Hadamard States from Light-like Hypersurfaces

Dappiaggi, Claudio; Moretti, Valter; Pinamonti, Nicola

doi:10.1007/978-3-319-64343-4

Cited by 19 publications

(19 citation statements)

References 115 publications

(250 reference statements)

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“…Having defined Hadamard states, we must ask whether (a) such states exist, at least on globally hyperbolic spacetimes, and (b) how to construct/characterize concretely Hadamard states on given spacetime representing given physical setups. (a) has been established by several rigorous methods, see [59,60,61,62,63]. In particular, given any one Hadamard state Ψ, we may go to a representation of the field algebra by operators on a Hilbert space, in which the state is represented by a vector.…”

Section: Hadamard States From Null Surfacesmentioning

confidence: 99%

Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime

Hollands

Wald

Zahn

2020

Class. Quantum Grav.

125

134

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In classical General Relativity, the values of fields on spacetime are uniquely determined by their values at an initial time within the domain of dependence of this initial data surface. However, it may occur that the spacetime under consideration extends beyond this domain of dependence, and fields, therefore, are not entirely determined by their initial data. This occurs, for example, in the well-known (maximally) extended Reissner-Nordström or Reissner-Nordström-deSitter (RNdS) spacetimes. The boundary of the region determined by the initial data is called the "Cauchy horizon." It is located inside the black hole in these spacetimes. The strong cosmic censorship conjecture asserts that the Cauchy horizon does not, in fact, exist in practice because the slightest perturbation (of the metric itself or the matter fields) will become singular there in a sufficiently catastrophic way that solutions cannot be extended beyond the Cauchy horizon. Thus, if strong cosmic censorship holds, the Cauchy horizon will be converted into a "final singularity," and determinism will hold. Recently, however, it has been found that, classically this is not the case in RNdS spacetimes in a certain range of mass, charge, and cosmological constant. In this paper, we consider a quantum scalar field in RNdS spacetime and show that quantum theory comes to the rescue of strong cosmic censorship. We find that for any state that is nonsingular (i.e., Hadamard) within the domain of dependence, the expected stress-tensor blows up with affine parameter, V , along a radial null geodesic transverse to the Cauchy horizon as T V V ∼ C/V 2 with C independent of the state and C = 0 generically in RNdS spacetimes. This divergence is stronger than in the classical theory and should be sufficient to convert the Cauchy horizon into a singularity through which the spacetime cannot be extended as a (weak) solution of the semiclassical Einstein equation. This behavior is expected to be quite general, although it is possible to have C = 0 in certain special cases, such as the BTZ black hole. arXiv:1912.06047v1 [gr-qc] 12 Dec 2019As we shall show in sec. 3, the behavior of the quantum stress tensor is determined by the trace anomaly and the continuity equation alone. We find that eq. (5) holds, with C ∼ κ 2 c − κ 2 − . Our arguments are technically rather 3 Our arguments rest on the assumption of regularity across the cosmological horizon, or the weaker condition that Tvv vanishes on the cosmological horizon, with v the "Killing parameter". This condition has a natural limit in RN, that of absence of radiation infalling from spatial infinity. It seems natural to discuss the behavior at the Cauchy horizon in terms of such asymptotic conditions, as without asymptotic conditions also the classical consideration of strong cosmic censorship becomes meaningless, as discussed for example in the Introduction to [12]: The domain of dependence of a compact achronal set in any globally hyperbolic spacetime has a Cauchy horizon, across which it is of course smoothly ex...

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Section: Hadamard States From Null Surfacesmentioning

confidence: 99%

Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime

Hollands

Wald

Zahn

2020

Class. Quantum Grav.

125

134

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“…11] constitute examples of divergence-free null hypersurfaces. One might therefore speculate that the simplicity of the representation formula in both cases will be a key ingredient in completely explaining the tantalising similarity between, on the one hand, the universal expressions derived in [KW91] for the two-point functions of isometry-invariant Hadamard states evaluated on pairs of solutions both in S A or in S B and, on the other hand, the expression of the two-point function of the distinguished state on future/past null infinity described in [DMP17] and in references therein. This ought to be further investigated.…”

Section: Final Remarks and Applicationsmentioning

confidence: 94%

On the global “two-sided” characteristic Cauchy problem for linear wave equations on manifolds

Lupo¹

2018

Lett Math Phys

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The global characteristic initial value problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown that, if geometrically well-motivated restrictions are placed on the supports of the (smooth) initial datum and of the (smooth) inhomogeneous term, then there exists a continuous global solution which is smooth "on each side" of the initial value hypersurface. A uniqueness result in Sobolev regularity H 1/2+ε loc is proved among solutions supported in the union of the causal past and future of the initial value hypersurface, and whose product with the indicator function of the causal future (resp. past) of the hypersurface is past compact (resp. future compact). An explicit representation formula for solutions is obtained, which prominently features an invariantly defined, densitised version of the null expansion of the hypersurface. Finally, applications to quantum field theory on curved spacetimes are briefly discussed.

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“…(iv) The factor 2 in the definition of the scaling transformations u λ has been introduced to match with the convention for 2-point functions on lightlike hyperplanes used in the literature, see e.g. [15]. See also the remark towards the end of Sect.…”

Section: Remark (I)mentioning

confidence: 99%

“…As is common in the operator algebraic approach to algebraic quantum field theory (cf. [24] and in the present context, see also [15,22,37,57]) one can introduce a family {W(G)} of C * algebras indexed by open, relatively compact subsets G of R × S * by defining W(G) as the C * -subalgebra generated by all W (ϕ) with supp(ϕ) ⊂ G.…”

Section: The Scaling Limit Theory On T * (And Its Extension)mentioning

confidence: 99%

“…Several important properties of ω have been established and are well known, from related contexts or from investigations of chiral conformal quantum field theory. We collect some of those properties here; proofs and further exposition can be found in [15,22,37,57]. For notational simplicity, the GNS representation of ω will be denoted by (H , π ,…”

Section: The Scaling Limit Theory On T * (And Its Extension)mentioning

confidence: 99%

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Temperature and entropy–area relation of quantum matter near spherically symmetric outer trapping horizons

2021

Self Cite

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We consider spherically symmetric spacetimes with an outer trapping horizon. Such spacetimes are generalizations of spherically symmetric black hole spacetimes where the central mass can vary with time, like in black hole collapse or black hole evaporation. While these spacetimes possess in general no timelike Killing vector field, they admit a Kodama vector field which in some ways provides a replacement. The Kodama vector field allows the definition of a surface gravity of the outer trapping horizon. Spherically symmetric spacelike cross sections of the outer trapping horizon define in- and outgoing lightlike congruences. We investigate a scaling limit of Hadamard 2-point functions of a quantum field on the spacetime onto the ingoing lightlike congruence. The scaling limit 2-point function has a universal form and a thermal spectrum with respect to the time parameter of the Kodama flow, where the inverse temperature $$\beta = 2\pi /\kappa $$ β = 2 π / κ is related to the surface gravity $$\kappa $$ κ of the horizon cross section in the same way as in the Hawking effect for an asymptotically static black hole. Similarly, the tunnelling probability that can be obtained in the scaling limit between in- and outgoing Fourier modes with respect to the time parameter of the Kodama flow shows a thermal distribution with the same inverse temperature, determined by the surface gravity. This can be seen as a local counterpart of the Hawking effect for a dynamical horizon in the scaling limit. Moreover, the scaling limit 2-point function allows it to define a scaling limit theory, a quantum field theory on the ingoing lightlike congruence emanating from a horizon cross section. The scaling limit 2-point function as well as the 2-point functions of coherent states of the scaling limit theory is correlation-free with respect to separation along the horizon cross section; therefore, their relative entropies behave proportional to the cross-sectional area. We thus obtain a proportionality of the relative entropy of coherent states of the scaling limit theory and the area of the horizon cross section with respect to which the scaling limit is defined. Thereby, we establish a local counterpart, and microscopic interpretation in the setting of quantum field theory on curved spacetimes, of the dynamical laws of outer trapping horizons, derived by Hayward and others in generalizing the laws of black hole dynamics originally shown for stationary black holes by Bardeen, Carter and Hawking.

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Hadamard States from Light-like Hypersurfaces

Cited by 19 publications

References 115 publications

Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime

Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime

On the global “two-sided” characteristic Cauchy problem for linear wave equations on manifolds

Temperature and entropy–area relation of quantum matter near spherically symmetric outer trapping horizons

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