Hadamard states on spherically symmetric characteristic surfaces, the semi-classical Einstein equations and the Hawking effect
Daan W. Janssen, Rainer Verch
备注：
34 pages, 6 pages appendix, 3 figures. v2: typographical corrections, reordered bibliography, numbered equations, additional references added, more detail added on certain definitions and derivations in appendix, expanded discussion on Minkowski vacuum state
We investigate quasi-free Hadamard states defined via characteristic initial data on null cones centred at the axis of symmetry in spherically symmetric space-times. We characterize the necessary singular behaviour of null boundary two-point functions such that one can define non-linear observables at this null boundary and give formulas for the calculation of these observables. These results extend earlier characterizations of null boundary states defining Hadamard states in the bulk of the null cone. As an application of our derived formulas, we consider their implications for the semi-classical Einstein equations and calculate the vacuum polarization associated with Hawking radiation near a collapsing body.
We consider a quantum field which is in a Hartle-Hawking state propagating in a spherically symmetric black hole space-time. We calculate the components of the stress tensor, renormalized using the Hadamard form of the Green's function, in the exterior region of this space-time. We then specialize these results to the case of the `lukewarm' Riessner-Nordstrom-de Sitter black hole.
We consider a quantum field which is in a Hartle-Hawking state propagating in a general spherically symmetric black hole space-time. We make use of uniform approximations to the radial equation to calculate the components of the stress tensor, renormalized using the Hadamard form of the Green's function, on the horizons of this space-time. We then specialize these results to the case of the `lukewarm' Reissner-Nordstrom-de Sitter black hole and derive some conditions on the stress tensor for the regularity of the Hartle-Hawking state.
Using tunnelling approach, Hawking radiation is derived for a general class of non-static spherically symmetric space time. The standard tunnelling rate formula is obtained using the unified first law of thermodynamics on the trapping horizon and Hawking temperature is measured by an Kodama observer inside the horizon.
We analyse the local behaviour of the two-point correlation function of a quantum state for a scalar field in a neighbourhood of a Killing horizon in a $2+1$-dimensional spacetime, extending the work of Moretti and Pinamonti in a $3+1$-dimensional scenario. In particular we show that, if the state is of Hadamard form in such neighbourhood, similarly to the $3+1$-dimensional case, under a suitable scaling limit towards the horizon, the two-point correlation function exhibits a thermal behaviour at the Hawking temperature. Since the whole analysis rests on the assumption that a Hadamard state ex…
By studying the Hawking radiation of the most general static spherically symmetric black hole arising from scalar and Dirac particles tunnelling, we find the Hawking temperature is invariant in the general coordinate representation (\ref{arbitrary1}), which satisfies two conditions: a) its radial coordinate transformation is regular at the event horizon; and b) there is a time-like Killing vector.
In this paper we present a simple but non-simplistic model of gravitational collapse with thermal emission of pre-Hawking radiation. We apply Einstein equations to a time-dependant spherically symmetric metric and an ultrarelativistic stress-energy tensor. In our model, particles either radially approach the center of the star as collapsing matter, or radially flee from it.
Sijie Gao has recently investigated Hawking radiation from spherically symmetrical gravitational collapse to an extremal R-N black hole for a real scalar field. Especially he estimated the upper bound for the expected number of particles in any wave packet belonging to $\mathcal{H}_{out}$ spontaneously produced from the state $|0>_{in}$, which confirms the traditional belief that extremal black holes do not radiate particles. Making some modifications, we demonstrate that the analysis can go through for a charged scalar field.
We present a derivation of Hawking radiation through tunneling mechanism for a general class of asymptotically flat, spherically symmetric spacetimes. The tunneling rate $Γ\sim \exp{(ΔS)}$ arises as a consequence of the first law of thermodynamics, TdS=dE + PdV. Therefore, this approach demonstrates how tunneling is intimately connected with the first law of thermodynamics through the principle of conservation of energy. The analysis is also generally applicable to any reasonable theory of gravity so long as the first law of thermodynamics for horizons holds in the form, TdS=dE + PdV.
We present a solution of Einstein equations with quintessential matter surrounding a $d$-dimensional black hole, whose asymptotic structures are determined by the state of the quintessential matter. We examine the thermodynamics of this black hole and find that the mass of the black hole depends on the equation of state of the quintessence, while the first law is universal. Investigating the Hawking radiation in this black hole background, we observe that the Hawking radiation dominates on the brane in the low-energy regime. For different asymptotic structures caused by the equation of state o…
The discussions of Hawking radiation via tunneling method have been performed extensively in the case of scalar particles. Moreover, there are also several works in discussing the tunneling method for Hawking radiation by using higher spins, e.g. neutrino, photon, and gravitino, in the background of static black holes. Interestingly, it is found that the Hawking temperature for static black holes using the higher spins particles has no difference compared to the one computed using scalars. In this paper, we study the Hawking radiation for a spherically symmetric and time dependent black holes …
Motivated by the recent work of Robinson and Wilczek, we evaluate the gravitational anomaly of a chiral scalar field in a Vaidya spacetime of arbitrary mass function, and thus the outgoing flux from the time-dependent horizon in that spacetime. We show that this flux differs from that of a perfect blackbody at a fixed temperature. When this flux is taken into account, general covariance in that spacetime is restored. We also generalize their results to the most general static, and spherically symmetric spacetime.
Whenever real particle production occurs in quantum field theory, the imaginary part of the Hadamard Elementary function $G^{(1)}$ is non-vanishing. A method is presented whereby the imaginary part of $G^{(1)}$ may be calculated for a charged scalar field in a static spherically symmetric spacetime with arbitrary curvature coupling and a classical electromagnetic field $A^μ$. The calculations are performed in Euclidean space where the Hadamard Elementary function and the Euclidean Green function are related by $(1/2)G^{(1)}=G_{E}$. This method uses a $4^{th}$ order WKB approximation for the Eu…
In the companion paper [Phys. Rev. D 103 (2021) 10, [2101.02951]] we have derived the short-ranged potentials for the Teukolsky equations for massless spins $(0,1/2,1,2)$ in general spherically-symmetric and static metrics. Here we apply these results to numerically compute the Hawking radiation spectra of such particles emitted by black holes (BHs) in three different ansatz: charged BHs, higher-dimensional BHs, and polymerized BHs arising from models of quantum gravity. In order to ensure the robustness of our numerical procedure, we show that it agrees with newly derived analytic formulas fo…
In the context of the dynamics and stability of black holes in modified theories of gravity, we derive the Teukolsky equations for massless fields of all spins in general spherically-symmetric and static metrics. We then compute the short-ranged potentials associated with the radial dynamics of spin 1 and spin 1/2 fields, thereby completing the existing literature on spin 0 and 2. These potentials are crucial for the computation of Hawking radiation and quasi-normal modes emitted by black holes. In addition to the Schwarzschild metric, we apply these results and give the explicit formulas for …
We aim to study the thermodynamic properties of the spherically symmetric reference frames with uniform acceleration, including the spherically symmetric generalization of Rindler reference frame and the new kind of uniformly accelerated reference frame. We find that, unlike the general studies about the horizon thermodynamics, one cannot obtain the laws of thermodynamics for their horizons in the usual approaches, despite that one can formally define an area entropy (Bekenstein-Hawking entropy). In fact, the common horizon for a set of uniformly accelerated observers is not always exist, even…
Thermal radiations from spherically symmetric black holes have been studied from the point of view of quantum tunneling. In this paper we extend this approach to study radiation of fermions from charged and rotating black strings. Using WKB approximation and Hamilton-Jacobi method we work out the tunneling probabilities of incoming and outgoing fermions and find the correct Hawking temperature for these objects. We show that in appropriate limits the results reduce to those for the uncharged and non-rotating black strings.
We obtain the solution corresponding to a static and spherically symmetric black hole with a cloud of strings (Letelier spacetime) immersed in a quintessential fluid. We discuss some aspects of its thermodynamics. We also present a discussion about Hawking radiation of particles, in the background under consideration.
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…
We survey many of the important properties of spherically symmetric spacetimes as follows. We present several different ways of describing a spherically symmetric spacetime and the resulting metrics. We then focus our discussion on an especially useful form of the metric of a spherically symmetric spacetime in polar-areal coordinates and its properties. In particular, we show how the metric component functions chosen are extremely compatible with notions in Newtonian mechanics. We also show the monotonicity of the Hawking mass in these coordinates. As an example, we discuss how these coordinat…
A static spherically symmetric black hole usually turns out to be either a Schwarzschild black hole or a Reissner-Nordström black hole. This result was summarised by Ruffini and Wheeler as the so-called no hair conjecture which states that for a spherically symmetric black hole only the information about mass ($M$) and electric charge ($e$) of the black hole is available for an external observer. In this work, we calculate the emission rate of charged particles from an asymptotically flat charged spherically symmetric black hole endowed with a scalar hair using a semi-classical tunneling forma…
We investigate the scalar Green function for spherically symmetric spacetimes expressed as a coordinate series expansion in the separation of the points. We calculate the series expansion of the function $V(x,x')$ appearing in the Hadamard parametrix of the scalar Green function to very high order. This expansion is then used to investigate the convergence properties of the series and to estimate its radius of convergence. Using the method of Pade approximants, we show that the series can be extended beyond its radius of convergence to within a short distance of the normal neighborhood boundar…
We generalize the work of Deser and Levin on the unified description of Hawking radiation and Unruh effect to general stationary motions in spherically symmetric black holes. We have also matched the chemical potential term of the thermal spectrum of the two sides for uncharged black holes.
It is well known that spherically symmetric reduction of General Relativity (SSG) leads to non-minimally coupled scalar matter. We generalize (and correct) recent results to Hawking radiation for a class of dilaton models which share with the Schwarzschild black hole non-minimal coupling of scalar fields and the basic global structure. An inherent ambiguity of such models (if they differ from SSG) is discussed. However, for SSG we obtain the rather disquieting result of a negative Hawking flux at infinity, if the usual recipe for such calculations is applied.
The conformal transformation of the Hawking-Hayward quasi-local mass is reexamined. It has been found recently that the conformal transformation of the latter exhibits the 'wrong' conformal factor compared to the way usual masses transform under conformal transformations of spacetime. We show, in analogy with what was found recently for the Misner-Sharp mass, that unlike the purely geometric definition of the Hawking-Hayward mass, the latter exhibits the 'right' conformal factor whenever expressed in terms of its material content via the field equations. The case of conformally invariant scala…
We give a general derivation, for any static spherically symmetric metric, of the relation $T_h=\frac{\cal K}{2π}$ connecting the black hole temperature ($T_h$) with the surface gravity ($\cal K$), following the tunneling interpretation of Hawking radiation. This derivation is valid even beyond the semi classical regime i. e. when quantum effects are not negligible. The formalism is then applied to a spherically symmetric, stationary noncommutative Schwarzschild space time. The effects of back reaction are also included. For such a black hole the Hawking temperature is computed in a closed for…
In this paper, we investigate effects of the minimal length on quantum tunnelling from spherically symmetric black holes using the Hamilton-Jacobi method incorporating the minimal length. We first derive the deformed Hamilton-Jacobi equations for scalars and fermions, both of which have the same expressions. The minimal length correction to the Hawking temperature is found to depend on the black hole's mass and the mass and angular momentum of emitted particles. Finally, we calculate a Schwarzschild black hole's luminosity and find the black hole evaporates to zero mass in infinite time.
We review recent developments in the treatment of spherically symmetric black holes in loop quantum gravity. In particular, we discuss an exact solution to the quantum constraints that represents a black hole and is free of singularities. We show that new observables that are not present in the classical theory arise in the quantum theory. We also discuss Hawking radiation by considering the quantization of a scalar field on the quantum spacetime.
Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a non-negative effective energy density (EED). The SEC ensures the timelike convergence property. However, for both classical and quantum fields, violations of the SEC can be observed even in the simplest of cases, like the Klein-Gordon field. Therefore there is a need to develop theorems with weaker restrictions, namely energy conditions averaged over an entire geodesic and weighted local averages of energy densities such as quantum energy inequalities (QEIs). We …
A simple ordinary differential equation is derived governing the red-shifts of wave-fronts propagating through a non-stationary spherically symmetric space-time. Approach to an event horizon corresponds to approach to a fixed point; in general, the phase portrait of the equation illuminates the qualitative features of the geometry. In particular, the asymptotics of the red-shift as a horizon is approached, a critical ingredient of Hawking's prediction of radiation from black holes, are easily brought out. This asympotic behavior has elements in common with the universal behavior near phase tra…
In this study we obtain the solution of the spherically symmetric de Sitter solution of black holes using a general form of distribution functions which include Gaussian, Rayleigh, and Maxwell-Boltzmann distribution as a special case. We investigate the properties of thermodynamics variables such as the Hawking temperature, the entropy, the mass and the heat capacity of black holes. Moreover, we show that the strong energy condition which includes the null energy condition is satisfied. Finally, we show the regularity of the solution by calculating the scalar curvature and invariant curvature …
We study the greybody factor and power spectra of Hawking radiation for the minimally or nonminimally coupled scalar field with exact numerical method in spherically symmetric Einstein-Gauss-Bonnet-de Sitter black hole spacetime. The effects of scalar coupling constant, angular momentum number of scalar, spacetime dimension, cosmological constant and Gauss-Bonnet coupling constant on the Hawking radiation are studied in detail. Specifically, the Gauss-Bonnet coupling constant always increases the greybody factor in the entire energy regime. Different from the case of Schwarzschild-de Sitter bl…
We study the greybody factors of the scalar fields in spherically symmetric Einstein-Gauss-Bonnet-de Sitter black holes in higher dimensions. We derive the greybody factors analytically for both minimally and non-minimally coupled scalar fields. Moreover, we discuss the dependence of the greybody factor on various parameters including the angular momentum number, the non-minimally coupling constant, the spacetime dimension, the cosmological constant and the Gauss-Bonnet coefficient in detail. We find that the non-minimal coupling may suppress the Hawking radiation, while the Gauss-Bonnet coupl…
In a previous paper we studied the collapse of a spherically symmetric dust distribution (marginally bound LTB) in d-dimensional AdS spacetime and obtained the condition for the formation of trapped surfaces. Here we extend the analysis by giving the canonical theory for the same and subsequently quantize the system by solving the Wheeler-DeWitt equation. We show that for the case of small dust perturbations around a black hole the wave functionals so obtained describe an AdS-Schwarzschild black hole in equilibrium with a thermal bath at Hawking temperature and show the non-trivial dependence …
We compute the black hole horizon entanglement entropy for a massless scalar field in the brick wall model by incorporating the minimal length. Taking the minimal length effects on the occupation number $n(ω,l)$ and the Hawking temperature into consideration, we obtain the leading UV divergent term and the subleading logarithmic term in the entropy. The leading divergent term scales with the horizon area. The subleading logarithmic term is the same as that in the usual brick wall model without the minimal length.
The thermodynamic properties of a shell of bosons with the inner surface locating at Planck length away from the horizon of Schwarzschild black holes by using statistical mechanics are studied. The covariant partition function of bosons is obtained, from which the Bose-Einstein condensation of bosons is found at a non-zero temperature in the curved spacetimes. As a special case of bosons, we analyze the entropy of photon gas near the horizon of the Schwarzschild black hole, which shows an area dependence similar to the Bekenstein-Hawking entropy. The results may offer new perspectives on the s…
Hawking radiation of a non-asymptotically flat (NAF) 4-dimensional spherically symmetric and static dilatonic black hole (BH) via the Hamilton-Jacobi (HJ) method has been studied. In addition to the naive coordinates, we have used four more different coordinate systems which are well-behaved at the horizon. Except the isotropic coordinates, direct computation of the HJ method leads us the standard Hawking temperature for all coordinate systems. The isotropic coordinates render possible to get the index of refraction extracting from the Fermat metric. It is explicitly shown that the index of re…
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…
We study the formation and the evaporation of a spherically symmetric black hole in conformal gravity. From the collapse of a spherically symmetric thin shell of radiation, we find a singularity-free non-rotating black hole. This black hole has the same Hawking temperature as a Schwarzschild black hole with the same mass, and it completely evaporates either in a finite or in an infinite time, depending on the ensemble. We consider the analysis both in the canonical and in the micro-canonical statistical ensembles. Last, we discuss the corresponding Penrose diagram of this physical process.
The present thesis attempts to study the effect of scalar fields on Hawking radiation and quasinormal modes of black holes. We selected a static, spherically symmetric electrically charged black hole with an additional scalar 'hair' for our analysis. The scalar 'hair' is sourced by a scalar field, conformally coupled to the Einstein-Hilbert action. The scalar field can survive even in the absence of the black hole's electric charge and is characterized as a 'primary hair'. This scalar field changes the gravitational constant, and hence modifies the ADM (Arnowitt-Deser-Missner) mass of the blac…
Spherically, plane, or hyperbolically symmetric spacetimes with an additional hypersurface orthogonal Killing vector are often called ``static'' spacetimes even if they contain regions where the Killing vector is non-timelike. It seems to be widely believed that an energy-momentum tenor for a matter field compatible with these spacetimes in general relativity is of the Hawking-Ellis type I everywhere. We show in arbitrary $n(\ge 3)$ dimensions that, contrary to popular belief, a matter field on a Killing horizon is not necessarily of type I but can be of type II. Such a type-II matter field on…
Understood in terms of pure states evolving into mixed states, the possibility of information loss in black holes is closely related to the global causal structure of spacetime, as is the existence of event horizons. However, black holes need not be defined by event horizons, and in fact we argue that in order to have a fully unitary evolution for black holes, they should be defined in terms of something else, such as a trapping horizon. The Misner-Sharp mass in spherical symmetry shows very simply how trapping horizons can give rise to black hole thermodynamics, Hawking radiation and singular…
We consider a D-dimensional self-gravitating spherically symmetric configuration of a generalized electro-magnetic n-form F and a dilatonic scalar field, admitting an interpretation in terms of intersecting p-branes. For theories with multiple times, selection rules are obtained, which obstruct the existence of p-branes in certain subspaces. General static solutions are obtained under a specific restriction on the model parameters, which corresponds to the known "intersection rules". More special families of solutions (with equal charges for some of the F-field components) are found with weake…
In this paper, we study the global behavior of solutions to the spherically symmetric coupled Einstein-Klein-Gordon (EKG) system in the presence of a negative cosmological constant. We prove that the Schwarzschild-AdS spacetimes (the trivial black hole solutions of the EKG system for which $ϕ=0$ identically) are asymptotically stable: Small perturbations of Schwarzschild-AdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching a Schwarzschild-AdS spacetime. The main difficulties in the proof arise from the lack of monotonicity for the Hawking m…
We numerically compute renormalized expectation values of quadratic operators in a quantum field theory (QFT) of free Dirac fermions in curved two-dimensional (Lorentzian) spacetime. First, we use a staggered-fermion discretization to generate a sequence of lattice theories yielding the desired QFT in the continuum limit. Numerically-computed lattice correlators are then used to approximate, through extrapolation, those in the continuum. Finally, we use so-called point-splitting regularization and Hadamard renormalization to remove divergences, and thus obtain finite, renormalized expectation …
Spherically symmetric black holes in Gravity from Entropy and spontaneous emission
Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University（扬州大学物理科学与技术学院引力与宇宙学中心）；Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet（吕勒奥理工学院工程技术与数学系）；National Institute for Theoretical and Computational Sciences (NITheCS)（南非国家理论与计算科学研究所）
We investigate the phonon radiation from a spherically symmetrical, stationary, viscid-free son-ic black hole by using a semi-classical method. The backreaction of the radiated phonon is taken into account. We obtain the phonon emission temperature, and it is consistent with the Hawking's formula.
We consider the problem of picking a physically motivated vacuum state on a spherically symmetric spacetime with an extra conformal Killing vector, as opposed to an extra Killing vector as in the Schwarzschild case. Considering a conformal symmetry instead of a symmetry allows us to consider spacetimes that are dynamical and not static (like Schwarzschild). The extra conformal symmetry allows us to calculate the response of particle detectors however. We look at the specific example of a self-similar LTB spacetime that represents a spherically symmetric but inhomogeneous cosmology. We remark t…
Classical (quasinormal) and quantum (Hawking) radiations are investigated for test fields in the background of a four dimensional, spherically symmetric and asymptotically flat black hole in the Einstein-dilaton-Gauss-Bonnet (EdGB) theory. The geometry of the EdGB black hole deviates from the Schwarzschild geometry only slightly. Therefore, here we observe that the quasinormal spectrum also deviates from its Schwarzschild limit at most moderately, allowing for a $9\%$ decrease in the damping rate and up to a $6\%$ decrease in the real oscillation frequency. However, the intensity of Hawking ra…
Quantum gravity effects are likely to play a crucial role in determining the outcome of gravitational collapse during its final stages. In this contribution we will outline a canonical quantization of the LeMaitre-Tolman-Bondi models, which describe the collapse of spherical, inhomogeneous, non-rotating dust. Although there are many models of gravitational collapse, this particular class of models stands out for its simplicity and the fact that both black holes and naked singularity end states may be realized on the classical level, depending on the initial conditions. We will obtain the appro…
We study the hamiltonian and constraints of spherically symmetric dilaton gravity model. We find the ADM mass of the solution representing the Schwarzchild black hole in thermal equilibrium with the Hawking radiation.
The tunneling formalism in the Hamilton-Jacobi approach is adopted to study Hawking radiation of massless Dirac particles from spherically symmetric black hole spacetimes incorporating the effects of the generalized uncertainty principle. The Hawking temperature is found to contain corrections from the generalized uncertainty principle. Further, we show from this result that the ratio of the GUP corrected energy of the particle to the GUP corrected Hawking temperature is equal to the ratio of the corresponding uncorrected quantities. This result is then exploited to compute the Hawking tempera…
We introduce a new attractor mechanism to find the entropy for spherically symmetric extremal black holes. The key ingredient is to find a two-dimensional (2D) dilaton gravity with the dilaton potential $V(ϕ)$. The condition of an attractor is given by $\nabla^2ϕ=V(ϕ_0)$ and $\bar{R}_2=-V^{\prime}(ϕ_0)$ and for a constant dilaton $ ϕ=ϕ_0$, these are also used to find the location of the degenerate horizon $r=r_{e}$ of an extremal black hole. As a nontrivial example, we consider an extremal regular black hole obtained from the coupled system of Einstein gravity and nonlinear electrodynamics. Th…
