Temperature and entropy-area relation of quantum matter near spherically symmetric outer trapping horizons
Fiona Kurpicz, Nicola Pinamonti, Rainer Verch
备注：
latex2e, 44 pages, 1 Figure. v2: Typos removed, minor errors corrected, references added. This version matches the journal publication in Lett. Math. Phys. up to formatting
发表信息：
Letters in Mathematical Physics 111 (2021) No 110 (open access)
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. These spacetimes possess in general no timelike Killing vector field, but admit a Kodama vector field which provides a replacement. 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 is related to the surface gravity of the horizon cross-section in the same way as in the Hawking effect for an asymptotically static black hole. Similarly, the tunneling probability in the scaling limit between in- and outgoing Fourier modes with respect to the the Kodama time 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. The scaling limit 2-point function as well as the 2-point functions of coherent states of the scaling-limit-theory have relative entropies behaving proportional to the cross-sectional horizon area. 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. (Extended abstract in the article.)
We prove that, in a class of spherically symmetric spacetimes exhibiting stable trapping of null geodesics, linear waves cannot (uniformly) decay faster than logarithmically. When these linear waves are treated as a model for nonlinear perturbations, this slow decay is highly suggestive of nonlinear instability. We also prove that, in a large class of asymptotically flat, spherically symmetric spacetimes, logarithmic decay actually holds as a uniform upper bound. In the presence of stable trapping, this result is therefore the best one can obtain. In addition, we provide an application of thes…
We examine potential deformations of inner black hole and cosmological horizons in Reissner-Nordström de-Sitter spacetimes. While the rigidity of the outer black hole horizon is guaranteed by theorem, that theorem applies to neither the inner black hole nor past cosmological horizon. Further for pure deSitter spacetime it is clear that the cosmological horizon can be deformed (by translation). For specific parameter choices, it is shown that both inner black hole and cosmological horizons can be infinitesimally deformed. However these do not extend to finite deformations. The corresponding res…
We present a generalization of the Hawking effect for dynamical trapping horizons by calculating the tunneling rate in the Hamilton-Jacobi formalism. It turns out that all horizons classified by Hayward are subjected to thermal quantum effects. While the Hawking effect for future outer and past inner trapping horizons is given as a particle emission, we show that the Hawking effect for future inner and past outer trapping horizons translates to an absorption. The universality of the treatment allows a natural transfer to the static case.
Region of trapped null geodesics hidden inside of extremely compact objects is of astrophysical importance because of trapping of gravitational waves, or neutrinos. The trapping effect of null geodesics was extensively studied for spherically symmetric extremely compact objects. Recently, influence of rotation of the extremely compact objects on the trapping of null geodesics was treated in the simplest possible model of internal linearised \mbox{Hartle--Thorne} spacetime with uniform energy density distribution and uniform emissivity distribution of null geodesics. Here we extend the study of…
In this article, we shed some light on the field theoretic aspect of the spherically symmetric trapping horizon in Vaidya spacetime. The effective field equations are that of a Chern-Simons theory coupled to bulk sources through two different couplings, one is purely geometric and the other is matter dependent. This is an effective generalization of the equilibrium horizon scenario where the Chern-Simons theory is coupled to the bulk geometry through a constant matter-independent coupling. Further, we note that the field equations pulled-back to a cross-section of the horizon is inadequate to …
We work with the notion of apparent/trapping horizons for spherically symmetric, dynamical spacetimes: these are quasi-locally defined, simply based on the behaviour of congruence of light rays. We show that the sign of the dynamical Hayward-Kodama surface gravity is dictated by the inner/outer nature of the horizon. Using the tunneling method to compute Hawking Radiation, this surface gravity is then linked to a notion of temperature, up to a sign that is dictated by the future/past nature of the horizon. Therefore two sign effects are conspiring to give a positive temperature for the black h…
In this Letter, we have examined the thermodynamic volume products for spherically symmetric and axisymmetric spacetimes in the framework of \emph{extended phase space}. Such volume products usually formulated in terms of the outer horizon~(${\cal H}^{+}$) and the inner horizon~(${\cal H}^{-}$) of black hole ~ (BH) spacetime. Besides volume product, the other thermodynamic formulations like \emph{volume sum, volume minus and volume division} are considered for a wide variety of spherically symmetric spacetime and axisymmetric spacetimes. Like area~(or entropy) product of multihorizons, the mas…
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Extremely compact objects trap gravitational waves or neutrinos, assumed to move along null geodesics in the trapping regions. The trapping of neutrinos was extensively studied for spherically symmetric extremely compact objects constructed under the simplest approximation of the uniform energy density distribution, with radius located under the photosphere of the external spacetime; in addition, uniform emissivity distribution of neutrinos was assumed in these studies. Here we extend the studies of the neutrino trapping for the case of the extremely compact Tolman VII objects representing the…
We consider dynamical black hole formation from a collapsing fluid described by a symmetric and flat FRW metric. Using the Hamilton-Jacobi method the local Hawking temperature for the formed trapping/apparent horizon is calculated. The local Hawking temperature depends on the tunneling path, which we take to be along a null direction $(Δs=0)$. We find that the local Hawking temperature depends directly on the equation of state of the collapsing fluid. We argue that Hawking radiation by quantum tunnelling from future inner and future outer trapping horizons is possible. However, only radiation …
We prove a new, large family of area laws in general relativity, which apply to certain classes of untrapped surfaces that we dub generalized holographic screens. Our family of area laws contains, as special cases, the area laws for marginally-trapped surfaces (holographic screens) and the event horizon (Hawking's area theorem). In addition to these results in general relativity, we show that in the context of holography the geometry of a generalized holographic screen is related to the outer entropy of the screen. Specifically, we show for spherically-symmetric spacetimes that the area of the…
We investigate toroidal Marginally Outer Trapped Surfaces (MOTS) and Marginally Outer Trapped Tubes (MOTT) in closed Friedmann-Lemaitre-Robertson-Walker (FLRW) geometries. They are constructed by embedding Constant Mean Curvature (CMC) Clifford tori in a FLRW spacetime. This construction is used to assess the quality of certain isoperimetric inequalities, recently proved in axial symmetry. Similarly to spherically symmetric MOTS existing in FLRW spacetimes, the toroidal ones are also unstable.
Matching collapse and expansion across Matter Trapping surfaces in inhomogeneous $Λ$CDM models
Universidade Federal do ABC（圣保罗联邦大学ABC分校）；Lanzhou University（兰州大学）；Instituto de Astrofísica e Ciências do Espaço, Universidade de Lisboa, Faculdade de Ciências（里斯本大学天体物理与空间科学研究所、理学院）；Université de Paris-Cité, APC-Astroparticule et Cosmologie (UMR-CNRS 7164)（巴黎西岱大学，APC-粒子天体物理学与宇宙学（UMR-CNRS 7164））
We consider the problem of determination of a volume of some bounded space-like hypersurfaces in the case of spherically symmetric spacetimes. In the case when the hypersurfaces is cut or bounded by a light-like hypersurface the problem may not be well defined. In order to define properly the volume we introduce the volume forms related to the given foliation (observer) of the considered spacetime. In the case of the constant curvature slice the volume of the hypersurface cut by the horizon (light-like surface) becomes composed of the two parts, outer and inner, treated differently. We compute…
Non-continuous "jumps" of Apparent Horizons occur generically in 3+1 (binary) black hole evolutions. The dynamical trapping horizon framework suggests a spacetime picture in which these "Apparent Horizon jumps" are understood as spatial cuts of a single spacetime hypersurface foliated by (compact) marginally outer trapped surfaces. We present here some work in progress which makes use of uni-parametric sequences of (axisymmetric) binary black hole initial data for exploring the plausibility of this spacetime picture. The modelling of Einstein evolutions by sequences of initial data has proved …
Quasi--local scalar variables approach is applied to a spherically symmetric inhomogeneous Lema\^ıtre--Tolman--Bondi metric containing a mixture of non-relativistic cold dark matter and coupled dark energy with constant equation of state. The quasi--local coupling term considered is proportional to the quasi--local cold dark matter energy density and a quasi--local Hubble factor-like scalar via a coupling constant $α$. The autonomous numerical system obtained from the evolution equations is classified for different choices of the free parameters: the adiabatic constant of the dark energy $w$ a…
We start investigating the extension of the Horizon Quantum Mechanics to the case of spheroidal sources. We first study the location of trapping surfaces in space-times resulting from an axial deformation of static isotropic systems, and show that the Misner-Sharp mass evaluated on the corresponding undeformed spherically symmetric space provides the correct gravitational radius to locate the horizon. We finally propose a way to determine the deformation parameter in the quantum theory.
Assuming the standard effective-field-theoretic formulation of Hawking radiation, we show explicitly how a generic effective theory predicts a firewall from an initially uneventful horizon for a spherically symmetric, uncharged black hole in $D$ dimensions for $D \geq 4$. The firewall is created via higher-derivative interactions within the scrambling time after the collapsing matter enters the trapping horizon. This result manifests the trans-Planckian problem of Hawking radiation and demonstrates the incompatibility between Hawking radiation and the uneventful horizon.
Universal Bounds on Horizons, Photon Spheres, and Shadows: The Role of Energy Conditions in Spherically Symmetric Black Holes
Herzen state Pedagogical University of Russia（俄罗斯赫尔岑国立师范大学）；Center for Theoretical Physics, Khazar University（哈扎尔大学理论物理中心）；SPB branch of SAO RAS（俄罗斯科学院特别天体物理台圣彼得堡分部）
Gravitational collapse of a massless scalar field with the periodic boundary condition in a cubic box is reported. This system can be regarded as a lattice universe model. We construct the initial data for a Gaussian like profile of the scalar field taking the integrability condition associated with the periodic boundary condition into account. For a large initial amplitude, a black hole is formed after a certain period of time. While the scalar field spreads out in the whole region for a small initial amplitude. It is shown that the expansion law in a late time approaches to that of the radia…
Recent research shows that Hawking radiation (HR) is also possible around the trapping horizon of a wormhole. In this article, we show that the HR of gravitino (spin-$3/2$) particles from the traversable Lorentzian wormholes (TLWH) reveals a negative Hawking temperature (HT). We first introduce the TLWH in the past outer trapping horizon geometry (POTHG). Next, we derive the Rarita-Schwinger equations (RSEs) for that geometry. Then, using both the Hamilton-Jacobi (HJ) ansätz and the WKB approximation in the quantum tunneling method, we obtain the probabilities of the emission/absorption modes.…
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We derive precise late-time asymptotics for solutions to the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes including as special cases the Schwarzschild and Reissner-Nordstrom families of black holes. We also obtain late-time asymptotics for the time derivatives of all orders and for the radiation field along null infinity. We show that the leading-order term in the asymptotic expansion is related to the existence of the conserved Newman-Penrose quantities on null infinity. As a corollary we obtain a characterization of all solutions which satisfy Price's…
A stationary and spherically symmetric black hole (For example, Reissner-Nordstrom black hole or Kerr-Newman black hole) has at most one singularity and two horizons. One horizon is the outer event horizon and the other is the inner Cauchy horizon. Can we construct static and spherically symmetric black hole solutions with N horizons and M singularities? De Sitter cosmos has only one apparent horizon. Can we construct cosmos solutions with N horizons? In this article, we present the static and spherically symmetric black hole and cosmos solutions with N horizons and M singularities in the vect…
We consider spherically-symmetric black holes in semiclassical gravity. For a collapsing radiating thin shell we derive a sufficient condition on the exterior geometry that ensures that a black hole is not formed. This is also a sufficient condition for an infalling test particle to avoid the apparent horizon of an existing black hole and approach it only within a certain minimal distance. Taking the presence of a trapped region and its outer boundary --- the apparent horizon--- as the defining feature of black holes, we explore the consequences of their finite time of formation according to a…
