16 research outputs found

    Green operators for low regularity spacetimes

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    In this paper we define and construct advanced and retarded Green operators for the wave operator on spacetimes with low regularity. In order to do so we require that the spacetime satisfies the condition of generalised hyperbolicity which is equivalent to well- posedness of the classical inhomogeneous problem with zero initial data where weak solutions are properly supported. Moreover, we provide an explicit formula for the kernel of the Green operators in terms of an arbitrary eigenbasis of H 1 and a suitable Green matrix that solves a system of second order ODEs

    Generalised hyperbolicity in spacetimes with string-like singularities

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    In this paper we present well-posedness results of the wave equation in H1H^{1} for spacetimes that contain string-like singularities. These results extend a framework able to characterise gravitational singularities as obstruction to the dynamics of test fields rather than point particles. In particular, we discuss spacetimes with cosmic strings and the relation of our results to the Strong Cosmic Censorship Conjecture.Comment: Accepted for publication in Classical and Quantum Gravit

    Adiabatic Ground States in Non-Smooth Spacetimes

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    Ground states are a well-known class of Hadamard states in smooth spacetimes. In this paper we show that the ground state of the Klein-Gordon field in a non-smooth ultrastatic spacetime is an adiabatic state. The order of the state depends linearly on the regularity of the metric. We obtain the result by combining microlocal estimates for the causal propagator, propagation of singularities results for non-smooth pseudodifferential operators, and eigenvalue asymptotics for elliptic operators of low regularity.Comment: arXiv admin note: text overlap with arXiv:2203.0436

    Adiabatic Ground States in Non-smooth Spacetimes

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    Ground states are a well-known class of Hadamard states in smooth spacetimes. In this paper, we show that the ground state of the Klein–Gordon field in a non-smooth ultrastatic spacetime is an adiabatic state. The order of the state depends linearly on the regularity of the metric. We obtain the result by combining a propagation of singularities result for non-smooth pseudodifferential operators, properties of the causal propagator, and eigenvalue asymptotics for elliptic operators of low regularity

    An Inquiry into the nature of gravitational singularities

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    Generalised hyperbolicity in spacetimes with Lipschitz regularity

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    In this paper we obtain general conditions under which the wave equation is wellposed in spacetimes with metrics of Lipschitz regularity. In particular, the results can be applied to spacetimes where there is a loss of regularity on a hypersurface such as shell-crossing singularities, thin shells of matter, and surface layers. This provides a framework for regarding gravitational singularities not as obstructions to the world lines of point-particles, but rather as obstruction to the dynamics of test fields

    Green operators in low regularity spacetimes and quantum field theory

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    In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field φ on a globally hyperbolic spacetime M with C1,1 metric g. This first entails showing that the (classical) Cauchy problem for the wave equation is well-posed for initial data and sources in Sobolev spaces and then constructing low-regularity advanced and retarded Green operators as maps between suitable function spaces. In specifying the relevant function spaces we need to control the norms of both φ and gφ in order to ensure that g ◦ G± and G± ◦ g are the identity maps on those spaces. The causal propagator G = G+ − G− is then used to define a symplectic form ω on a normed space V(M) which is shown to be isomorphic to ker(g). This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local C∗-algebras
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