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Journal articles on the topic 'Non-Globally Hyperbolic Spacetimes'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

KAY, BERNARD S. "THE PRINCIPLE OF LOCALITY AND QUANTUM FIELD THEORY ON (NON GLOBALLY HYPERBOLIC) CURVED SPACETIMES." Reviews in Mathematical Physics 04, spec01 (December 1992): 167–95. http://dx.doi.org/10.1142/s0129055x92000194.

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In the context of a linear model (the covariant Klein Gordon equation) we review the mathematical and conceptual framework of quantum field theory on globally hyperbolic spacetimes, and address the question of what it might mean to quantize a field on a non globally hyperbolic spacetime. Our discussion centres on the notion of F-locality which we introduce and which asserts there is a net of local algebras such that every neighbourhood of every point contains a globally hyperbolic subneighbourhood of that point for which the field algebra coincides with the algebra one would obtain were one to regard the subneighbourhood as a spacetime in its own right and quantize — with some choice of time-orientation — according to the standard rules for quantum field theory on globally hyperbolic spacetimes. We show that F-locality is a property of the standard field algebra construction for globally hyperbolic spacetimes, and argue that it (or something similar) should be imposed as a condition on any field algebra construction for non globally hyperbolic spacetimes. We call a spacetime for which there exists a field algebra satisfying F-locality F-quantum compatible and argue that a spacetime which did not satisfy something similar to this condition could not arise as an approximate classical description of a state of quantum gravity and would hence be ruled out physically. We show that all F-quantum compatible spacetimes are time orientable. We also raise the issue of whether chronology violating spacetimes can be F-quantum compatible, giving a special model — a massless field theory on the “four dimensional spacelike cylinder” — which is F-quantum compatible, and a (two dimensional) model — a massless field theory on Misner space — which is not. We discuss the possible relevance of this latter result to Hawking’s recent Chronology Protection Conjecture.

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2

Seggev, Itai. "Dynamics in stationary, non-globally hyperbolic spacetimes." Classical and Quantum Gravity 21, no. 11 (April 29, 2004): 2651–68. http://dx.doi.org/10.1088/0264-9381/21/11/010.

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3

Ishibashi, Akihiro, and Robert M. Wald. "Dynamics in non-globally-hyperbolic static spacetimes: III. Anti-de Sitter spacetime." Classical and Quantum Gravity 21, no. 12 (May 19, 2004): 2981–3013. http://dx.doi.org/10.1088/0264-9381/21/12/012.

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4

Fewster, C. J., and A. Higuchi. "Quantum field theory on certain non-globally hyperbolic spacetimes." Classical and Quantum Gravity 13, no. 1 (January 1, 1996): 51–61. http://dx.doi.org/10.1088/0264-9381/13/1/006.

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5

BULLOCK, DAVID M. A. "KLEIN–GORDON SOLUTIONS ON NON-GLOBALLY HYPERBOLIC STANDARD STATIC SPACETIMES." Reviews in Mathematical Physics 24, no. 10 (November 2012): 1250028. http://dx.doi.org/10.1142/s0129055x12500286.

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We construct a class of solutions to the Cauchy problem of the Klein–Gordon equation on any standard static spacetime. Specifically, we have constructed solutions to the Cauchy problem based on any self-adjoint extension (satisfying a technical condition: "acceptability") of (some variant of) the Laplace–Beltrami operator defined on test functions in an L2-space of the static hypersurface. The proof of the existence of this construction completes and extends work originally done by Wald. Further results include: the uniqueness of these solutions; their support properties; the construction of the space of solutions and the energy and symplectic form on this space; an analysis of certain symmetries on the space of solutions; and various examples of this method, including the construction of a non-bounded below acceptable self-adjoint extension generating the dynamics.

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6

Duggal, K. L. "Space time manifolds and contact structures." International Journal of Mathematics and Mathematical Sciences 13, no. 3 (1990): 545–53. http://dx.doi.org/10.1155/s0161171290000783.

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A new class of contact manifolds (carring a global non-vanishing timelike vector field) is introduced to establish a relation between spacetime manifolds and contact structures. We show that odd dimensional strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular contact structure. As examples, we present a causal spacetime with a non regular contact structure and a physical model [Gödel Universe] of Homogeneous contact manifold. Finally, we construct a model of 4-dimensional spacetime of general relativity as a contact CR-submanifold.

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7

Yurtsever, Ulvi. "Algebraic approach to quantum field theory on non-globally-hyperbolic spacetimes." Classical and Quantum Gravity 11, no. 4 (April 1, 1994): 999–1012. http://dx.doi.org/10.1088/0264-9381/11/4/016.

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8

Rovenski, Vladimir. "Einstein-Hilbert type action on spacetimes." Publications de l'Institut Math?matique (Belgrade) 103, no. 117 (2018): 199–210. http://dx.doi.org/10.2298/pim1817199r.

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The mixed gravitational field equations have been recently introduced for codimension one foliated manifolds, e.g. stably causal and globally hyperbolic spacetimes. These Euler-Lagrange equations for the total mixed scalar curvature (as analog of Einstein-Hilbert action) involve a new kind of Ricci curvature (called the mixed Ricci curvature). In the work, we derive Euler-Lagrange equations of the action for any spacetime, in fact, for a pseudo-Riemannian manifold endowed with a non-degenerate distribution. The obtained equations are presented in the classical form of Einstein field equation with the new Ricci type curvature instead of Ricci curvature

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9

Garfinkle, David, and Steven G. Harris. "Ricci fall-off in static and stationary, globally hyperbolic, non-singular spacetimes." Classical and Quantum Gravity 14, no. 1 (January 1, 1997): 139–51. http://dx.doi.org/10.1088/0264-9381/14/1/015.

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10

Dappiaggi, Claudio, Giuseppe Ruzzi, and Ezio Vasselli. "Aharonov–Bohm superselection sectors." Letters in Mathematical Physics 110, no. 12 (October 17, 2020): 3243–78. http://dx.doi.org/10.1007/s11005-020-01335-4.

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AbstractWe show that the Aharonov–Bohm effect finds a natural description in the setting of QFT on curved spacetimes in terms of superselection sectors of local observables. The extension of the analysis of superselection sectors from Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the presence of a new quantum number labelling charged superselection sectors. In the present paper, we show that this “topological” quantum number amounts to the presence of a background flat potential which rules the behaviour of charges when transported along paths as in the Aharonov–Bohm effect. To confirm these abstract results, we quantize the Dirac field in the presence of a background flat potential and show that the Aharonov–Bohm phase gives an irreducible representation of the fundamental group of the spacetime labelling the charged sectors of the Dirac field. We also show that non-Abelian generalizations of this effect are possible only on spacetimes with a non-Abelian fundamental group.

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11

MORETTI, VALTER. "ASPECTS OF NONCOMMUTATIVE LORENTZIAN GEOMETRY FOR GLOBALLY HYPERBOLIC SPACETIMES." Reviews in Mathematical Physics 15, no. 10 (December 2003): 1171–217. http://dx.doi.org/10.1142/s0129055x03001886.

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Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally-hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a C*-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of C*-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called loci, are realized as the elements of the inductive limit of the spaces of the algebraic states on the C*-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the role of a Lorentzian metric. Specializing back the formalism to the usual globally-hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events.

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12

de Lorenci, V. A., and E. S. Moreira. "Lessons from the Casimir Effect on a Spinning Circle." International Journal of Modern Physics A 18, no. 12 (May 10, 2003): 2073–76. http://dx.doi.org/10.1142/s0217751x03015507.

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This work is a pedagogical account on the diagnosis of certain pathologies which arise when standard canonical quantization procedures are improperly used in the context of non globally hyperbolic spacetimes. The improper use of "helical times" to implement quantization is also addressed.

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13

GUIDO, D., R. LONGO, J. E. ROBERTS, and R. VERCH. "CHARGED SECTORS, SPIN AND STATISTICS IN QUANTUM FIELD THEORY ON CURVED SPACETIMES." Reviews in Mathematical Physics 13, no. 02 (February 2001): 125–98. http://dx.doi.org/10.1142/s0129055x01000557.

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The first part of this paper extends the Doplicher–Haag–Roberts theory of superselection sectors to quantum field theory on arbitrary globally hyperbolic spacetimes. The statistics of a superselection sector may be defined as in flat spacetime and each charge has a conjugate charge when the spacetime possesses non-compact Cauchy surfaces. In this case, the field net and the gauge group can be constructed as in Minkowski spacetime. The second part of this paper derives spin-statistics theorems on spacetimes with appropriate symmetries. Two situations are considered: First, if the spacetime has a bifurcate Killing horizon, as is the case in the presence of black holes, then restricting the observables to the Killing horizon together with "modular covariance" for the Killing flow yields a conformally covariant quantum field theory on the circle and a conformal spin-statistics theorem for charged sectors localizable on the Killing horizon. Secondly, if the spacetime has a rotation and PT symmetry like the Schwarzschild–Kruskal black holes, "geometric modular action" of the rotational symmetry leads to a spin-statistics theorem for charged covariant sectors where the spin is defined via the SU(2)-covering of the spatial rotation group SO(3).

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14

Ishibashi, Akihiro, and Robert M. Wald. "Dynamics in non-globally-hyperbolic static spacetimes: II. General analysis of prescriptions for dynamics." Classical and Quantum Gravity 20, no. 16 (July 31, 2003): 3815–26. http://dx.doi.org/10.1088/0264-9381/20/16/318.

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15

Stoica, Ovidiu-Cristinel. "Spacetimes with singularities." Analele Universitatii "Ovidius" Constanta - Seria Matematica 20, no. 2 (June 1, 2012): 213–38. http://dx.doi.org/10.2478/v10309-012-0050-3.

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Abstract We report on some advances made in the problem of singularities in general relativity.First is introduced the singular semi-Riemannian geometry for metrics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like the Riemann curvature are usually prohibited by the fact that the metric is not invertible. The things become even worse at the points where the signature changes. We show that we can still do many of these operations, in a different framework which we propose. This allows the writing of an equivalent form of Einstein's equation, which works for degenerate metric too.Once we make the singularities manageable from mathematical viewpoint, we can extend analytically the black hole solutions and then choose from the maximal extensions globally hyperbolic regions. Then we find space-like foliations for these regions, with the implication that the initial data can be preserved in reasonable situations. We propose qualitative models of non-primordial and/or evaporating black holes.We supplement the material with a brief note reporting on progress made since this talk was given, which shows that we can analytically extend the Schwarzschild and Reissner-Nordström metrics at and beyond the singularities, and the singularities can be made degenerate and handled with the mathematical apparatus we developed.

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16

Wrochna, Michał, and Jochen Zahn. "Classical phase space and Hadamard states in the BRST formalism for gauge field theories on curved spacetime." Reviews in Mathematical Physics 29, no. 04 (May 2017): 1750014. http://dx.doi.org/10.1142/s0129055x17500143.

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We investigate linearized gauge theories on globally hyperbolic spacetimes in the BRST formalism. A consistent definition of the classical phase space and of its Cauchy surface analogue is proposed. We prove that it is isomorphic to the phase space in the ‘subsidiary condition’ approach of Hack and Schenkel in the case of Maxwell, Yang–Mills, and Rarita–Schwinger fields. Defining Hadamard states in the BRST formalism in a standard way, their existence in the Maxwell and Yang–Mills case is concluded from known results in the subsidiary condition (or Gupta–Bleuler) formalism. Within our framework, we also formulate criteria for non-degeneracy of the phase space in terms of BRST cohomology and discuss special cases. These include an example in the Yang–Mills case, where degeneracy is not related to a non-trivial topology of the Cauchy surface.

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17

Mondal, Puskar. "On the non-blow up of energy critical nonlinear massless scalar fields in ‘3+1’ dimensional globally hyperbolic spacetimes: light cone estimates." Annals of Mathematical Sciences and Applications 6, no. 2 (2021): 227–308. http://dx.doi.org/10.4310/amsa.2021.v6.n2.a5.

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18

Choudhury, Binayak S., and Himadri S. Mondal. "Continuous representation of a globally hyperbolic spacetime with non-compact Cauchy surfaces." Analysis and Mathematical Physics 5, no. 2 (October 24, 2014): 183–91. http://dx.doi.org/10.1007/s13324-014-0093-x.

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19

HOLLANDS, STEFAN. "RENORMALIZED QUANTUM YANG–MILLS FIELDS IN CURVED SPACETIME." Reviews in Mathematical Physics 20, no. 09 (October 2008): 1033–172. http://dx.doi.org/10.1142/s0129055x08003420.

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We present a proof that the quantum Yang–Mills theory can be consistently defined as a renormalized, perturbative quantum field theory on an arbitrary globally hyperbolic curved, Lorentzian spacetime. To this end, we construct the non-commutative algebra of observables, in the sense of formal power series, as well as a space of corresponding quantum states. The algebra contains all gauge invariant, renormalized, interacting quantum field operators (polynomials in the field strength and its derivatives), and all their relations such as commutation relations or operator product expansion. It can be viewed as a deformation quantization of the Poisson algebra of classical Yang–Mills theory equipped with the Peierls bracket. The algebra is constructed as the cohomology of an auxiliary algebra describing a gauge fixed theory with ghosts and anti-fields. A key technical difficulty is to establish a suitable hierarchy of Ward identities at the renormalized level that ensures conservation of the interacting BRST-current, and that the interacting BRST-charge is nilpotent. The algebra of physical interacting field observables is obtained as the cohomology of this charge. As a consequence of our constructions, we can prove that the operator product expansion closes on the space of gauge invariant operators. Similarly, the renormalization group flow is proved not to leave the space of gauge invariant operators. The key technical tool behind these arguments is a new universal Ward identity that is formulated at the algebraic level, and that is proven to be consistent with a local and covariant renormalization prescription. We also develop a new technique to accomplish this renormalization process, and in particular give a new expression for some of the renormalization constants in terms of cycles.

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20

Barbado, Luis C., Ana L. Báez-Camargo, and Ivette Fuentes. "Evolution of confined quantum scalar fields in curved spacetime. Part I." European Physical Journal C 80, no. 8 (August 2020). http://dx.doi.org/10.1140/epjc/s10052-020-8369-9.

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Abstract We develop a method for computing the Bogoliubov transformation experienced by a confined quantum scalar field in a globally hyperbolic spacetime, due to the changes in the geometry and/or the confining boundaries. The method constructs a basis of modes of the field associated to each Cauchy hypersurface, by means of an eigenvalue problem posed in the hypersurface. The Bogoliubov transformation between bases associated to different times can be computed through a differential equation, which coefficients have simple expressions in terms of the solutions to the eigenvalue problem. This transformation can be interpreted physically when it connects two regions of the spacetime where the metric is static. Conceptually, the method is a generalisation of Parker’s early work on cosmological particle creation. It proves especially useful in the regime of small perturbations, where it allows one to easily make quantitative predictions on the amplitude of the resonances of the field, providing an important tool in the growing research area of confined quantum fields in table-top experiments. We give examples within the perturbative regime (gravitational waves) and the non-perturbative regime (cosmological particle creation). This is the first of two articles introducing the method, dedicated to spacetimes without boundaries or which boundaries remain static in some synchronous gauge.

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21

Bartolo, R., A. M. Candela, and E. Caponio. "Normal Geodesics Connecting two Non-necessarily Spacelike Submanifolds in a Stationary Spacetime." Advanced Nonlinear Studies 10, no. 4 (January 1, 2010). http://dx.doi.org/10.1515/ans-2010-0407.

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AbstractIn this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime ℳ. The proof is based on both variational and geometric arguments involving the causal structure of ℳ, the completeness of suitable Finsler metrics associated to it and some basic properties of a submersion. By this interaction, unlike previous results on the topic, also non-spacelike submanifolds can be handled.

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