We show that there exist steady states of the massless Einstein-Vlasov system which surround a Schwarzschild black hole. The steady states are (thick) shells with finite mass and compact support. Furthermore we prove that an arbitrary number of shells, necessarily well separated, can surround the black hole. To our knowledge this is the first result of static self-gravitating solutions to any massless Einstein-matter system which surround a black hole. We also include a numerical investigation about the properties of the shells.
It is shown that the generalized Collins--Stewart radiation and Milne solutions are attractors of the massless Einstein--Vlasov system for Bianchi types II and V spacetimes, respectively. The proof is based on an energy method and bootstrap argument which are used to determine the decay rates of the perturbations away from the attractors.
We analyse the Kantowski-Sachs cosmologies with Vlasov matter of massive and massless particles using dynamical systems analysis. We show that generic solutions are past and future asymptotic to the non-flat locally rotationally symmetric Kasner vacuum solution. Furthermore, we establish that solutions with massive Vlasov matter behave like solutions with massless Vlasov matter towards the singularities.
We construct a one-parameter family of static and spherically symmetric solutions to the Einstein-Vlasov system bifurcating from the Schwarzschild spacetime. The constructed solutions have the property that the spatial support of the matter is a finite, spherically symmetric shell located away from the black hole. Our proof is mostly based on the analysis of the set of trapped timelike geodesics and of the effective potential energy for static spacetimes close to Schwarzschild. This provides an alternative approach to the construction of static solutions to the Einstein-Vlasov system in a neig…
Minkowski space is shown to be globally stable as a solution to the Einstein--Vlasov system in the case when all particles have zero mass. The proof proceeds by showing that the matter must be supported in the "wave zone", and then proving a small data semi-global existence result for the characteristic initial value problem for the massless Einstein--Vlasov system in this region. This relies on weighted estimates for the solution which, for the Vlasov part, are obtained by introducing the Sasaki metric on the mass shell and estimating Jacobi fields with respect to this metric by geometric qua…
New developments related to our recent study of the accretion of the Vlasov gas onto a moving Schwarzschild black hole are presented. We discuss the low-temperature limit of the mas accretion rate and a simple Monte Carlo simulation used to check the results obtained in this limit. We also comment on several numerical aspects related with momentum integrals expressing the particle density current and the particle density.
We prove the stability of de Sitter space-time as a solution to the Einstein-Vlasov system with massless particles. The semi-global stability of Minkowski space-time is also addressed. The proof relies on conformal techniques, namely Friedrich's conformal Einstein field equations. We exploit the conformal invariance of the massless Vlasov equation on the cotangent bundle and adapt Kato's local existence theorem for symmetric hyperbolic systems to prove a long enough time of existence for solutions of the evolution system implied by the Vlasov equation and the conformal Einstein field equations…
We derive an exact, axially symmetric solution representing stationary accretion of the relativistic, collisionless Vlasov gas onto a moving Schwarzschild black hole. The gas is assumed to be in thermal equilibrium at infinity, where it obeys the Maxwell-Jüttner distribution. The Vlasov equation is solved analytically in terms of suitable action-angle variables. We provide explicit expressions for the particle current density and accretion rates. In the limit of infinite asymptotic temperature of the gas, we recover the qualitative picture known form the relativistic Bondi-Hoyle-Lyttleton accr…
The spherically symmetric Einstein-Vlasov system is considered in Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem is the issue of global existence for initial data without size restrictions. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to previous methods. We prove that global existence holds outside the centre in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in \cite{R…
Future stability of expanding spatially homogeneous FLRW solutions of the spherically symmetric Einstein--massless Vlasov system with spatial topology $\mathbb{R}^3$
In recent years, the AdS instability conjecture, put forward by Dafermos-Holzegel in 2006, has attracted a substantial amount of numerical and heuristic studies. Following the pioneering work of Bizon-Rostworowski, research efforts have been mainly focused on the study of the spherically symmetric Einstein-scalar field system. The first rigorous proof of the instability of AdS in the simplest spherically symmetric setting, namely for the Einstein-null dust system, was obtained in [G. Moschidis, A proof of the instability of AdS for the Einstein-null dust system with an inner mirror, 2017, arXi…
This paper continues the investigation of the formation of naked singularities in the collapse of collisionless matter initiated in [RV]. There the existence of certain classes of non-smooth solutions of the Einstein-Vlasov system was proved. Those solutions are self-similar and hence not asymptotically flat. To obtain solutions which are more physically relevant it makes sense to attempt to cut off these solutions in a suitable way so as to make them asymptotically flat. This task, which turns out to be technically challenging, will be carried out in this paper. [RV] A. D. Rendall and J. J. L…
In this paper, we initiate the study of the asymptotically AdS initial-boundary value problem for the Einstein-massless Vlasov system with $Λ<0$ in spherical symmetry. We will establish the existence and uniqueness of a maximal future development for the characteristic initial-boundary value problem in the case when smooth initial data are prescribed on a future light cone $\mathcal{C}^{+}$ emanating from a point at $\{r=0\}$ and a reflecting boundary condition is imposed on conformal infinity $\mathcal{I}$. We will then prove a number of continuation criteria for smooth solutions of the spher…
We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin $s=\pm \frac{1}{2}$ components of the Dirac field a uniform bound of a positive definite energy and an integrated local energy decay estimate from a symmetric hyperbolic wave system. Based on these estimates, we further show that these components have globally pointwise decay $fv^{-3/2-s}τ^{-5/2+s}$ as both an upper and a lower bound outside the black hole, with function $f$ finite and explicitly expressed in terms of the initial data and the coordinates. This establ…
We construct spherically symmetric, static solutions to the Einstein-Vlasov system with non-vanishing cosmological constant $Λ$. The results are divided as follows. For small $Λ>0$ we show existence of globally regular solutions which coincide with the Schwarzschild-deSitter solution in the exterior of the matter sources. For $Λ<0$ we show via an energy estimate the existence of globally regular solutions which coincide with the Schwarzschild-Anti-deSitter solution in the exterior vacuum region. We also construct solutions with a Schwarzschild singularity at the center regardless of the sign o…
We find confluent Heun solutions to the radial equations of two Halilsoy-Badawi metrics. For the first metric, we studied the radial part of the massless Dirac equation and for the second case, we studied the radial part of the massless Klein-Gordon equation.
In this paper the existence of a class of self-similar solutions of the Einstein-Vlasov system is proved. The initial data for these solutions are not smooth, with their particle density being supported in a submanifold of codimension one. They can be thought of as intermediate between smooth solutions of the Einstein-Vlasov system and dust. The motivation for studying them is to obtain insights into possible violation of weak cosmic censorship by solutions of the Einstein-Vlasov system. By assuming a suitable form of the unknowns it is shown that the existence question can be reduced to that …
We address the concerns raised by the author of "Comment: Some exact quasinormal frequencies of a massless scalar field in Schwarzschild spacetime" appeared in arXiv:1807.05940. In particular, we explain why the analysis in our paper [Phys. Rev. D{\bf{98}}, 024017 (2018)] is relevant and correct.
We present the manifestly covariant Lagrangian of a massless polarized particle, that implies all dynamic and algebraic equations as the conditions of extreme of this variational problem. The model allows for minimal interaction with a gravitational field, leading to the equations, coinciding with Maxwell equations in the geometrical optics approximation. The model allows also a wide class of non minimal interactions, which suggests an alternative way to study the electromagnetic radiation beyond the leading order of geometrical optics. As a specific example, we construct a curvature-dependent…
Coordinate Independence of the Schwarzschild Black Hole Accretion Vlasov Gas Model
College of Mathematics and Physics, Hunan University of Arts and Sciences（常德学院数学与物理学院）；Hunan Province Key Laboratory Integration of Photoelectric Information Integration and Optical Manufacturing Technology（湖南省光电信息集成与光学制造技术重点实验室）；Center for Astrophysics, Guangzhou University（广州大学天体物理中心）
A short derivation is given of the weak gravitational field approximation to the scalar massless propagator in Schwarzschild spacetime obtained by Paszko using the path-integral approach. The contribution from the direct coupling of the quantum field to the scalar curvature is explictly included. The propagator complies with Hadamard's pattern, and the vacuum state is consistent with the perturbative version of the Boulware vacuum. The momentum space propagator is computed for massless or massive particles to the same perturbative order. The renormalized value of $\ < ϕ^2(x)\ >$ for the massle…
In this paper, we show that massless Dirac waves in the Schwarzschild geometry decay to zero at a rate $t^{-2λ}$, where $λ=1, 2,...$ is the angular momentum. Our technique is to use Chandrasekhar's separation of variables whereby the Dirac equations split into two sets of wave equations. For the first set, we show that the wave decays as $t^{-2λ}$. For the second set, in general, the solutions tend to some explicit profile at the rate $t^{-2λ}$. The decay rate of solutions of Dirac equations is achieved by showing that the coefficient of the explicit profile is exactly zero. The key ingredient…
A new branch of quasinormal modes for a massless scalar field propagating on the Schwarzschild spacetime was recently announced [1807.04513]. We review the quasinormal modes characterisation and arguments and identify the flaws in their proof. Then, we preset explicit counterexamples to such arguments. Finally, we study the modes via alternative methods and do not find the new branch. We conclude against their interpretation.
We solve the coupled Einstein-Vlasov system in spherical symmetry using direct numerical integration of the Vlasov equation in phase space. Focusing on the case of massless particles we study critical phenomena in the model, finding strong evidence for generic type I behaviour at the black hole threshold that parallels what has previously been observed in the massive sector. For differing families of initial data we find distinct critical solutions, so there is no universality of the critical configuration itself. However we find indications of at least a weak universality in the lifetime scal…
We reduce the equations governing the spherically symmetric perturbations of static spherically symmetric solutions of the Einstein-Vlasov system (with either massive or massless particles) to a single stratified wave equation $-ψ_{,tt}=Hψ$, with $H$ containing second derivatives in radius, and integrals over energy and angular momentum. We identify an inner product with respect to which $H$ is symmetric, and use the Ritz method to approximate the lowest eigenvalues of $H$ numerically. For two representative background solutions with massless particles we find a single unstable mode with a gro…
We express the Einstein-Vlasov system in spherical symmetry in terms of a dimensionless momentum variable $z$ (radial over angular momentum). This regularises the limit of massless particles, and in that limit allows us to obtain a reduced system in independent variables $(t,r,z)$ only. Similarly, in this limit the Vlasov density function $f$ for static solutions depends on a single variable $Q$ (energy over angular momentum). This reduction allows us to show that any given static metric which has vanishing Ricci scalar, is vacuum at the centre and for $r>3M$ and obeys certain energy condition…
The scalar scattering cross section is computed for Schwarzschild-Tangherlini black holes on the brane. The cross sections are computed numerically via the partial-wave method. The phase shifts are found numerically and also via the Born approximation. It is shown that spacetimes with more than six dimensions present a finite scattering cross section in all directions.
We show that not all quasinormal modes of a massless scalar field in the Schwarzschild metric are encoded in the corresponding characteristic equation expressed by means of a continued fraction. We provide an analytical formula for these new hitherto missing quasinormal modes, and in doing that we also construct a generalization of the Gauss convergence criterion.
In this paper we are studying the scattering of massless Dirac fermions by Schwarzschild and Reissner-Nordström black holes. This is done by applying the partial wave analysis to the scattering modes obtained after solving the massless Dirac equation in the asymptotic regions of the two black hole geometries. We succeed to obtain analytic phase shifts with the help of which the scattering cross section is computed. The glory and spiral scattering phenomena are showed to be present like in the case of massive fermion scattering by black holes.
We prove global stability of the Minkowski spacetime in the wave coordinates system for the massive Einstein-Vlasov system. In particular, compared with previous results by Lindblad-Taylor, in which the Vlasov part is assumed to have compact support assumption, and Fajman-Joudioux-Smulevci, in which the spacetime is assumed to be exact Schwarzschild in the exterior region, we do not impose any compact support condition for the Vlasov part and allow a large class of non-isotropic perturbations for the metric part, which decay at rate $\langle r\rangle^{-1+}$ towards space infinity.
Black holes are among the most intriguing objects in nature. They are believed to be fully described by General Relativity (GR), and the astrophysical black holes are expected to belong to the Kerr family, obeying the no-hair theorems. Alternative theories of gravity or parameterized deviations of GR allow black hole solutions, which have additional parameters other than mass and angular momentum. We analyze a Schwarzschild-like metric, proposed by Johannsen and Psaltis, characterized by its mass and a deformation parameter. We compute the absorption cross section of massless scalar waves for …
In Schwarzschild spacetime, the gravitational spin-orbit couplings of the massless Dirac field and the photon field can be studied in a unified way. In contrary to the previous investigations presented mainly at the quantum-mechanical level, our work is presented at the level of quantum field theory without resorting to the Foldy-Wouthuysen transformation. If massless Dirac particles and photons have the same momentums, their energy-level splittings due to the gravitational spin-orbit couplings are the same. Massless Dirac particles and photons coming from the Hawking radiations are partially …
Stochastic motion of massless particles near event horizons in Schwarzschild and Reissner-Nordström spacetimes
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；School of Mathematics, Guangxi University（广西大学数学学院）；Guangxi Center for Mathematical Research, Guangxi University（广西大学广西数学研究中心）；Ningde No.5 Middle School of Fujian（福建省宁德第五中学）
We provide a systematic and comprehensive derivation of the linearized dynamics of massive and partially massless spin-2 particles in a Schwarzschild (anti) de Sitter black hole background, in four and higher spacetime dimensions. In particular, we show how to obtain the quadratic actions for the propagating modes and recast the resulting equations of motion in a Schrödinger-like form. In the case of partially massless fields in Schwarzschild de Sitter spacetime, we study the isospectrality between modes of different parity. In particular, we prove isospectrality analytically for modes with mu…
We employ quantum kinetic theory to investigate local quantum physics in the background of spherically symmetric and neutral black holes formed through the gravitational collapse. For this purpose in mind, we derive and study the covariant Wigner distribution function $\mathcal{W}(x,p)$ near to and far away from the black-hole horizon. We find that the local density of the particle number is negative in the near-horizon region, while the entropy density is imaginary. These pose a question whether kinetic theory is applicable in the near-horizon region. We elaborate on that and propose a possib…
