and spacelike singularities. These are much simpler, as they admit a unique connection from a similar limiting process (i.e., no gauge freedom); and that connection is complete.

Ernst formulation of axisymmetric fields inf(R)gravity: Applications to neutron stars and gravitational waves

The Ernst formulation of the Einstein equations is generalised to accommodate $f(R)$ theories of gravity. It is shown that, as in general relativity, the axisymmetric $f(R)$ field equations for a vacuum spacetime that is either stationary or cylindrically symmetric reduce to a single, non-linear differential equation for a complex-valued scalar function. As a worked example, we apply the generalised Ernst equations to derive a $f(R)$ generalisation of the Zipoy-Voorhees metric, which may be used to describe the gravitational field outside of an ellipsoidal neutron star. We also apply the theory to investigate the phase speed of large-amplitude gravitational waves in $f(R)$ gravity in the context of soliton-like solutions that display shock-wave behaviour across the causal boundary.

Radiation from a D-dimensional collision of shock waves: Two-dimensional reduction and Carter–Penrose diagram

We analyze the causal structure of the two-dimensional (2D) reduced background used in the perturbative treatment of a head-on collision of two [Formula: see text]-dimensional Aichelburg–Sexl gravitational shock waves. After defining all causal boundaries, namely the future light-cone of the collision and the past light-cone of a future observer, we obtain characteristic coordinates using two independent methods. The first is a geometrical construction of the null rays which define the various light cones, using a parametric representation. The second is a transformation of the 2D reduced wave operator for the problem into a hyperbolic form. The characteristic coordinates are then compactified allowing us to represent all causal light rays in a conformal Carter–Penrose diagram. Our construction holds to all orders in perturbation theory. In particular, we can easily identify the singularities of the source functions and of the Green’s functions appearing in the perturbative expansion, at each order, which is crucial for a successful numerical evaluation of any higher order corrections using this method.

Asymptotic behavior of a relativistic diffusion in Robertson–Walker space–times

Nous determinons le comportement asymptotique en temps long d’une diffusion relativiste a valeurs dans le fibre tangent unitaire d’un espace de Robertson–Walker. On montre en particulier qu’au voisinage du temps d’explosion de la diffusion, sa projection sur la variete de base converge presque surement vers un point aleatoire de la frontiere causale et nous decrivons le comportement du vecteur tangent au voisinage de ce point limite.

Hausdorff separability of the boundaries for spacetimes and sequential spaces

There are several ideal boundaries and completions in general relativity sharing the topological property of being sequential, i.e., determined by the convergence of its sequences and, so, by some limit operator L. As emphasized in a classical article by Geroch, Liang, and Wald, some of them have the property, commonly regarded as a drawback, that there are points of the spacetime M non-T1-separated from points of the boundary ∂M. Here, we show that this problem can be solved from a general topological viewpoint. In particular, there is a canonical minimum refinement of the topology in the completion M¯ which T2-separates the spacetime M and its boundary ∂M — no matter the type of completion one chooses. Moreover, we analyze the case of sequential spaces and show how the refined T2-separating topology can be constructed from a modification L∗ of the original limit operator L. Finally, we particularize this procedure to the case of the causal boundary and show how the separability of M and ∂M can be introduced as an abstract axiom in its definition.

Which spacetimes admit conformal compactifications?

We consider the future causal boundary as a tool to find obstructions to conformal extensions, the latter being a slight generalization to conformal compactifications.

Cross-lagged relations between mentoring received from supervisors and employee OCBs: Disentangling causal direction and identifying boundary conditions.

Although mentoring has documented relationships with employee attitudes and outcomes of interest to organizations, neither the causal direction nor boundary conditions of the relationship between mentoring and organizational citizenship behaviors (OCBs) has been fully explored. On the basis of Social Learning Theory (SLT; Bandura, 1977, 1986), we predicted that mentoring received by supervisors would causally precede OCBs, rather than employee OCBs resulting in the receipt of more mentoring from supervisors. Results from cross-lagged data collected at 2 points in time from 190 intact supervisor-employee dyads supported our predictions; however, only for OCBs directed at individuals (OCB-Is) and not for OCBs directed at the organization (OCB-Os). Further supporting our theoretical rationale for expecting mentoring to precede OCBs, we found that coworker support operates as a substitute for mentoring in predicting OCB-Is. By contrast, no moderating effects were found for perceived organizational support. The results are discussed in terms of theoretical implications for mentoring and OCB research, as well as practical suggestions for enhancing employee citizenship behaviors.

On the usage of the 2D-AR-model in texture completion scenarios with causal boundary conditions: A tutorial

In recent years, significant progress has been witnessed in several image and video completion scenarios. Given a specific application, these methods can produce, reproduce or extend a given texture sample. While there are many promising algorithms available, there is still a lack of theoretical understanding on how some of them are designed and under which conditions they perform. For that, we analyze and describe the technique behind one of the most popular parametric completion algorithms: the autoregressive (AR) model. Furthermore, we address important implementation details, complexity issues and restrictions of the model. Beyond that, we explain how the performance of the AR model can be significantly improved. In summary, this paper aims to achieve three major goals: (1) to provide a comprehensive tutorial for experienced and non- experienced readers, (2) to propose novel methods that improve the performance of the 2D-AR completion, and (3) to motivate and guide researchers that are interested in the usage of the AR model for texture completion tasks.

Modular localization and the holistic structure of causal quantum theory, a historical perspective

Recent insights into the conceptual structure of localization in QFT (modular localization) led to clarifications of old unsolved problems. The oldest one is the Einstein–Jordan conundrum which led Jordan in 1925 to the discovery of quantum field theory. This comparison of fluctuations in subsystems of heat bath systems (Einstein) with those resulting from the restriction of the QFT vacuum state to an open subvolume (Jordan) leads to a perfect analogy; the globally pure vacuum state becomes upon local restriction a strongly impure KMS state. This phenomenon of localization-caused thermal behavior as well as the vacuum-polarization clouds at the causal boundary of the localization region places localization in QFT into a sharp contrast with quantum mechanics and justifies the attribute “holstic”. In fact it positions the E–J Gedankenexperiment into the same conceptual category as the cosmological constant problem and the Unruh Gedankenexperiment. The holistic structure of QFT resulting from “modular localization” also leads to a revision of the conceptual origin of the crucial crossing property which entered particle theory at the time of the bootstrap S-matrix approach but suffered from incorrect use in the S-matrix settings of the dual model and string theory.The new holistic point of view, which strengthens the autonomous aspect of QFT, also comes with new messages for gauge theory by exposing the clash between Hilbert space structure and localization and presenting alternative solutions based on the use of stringlocal fields in Hilbert space. Among other things this leads to a reformulation of the Englert–Higgs symmetry breaking mechanism.

The bounded $$L^2$$ L 2 curvature conjecture

This is the main paper in a sequence in which we give a complete proof of the bounded \(L^2\) curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the \(L^2\)-norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the scaling of the Einstein equations, it is nevertheless critical with respect to its causal geometry. Indeed, \(L^2\) bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than \(1+1\) (based on Strichartz estimates) were obtained in Bahouri and Chemin (Am J Math 121:1337–1777, 1999; IMRN 21:1141–1178, 1999), Tataru (Am J Math 122:349–376, 2000; JAMS 15(2):419–442, 2002), Klainerman and Rodnianski (Duke Math J 117(1):1–124, 2003) and optimized in Klainerman and Rodnianski (Ann Math 161:1143–1193, 2005), Smith and Tataru (Ann Math 162:291–366, 2005), the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. To achieve our goals we recast the Einstein vacuum equations as a quasilinear \(so(3,1)\)-valued Yang–Mills theory and introduce a Coulomb type gauge condition in which the equations exhibit a specific new type of null structure compatible with the quasilinear, covariant nature of the equations. To prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure. These require a careful construction and control of parametrices including \(L^2\) error bounds which is carried out in Szeftel (Parametrix for wave equations on a rough background I: regularity of the phase at initial time, arXiv:1204.1768, 2012; Parametrix for wave equations on a rough background II: construction of the parametrix and control at initial time, arXiv:1204.1769, 2012; Parametrix for wave equations on a rough background III: space-time regularity of the phase, arXiv:1204.1770, 2012; Parametrix for wave equations on a rough background IV: control of the error term, arXiv:1204.1771, 2012), as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in Szeftel (Sharp Strichartz estimates for the wave equation on a rough background, arXiv:1301.0112, 2013). It is at this level that our problem is critical. Indeed, any known notion of a parametrix relies in an essential way on the eikonal equation, and our space-time possesses, barely, the minimal regularity needed to make sense of its solutions.