Generalised hyperbolicity in spacetimes with string-like singularities

Sanchez, Yafet Sanchez; Vickers, James

doi:10.1088/0264-9381/33/20/205002

Cited by 5 publications

(8 citation statements)

References 33 publications

(52 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since then, these requirements have been progressively reduced to the requirement that the second partial weak derivatives of g ab be square integrable and of K ab once differentiable [61][62][63][64][65][66][67][68]; the precise statement is that g ab is in the Sobolev space H 3/2+ loc and K ab in H 1/2+ loc for > 0. More recently, studies of low-regularity metrics in the context of junction conditions have produced limited proofs of local well-posedness for metrics with only first partial weak derivatives being square integrable: g ab ∈ H 1 loc and K ab ∈ H 0 loc [69][70][71][72][73]. This is precisely the regularity regime of our desired results, and we will assume existence, in accordance with expectations partially borne out in this class of cases for the Cauchy problem.…”

Section: Multiple Junctionsmentioning

confidence: 99%

Coarse graining holographic black holes

Engelhardt

Wall

2019

J. High Energ. Phys.

173

View full text Add to dashboard Cite

We expand our recent work on the outer entropy, a holographic coarsegrained entropy defined by maximizing the boundary entropy while fixing the classical bulk data outside some surface. When the surface is marginally trapped and satisfies certain "minimar" conditions, we prove that the outer entropy is exactly equal to a quarter the area (while for other classes of surfaces, the area gives an upper or lower bound). We explicitly construct the entropy-maximizing interior of a minimar surface, and show that it satisfies the appropriate junction conditions. This provides a statistical explanation for the area-increase law for spacelike holographic screens foliated by minimar surfaces. Our construction also provides an interpretation of the area for a class of non-minimal extremal surfaces.On the boundary side, we define an increasing simple entropy by maximizing the entropy subject to a set of "simple experiments" performed after some time. We show (to all orders in perturbation theory around equilibrium) that the simple entropy is the boundary dual to our bulk construction.

show abstract

Section: Multiple Junctionsmentioning

confidence: 99%

Coarse graining holographic black holes

Engelhardt

Wall

2019

J. High Energ. Phys.

173

View full text Add to dashboard Cite

show abstract

“…Proof. The results of [5,6,7] establish well-posedness. The only thing that remains to be shown is that the weak solutions have the correct support.…”

Section: The Lapse Function N Can Be Chosen Asmentioning

confidence: 87%

“…Proof The main idea is to use Galerkin's method. We sketch the proof below for a complete proof see [5,6,7].…”

Section: The Lapse Function N Can Be Chosen Asmentioning

confidence: 99%

“…To construct the map E + we will build a sequence of operators E + m that converge in the weak operator topology to E + . We will construct the operators E + m using Galerkin's method following the methods of [5,6,7]. First consider a complete orthogonal basis {w j (x)} j∈J in H 1 (Σ) which is also orthonormal in L 2 (Σ).…”

Section: Galerkin Type Approximation Of E +mentioning

confidence: 99%

See 1 more Smart Citation

Green operators for low regularity spacetimes

Sanchez

Vickers

2018

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

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 wellposedness 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.

show abstract

“…It also ensures that the solutions to the wave equation are in H 2 loc (M ) (see Appendix B for details of the function spaces we use) as shown in Theorem 4.7 below, which ensures we have enough regularity to define the quantisation functors we need. Although one can define solutions to the wave equation for metrics of lower regularity [51,52] there are difficulties in defining the corresponding advanced and retarded Green operators for these cases. In Section 3 we establish the results we need to prove existence and uniqueness of solutions to the forward (and backward) initial value problem for the wave equation on R n+1 for C 1,1 metrics.…”

Section: Introductionmentioning

confidence: 99%

Green operators in low regularity spacetimes and quantum field theory

Hoermann

Sanchez²,

Spreitzer

et al. 2020

Class. Quantum Grav.

View full text Add to dashboard Cite

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 C 1,1 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 g φ 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.

show abstract

Generalised hyperbolicity in spacetimes with string-like singularities

Cited by 5 publications

References 33 publications

Coarse graining holographic black holes

Coarse graining holographic black holes

Green operators for low regularity spacetimes

Green operators in low regularity spacetimes and quantum field theory

Contact Info

Product

Resources

About