Globally Hyperbolic Research Articles

A gravitational collapse singularity theorem consistent with black hole evaporation

The global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work I show that the causality conditions in Penrose's theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in the absence of predictability and hence compatibly with quantum field theory and black hole evaporation.

Boundary conditions and vacuum fluctuations in {mathrm {AdS}}_4

Initial conditions given on a spacelike, static slice of a non-globally hyperbolic spacetime may not define the fates of classical and quantum fields uniquely. Such lack of global hyperbolicity is a well-known property of the anti-de Sitter solution and led many authors to question how is it possible to develop a quantum field theory on this spacetime. Wald and Ishibashi took a step towards the healing of that causal issue when considering the propagation of scalar fields on AdS. They proposed a systematic procedure to obtain a physically consistent dynamical evolution. Their prescription relies on determining the self-adjoint extensions of the spatial component of the differential wave operator. Such a requirement leads to the imposition of a specific set of boundary conditions at infinity. We employ their scheme in the particular case of the four-dimensional AdS spacetime and compute the expectation values of the field squared and the energy-momentum tensor, which will then bear the effects of those boundary conditions. We are not aware of any laws of nature constraining us to prescribe the same boundary conditions to all modes of the wave equation. Thus, we formulate a physical setup in which one of those modes satisfy a Robin boundary condition, while all others satisfy the Dirichlet condition. Due to our unusual settings, the resulting contributions to the fluctuations of the expectation values will not respect AdS invariance. As a consequence, a back-reaction procedure would yield a non-maximally symmetric spacetime. Furthermore, we verify the violation of weak energy condition as a direct consequence of our prescription for dynamics.

Formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary

We study nonlinear hyperbolic conservation laws posed on a differential $$(n+1)$$-manifold with boundary referred to as a spacetime, and defined from a prescribed flux field of n-forms depending on a parameter (the unknown variable)—a class of equations proposed by LeFloch and Okutmustur (Far East J. Math. Sci. 31:49–83, 2008). Our main result is a proof of the convergence of the finite volume method for weak solutions satisfying suitable entropy inequalities. A main difference with previous work is that we allow for slices with a boundary and, in addition, introduce a new formulation of the finite volume method involving the notion of total flux functions. Under a natural global hyperbolicity condition on the flux field and the spacetime and by assuming that the spacetime admits a foliation by compact slices with boundary, we establish an existence and uniqueness theory for the initial and boundary value problem, and we prove a contraction property in a geometrically natural $$L^1$$-type distance.

On Completeness of Sliced Spaces under the Alexandrov Topology

We show that in a sliced spacetime ( V , g ) , global hyperbolicity in V is equivalent to T A -completeness of a slice, if and only if the product topology T P , on V, is equivalent to T A , where T A denotes the usual spacetime Alexandrov “interval” topology.

A Nonlinear Hyperbolic Model for Radiative Transfer Equation in Slab Geometry

Linear models for the radiative transfer equation have been well developed, while nonlinear models are seldom investigated even for slab geometry due to some essential difficulties. We have proposed a moment model in [Y. Fan, R. Li, and L. Zheng, J. Comput. Phys., 404 (2020), 109128] for slab geometry, which combines the ideas of the classical $P_N$ and $M_N$ model. Though the model is far from perfect, it was demonstrated to be quite efficient in numerically approximating the solution of the radiative transfer equation, and we are motivated to improve this model further. Consequently, we propose in this paper a new model following the chartmap in [Y. Fan, R. Li, and L. Zheng, J. Comput. Phys., 404 (2020), 109128] with some significant theoretic progress. The new model is derived with global hyperbolicity, and meanwhile some necessary physical properties are preserved. We give a complete analysis of the characteristic structure and propose a numerical scheme for the new model. Numerical examples are presented to demonstrate the numerical performance of the new model.

Globally Hyperbolic Spacetimes as Posets

It is well-known that a spacetime with its causal relation is a partially ordered set (poset for short). If it is globally hyperbolic, then it is a bicontinuous poset whose the interval topology is the manifold topology. In this work, we will state a new condition on a poset, which is called DS-FI cluster point condition and we show that when a causally simple spacetime ${\mathscr{M}}$ as a poset with the manifold topology satisfies this condition, then ≪= I+ (where by ≪, we mean the way-below relation on arbitrary poset and by I+, we mean the chronological relation on spacetime), and ${\mathscr{M}}$ is a bicontinuous poset whose the interval topology is the manifold topology. Furthermore, we show that, on a causally simple spacetime ${\mathscr{M}}$ as a poset, the global hyperbolicity condition is strictly stronger than the DS-FI cluster point condition.

Globally hyperbolic spacetimes can be defined without the ‘causal’ condition

Reasonable spacetimes are non-compact and of dimension larger than two. We show that these spacetimes are globally hyperbolic if and only if the causal diamonds are compact. That is, there is no need to impose the causality condition, as it can be deduced. We also improve the definition of global hyperbolicity for the non-regular theory (non C1,1 metric) and for general cone structures by proving the following convenient characterization for upper semi-continuous cone distributions: causality and the causally convex hull of compact sets is compact. In this case the causality condition cannot be dropped, independently of the spacetime dimension. Similar results are obtained for causal simplicity.

Causal Properties of Fibered Space-Time

On the space of a principal bundle, a Lorentzian metric and a time orientation are given that are invariant with respect to the action of the structure group. These objects form a fibered space-time and, in the case of spacelike fibers, induce the same structures on the base. The following causality conditions are discussed: chronology, causality, stable and strong causality, and global hyperbolicity. It is proved that if the base space-time satisfies one of the above conditions, then so does the fibered space-time.

Boundary conditions and renormalized stress-energy tensor on a Poincaré patch of AdS2

Quantum field theory on anti-de Sitter spacetime requires the introduction of boundary conditions at its conformal boundary, due essentially to the absence of global hyperbolicity. Here we calculate the renormalized stress-energy tensor $T_{\mu\nu}$ for a scalar field $\phi$ on the Poincar\'e patch of $\text{AdS}_2$ and study how it depends on those boundary conditions. We show that, except for the Dirichlet and Neumann cases, the boundary conditions break the maximal $\textrm{AdS}$ invariance. As a result, $\langle\phi^2\rangle$ acquires a space dependence and $\langle T_{\mu\nu}\rangle$ is no longer proportional to the metric. When the physical quantities are expanded in a parameter $\beta$ which characterizes the boundary conditions (with $\beta=0$ corresponding to Dirichlet and $\beta=\infty$ corresponding to Neumann), the singularity of the Green's function is entirely subtracted at zeroth order in $\beta$. As a result, the contribution of nontrivial boundary conditions to the stress-energy tensor is free of singular terms.

Causality theory for closed cone structures with applications

We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular differentiability assumptions. We prove the validity of most results of the regular Lorentzian causality theory including: causal ladder, Fermat’s principle, notable singularity theorems in their causal formulation, Avez–Seifert theorem, characterizations of stable causality and global hyperbolicity by means of (smooth) time functions. For instance, we give the first proof for these structures of the equivalence between stable causality, [Formula: see text]-causality and existence of a time function. The result implies that closed cone structures that admit continuous increasing functions also admit smooth ones. We also study proper cone structures, the fiber bundle analog of proper cones. For them, we obtain most results on domains of dependence. Moreover, we prove that horismos and Cauchy horizons are generated by lightlike geodesics, the latter being defined through the achronality property. Causal geodesics and steep temporal functions are obtained with a powerful product trick. The paper also contains a study of Lorentz–Minkowski spaces under very weak regularity conditions. Finally, we introduce the concepts of stable distance and stable spacetime solving two well-known problems (a) the characterization of Lorentzian manifolds embeddable in Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof that topology, order and distance (with a formula à la Connes) can be represented by the smooth steep temporal functions. The paper is self-contained, in fact we do not use any advanced result from mathematical relativity.

An exact LMI condition for the strong delay‐independent stability analysis of neutral delay systems

SummaryThis paper concentrates on strong delay‐independent stability of neutral linear time‐invariant delay systems with multiple commensurate time delays. The stability analysis of linear neutral systems is complicated by the need to locate the roots of a transcendental characteristic equation and to take into account the global hyperbolicity of an associated difference system. In this paper, we propose a convex necessary and sufficient condition for testing strong delay‐independent stability. This result mainly follows from Kronecker sum properties and the Kalman‐Yakubovich‐Popov lemma, which allows us to present the main result in terms of a single linear matrix inequality feasibility test. The paper is closed by showing numerical examples that illustrate the applicability and effectiveness of the proposed method.

Renormalization for Unimodal Maps with Non-integer Exponents

We define an analytic setting for renormalization of unimodal maps with an arbitrary critical exponent. We prove the global hyperbolicity of renormalization conjecture for unimodal maps of bounded type with a critical exponent which is sufficiently close to an even integer. Furthermore, we prove the global $$C^{1+\beta }$$ -rigidity conjecture for such maps, giving the first example of a smooth rigidity theorem for unimodal maps whose critical exponent is not an even integer.

Lyapounov Functions of Closed Cone Fields: From Conley Theory to Time Functions

We propose a theory "a la Conley" for cone fields using a notion of relaxed orbits based on cone enlargements, in the spirit of space time geometry. We work in the setting of closed (or equivalently semi-continuous) cone fields with singularities. This setting contains (for questions which are parametrization independent such as the existence of Lyapounov functions) the case of continuous vector-fields on manifolds, of differential inclusions, of Lorentzian metrics, and of continuous cone fields. We generalize to this setting the equivalence between stable causality and the existence of temporal functions. We also generalize the equivalence between global hyperbolicity and the existence of a steep temporal functions.

Scalar field dynamics in a BTZ background with generic boundary conditions

We revisit the dynamics of a massive scalar field in a Banados, Teitelboim, and Zanelli background taking into account the lack of global hyperbolicity of the spacetime. We approach this issue using the strategy of Ishibashi and Wald which finds a unique smooth solution as the causal evolution of initial data, each possible evolution corresponding to a positive self-adjoint extension of certain operator in a Hilbert space on the initial surface. Moreover, solutions obtained this way are the most general ones satisfying a few physically sensible requirements. This procedure is intimately related to the choice of boundary conditions and the existence of bound states. We find that the scalar field dynamics in the (effective) mass window -3/4le m_e^2ell ^2<0 can be well defined within a one-parametric family of distinct boundary conditions (-3/4 being the conformally coupled case), while for m_e^2ell ^2ge 0 the boundary condition is unique (only one self-adjoint extension is possible). It is argued that there is no sensible evolution possible for m_e^2ell ^2<-1, and also it is shown that in the range m_e^2ell ^2 in [-1,-3/4) there is a U(1) family of allowed boundary conditions, however, the positivity of the self-adjoint extensions is only motivated but not proven. We focus mainly on describing the dynamics of such evolutions given the initial data and all possible boundary conditions, and in particular we show the energy is always positive and conserved.

Dynamical Casimir effect and loop corrections

We calculate quantum loop corrections to the stress-energy flux caused by moving mirrors. We consider massless, self-interacting, $\phi^4$, real scalar theory. In these calculations we encounter a new and quite unexpected subtleties due to the absence of global hyperbolicity in the presence of mirrors. We attempt to clearly phrase as many as possible hidden assumptions and complications that appear along the course of the solution of the problem in question. On top of that we find that quantum loop corrections to the stress-energy flux are growing with time and are not suppressed in comparison with the semi-classical contributions. Thus, we observe the break down of the perturbation theory, discuss its physical origin and ways to deal with such a situation. As a byproduct we observe a similarity of the problem in question with that for the minimally coupled, massless scalar field in de Sitter space.

13-Moment System with Global Hyperbolicity for Quantum Gas

We point out that the quantum Grad's 13-moment system [R. Yano, Physica A: Statistical Mechanics and its Applications, 416:231-241, 2014] is lack of global hyperbolicity, and even worse, the thermodynamic equilibrium is not an interior point of the hyperbolicity region of the system. To remedy this problem, by fully considering Grad's expansion, we split the expansion into the equilibrium part and the non-equilibrium part, and propose a regularization for the system with the help of the new theory developed in [Z. Cai et al., SIAM J. Appl. Math., 75(5):2001-2023, 2015, Y. Fan, J. Stat. Phys., 161(4), 2015]. This provides us a new model which is hyperbolic for all admissible thermodynamic states, and meanwhile preserves the approximate accuracy of the original system. It should be noted that this procedure is not a trivial application of the theory in [Z. Cai et al., SIAM J. Appl. Math., 75(5):2001-2023, 2015, Y. Fan, J. Stat. Phys., 161(4), 2015].

Waves in the Witten Bubble of Nothing and the Hawking Wormhole

We investigate the propagation of the scalar waves in the Witten space-time called "bubble of nothing" and in its remarkable sub-manifold, the Lorentzian Hawking wormhole. Due to the global hyperbolicity, the global Cauchy problem is well-posed in the functional framework associated with the energy. We perform a complete spectral analysis that allows to get an explicit form of the solutions in terms of special functions. If the effective mass is non zero, the profile of the waves is asymptotically almost periodic in time. In contrast, the massless case is dispersive. We develop the scattering theory, classical as well as quantum. The quantized scattering operator leaves invariant the Fock vacuum: there is no creation of particles. The resonances can be defined in the massless case and they are purely imaginary.

On the hyperbolicity of one-dimensional models for transient two-phase flow in a pipeline

Characteristic properties of one-dimensionalmodels of transient gas-liquid two-phase flows in long pipelines are investigated. The methods for studying the hyperbolicity of the systems of equations of multi-fluid and drift-fluxmodels are developed. On the basis of analytical and numerical studies, the limits of the hyperbolicity domains in the space of governing dimensionless parameters are found, and the impact of the closure relations on the characteristic properties of the models is analyzed. The methods of ensuring the global unconditional hyperbolicity are proposed. Explicit formulas for the eigenvelocities of the system of the drift-flux model equations are obtained and the conclusions about their sign-definiteness are drawn.

Global Hyperbolicity for Spacetimes with Continuous Metrics

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the compactness of the space of causal curves and to the existence of a Cauchy hypersurface. Furthermore, global hyperbolicity implies causal simplicity, stable causality and the existence of maximal curves connecting any two causally related points.

Conical spacetimes and global hyperbolicity

Vickers and Wilson ([26]) have shown global hyperbolicity of the conical spacetime in the sense of well-posedness of the initial value problem for the wave equation in generalized functions. We add the aspect of metric splitting and preliminary thoughts on Cauchy hypersurfaces and causal curves. AMS Mathematics Subject Classi cation (2010): 46F30, 83C75