We study the late-time behavior of spherically symmetric solutions of the Yang-Mills equations on Minkowski and Schwarzschild backgrounds. Using nonlinear perturbation theory we show in both cases that solutions having smooth compactly supported initial data posses tails which decay as $t^{-4}$ at timelike infinity. Moreover, for small initial data on Minkowski background we derive the third-order formula for the amplitude of the tail and confirm numerically its accuracy.
One perceives same response of a single-atom detector when placed at a point outside the horizon in Schwarzschild spacetime to that of a static single-atom detector in thermal Minkowski spacetime. So one cannot distinguish Schwarzschild spacetime from thermal Minkowski spacetime by using a single-atom detector. We show that, for Schwarzschild spacetime, beyond a characteristic length scale which is proportional to the inverse of the surface gravity $κ$, the \emph{Resonance Casimir-Polder interaction (RCPI)} between two entangled atoms is characterized by a $1/L^2$ power-law provided the atoms …
The Kerr-Schild version of the Schwarzschild metric contains a Minkowski background which provides a definition of a boosted black hole. There are two Kerr-Schild versions corresponding to ingoing or outgoing principle null directions. We show that the two corresponding Minkowski backgrounds and their associated boosts have an unexpected difference. We analyze this difference and discuss the implications in the nonlinear regime for the gravitational memory effect resulting from the ejection of massive particles from an isolated system. We show that the nonlinear effect agrees with the lineariz…
We prove a sharp inequality for hypersurfaces in the n-dimensional Anti-deSitter-Schwarzschild manifold for general n greater or equal to 3. This inequality generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established by the first author in [3].
We present here the linear regime of the Einstein's field equations in the characteristic formulation. Through a simple decomposition of the metric variables in spin-weighted spherical harmonics, the field equations are expressed as a system of coupled ordinary differential equations. The process for decoupling them leads to a simple equation for $J$ - one of the Bondi-Sachs metric variables - known in the literature as the master equation. Then, this last equation is solved in terms of Bessel's functions of the first kind for the Minkowski's background, and in terms of the Heun's function in …
The holographic entanglement entropies (HEE) associated with four dimensional Schwarzschild and Reisner-Nordström black holes in asymptotically Minkowski spacetimes are investigated. Unlike the cases of asymptotically AdS spacetimes for which the boundaries are always taken at (timelike) conformal infinities, we take the boundaries at either large but finite radial coordinate (far boundary) or very close to the black hole event horizons (near horizon boundary). The reason for such choices is that such boundaries are similar to the conformal infinity of AdS spacetime in that they are all timeli…
We prove global existence for Einstein's equations with a charged scalar field for initial conditions sufficiently close to the Minkowski spacetime without matter. The proof relies on generalized wave coordinates adapted to the outgoing Schwarzschild light cones and the estimates for the massless Maxwell-Klein-Gordon system, on the background of metrics asymptotically approaching Schwarzschild at null infinity in such coordinates, by Kauffman. The generalized wave coordinates are obtained from a change of variables, introduced by Lindblad, to asymptotically Schwarzschild coordinates at null in…
We show that the response rate of (i) a static source interacting with Hawking radiation of massless scalar field in Schwarzschild spacetime (with the Unruh vacuum) and that of (ii) a uniformly accelerated source with the same proper acceleration in Minkowski spacetime (with the Minkowski vacuum) are equal. We show that this equality will not hold if the Unruh vacuum is replaced by the Hartle-Hawking vacuum. It is verified that the source responds to the Hawking radiation near the horizon as if it were at rest in a thermal bath in Minkowski spacetime with the same temperature. It is also verif…
We provide a Finslerian extension of the Schwarzschild metric based on heuristic arguments. The proposed metric asymptotically approaches not the Minkowski space-time but the Bogoslovsky locally anisotropic space-time which arises naturally as a deformation of very special relativity.
We show that the apparently periodic Charap-Duff Yang-Mills `instantons' in time-compactified Euclidean Schwarzschild space are actually time independent. For these solutions, the Yang-Mills potential is constant along the time direction (no barrier) and therefore, there is no tunneling. We also demonstrate that the solutions found to date are three dimensional monopoles and dyons. We conjecture that there are no time-dependent solutions in the Euclidean Schwarzschild background.
Exact radial stability threshold for a Minkowski-Schwarzschild thin shell
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México（墨西哥国立自治大学核科学研究所）；Fesenkov Astrophysical Institute（费森可夫天体物理研究所）；Al-Farabi Kazakh National University（阿里-法拉比哈萨克国立大学）
The Minkowski inequality is a classical inequality in differential geometry, giving a bound from below, on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving versions of this inequality for manifolds other than R^n; for example, such an inequality holds for surfaces in spatial Schwarzschild and AdS-Schwarzschild manifolds. In this note, we adapt a recent analysis of Y. Wei to prove a Minkowski-like inequality for general static asymptotically flat manifolds.
Entropy release from Minkowski breaking in regular Schwarzschild black holes
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa（里斯本大学理学院太空科学与天体物理研究所）；Departamento de Física, Faculdade de Ciências da Universidade de Lisboa（里斯本大学理学院物理系）；Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará（帕拉联邦大学物理学院物理研究生项目）；Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará（帕拉联邦大学精确科学与技术学院）
Post-Minkowskian expansion of the Prompt Response in a Schwarzschild background
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；CENTRA, Departamento de Fisica, Instituto Superior Técnico – IST, Universidade de Lisboa – UL（里斯本大学高等技术学院物理系CENTRA）
We prove uniform decay estimates in the entire exterior of the Schwarzschild black hole for gauge invariant norms on the Yang-Mills fields valued in the Lie algebra associated to the Lie group $SU(2)$. We assume that the initial data are spherically symmetric satisfying a certain Ansatz, and have small energy, which eliminates the stationary solutions which do not decay. In particular, there don't exist any Coulomb type solutions satisfying this Ansatz. We first prove a Morawetz type estimate for the Yang-Mills fields within this setting, using the Yang-Mills equations directly. We then adapt …
A Regge--Wheeler analysis is performed for a novel black hole mimicker `the regular black hole with asymptotically Minkowski core', followed by an approximation of the permitted quasi-normal modes for propagating waveforms. A first-order WKB approximation is computed for spin zero and spin one perturbations of the candidate spacetime. Subsequently, numerical results analysing the respective fundamental modes are compiled for various values of the $a$ parameter (which quantifies the distortion from Schwarzschild spacetime), and for various multipole numbers $\ell$. Both electromagnetic spin one…
Kruskal's extension solves the problem of the arrow of time of the ``Schwarzschild solution'' through combining two Hilbert manifolds by a singular coordinate transformation. We discuss the implications for the singularity problem and the definition of the mass point. The analogy set by Rindler between the Kruskal metric and the Minkowski spacetime is investigated anew. The question is answered, whether this analogy is limited to a similarity of the chosen "Bildräume'', or can be given a deeper, intrinsic meaning. The conclusion is reached by observing a usually neglected difference: the left …
The case of asymptotic Minkowskian space-times is considered. A special class of asymptotic rectilinear coordinates at the spatial infinity, related to a specific system of free falling observers, is chosen. This choice is applied in particular to the Schwarzschild metric, obtaining a vanishing energy for this space-time. This result is compared with the result of some known theorems on the uniqueness of the energy of any asymptotic Minkowskian space, showing that there is no contradiction between both results, the differences becoming from the use of coordinates with different operational mea…
We extend the non-commutative coordinates relationship into other than the Minkowski space-time. We clarify the non-commutativity dependency to the geometrical structure. As well as, we find an inverse map between Riemann's normal and global coordinates. Furthermore, we show that behavior of the corresponding coordinates non-commutativity like as a tensor. All results summarized for the Schwarzschild metric. And they summarized for the space-time in the presence of weak gravitational waves.
We consider the Cauchy problem for a spherically symmetric SU(2) Yang-Mills field propagating outside the Schwarzschild black hole. Although solutions starting from smooth finite energy initial data remain smooth for all times, not all of them scatter since there are non-generic solutions which asymptotically tend to unstable static solutions. We show that a static solution with one unstable mode appears as an intermediate attractor in the evolution of initial data near a border between basins of attraction of two different vacuum states. We study the saddle-point dynamics near this attractor,…
We study scalar perturbations of a recently found 3+1-dimensional FLRW quantum space-time solution in Yang-Mills matrix models. In particular, the linearized Schwarzschild metric is obtained as a solution. It arises from a quasi-static would-be massive graviton mode, and slowly decreases during the cosmic expansion. Along with the propagating graviton modes, this strongly suggests that 3+1 dimensional (quantum) gravity emerges from the IKKT matrix model on this background. For the dynamical scalar modes, non-linear effects must be taken into account. We argue that they lead to non-Ricci-flat m…
The paper deals with a special kind of problems that appear in solutions of Einstein's field equations for extended bodies: many structure-dependent terms appear in intermediate calculations that cancel exactly in virtue of the local equations of motion or can be eliminated by appropriate gauge transformations. For a single body at rest these problems are well understood for both the post-Newtonian and the post-Minkowskian cases. However, the situation is still unclear for approximations of higher orders. This paper discusses this problem for a "body" of spherical symmetry to post-linear order…
We prove that sufficiently regular solutions to the wave equation $\Box_gϕ=0$ on the exterior of the Schwarzschild black hole obey the estimates $|ϕ|\leq C_δv_+^{-{3/2}+δ}$ and $|\partial_tϕ|\leq C_δ v_+^{-2+δ}$ on a compact region of $r$ and along the event horizon. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite $r$ and along the event horizon.
We show that static sources coupled to a massless scalar field in Schwarzschild spacetime give rise to emission and absorption of zero-energy particles due to the presence of Hawking radiation. This is in complete analogy with the description of the bremsstrahlung by a uniformly accelerated charge from the coaccelerated observers' point of view. The response rate of the source is found to coincide with that in Minkowski spacetime as a function of its proper acceleration. This result may be viewed as restoration of the equivalence principle by the Hawking effect.
Solutions to the Yang-Mills field equations which describe exploding or imploding shells of gauge field and which may serve as approximations to the exploding sphaleron are discussed. The solutions are conformally related to $SO(2)\times SO(4)$ invariant Yang-Mills solutions. The behavior of fermions in the gauge field background and higher gauge groups are also considered.
The description of a point mass in general relativity (GR) is given in the framework of the field formulation of GR where all the dynamical fields, including the gravitational field, are considered in a fixed background spacetime. With the use of stationary (not static) coordinates non-singular at the horizon, the Schwarzschild solution is presented as a point-like field configuration in a whole background Minkowski space. The requirement of a stable $η$-causality stated recently in [J.B.Pitts and W.C.Schieve, Found. Phys., v. 34, 211 (2004)] is used essentially as a criterion for testing conf…
We study the peeling of Dirac and Maxwell fields on a Schwarzschild background following the approach developed by the authors in Mason-Nicolas 2009 for the wave equation. The method combines a conformal compactification with vector field techniques in order to work out the optimal space of initial data for a given transverse regularity of the rescaled field across null infinity. The results show that analogous decay and regularity assumptions in Minkowski and in Schwarzschild produce the same regularity across null infinity. The results are valid also for the classes of asymptotically simple …
From the field equations in the linear regime of the characteristic formulation of general relativity, Bishop, for a Schwarzschild's background, and Mädler, for a Minkowski's background, were able to show that it is possible to derive a fourth order ordinary differential equation, called master equation, for the $J$ metric variable of the Bondi-Sachs metric. Once $β$, another Bondi-Sachs potential, is obtained from the field equations, and $J$ is obtained from the master equation, the other metric variables are solved integrating directly the rest of the field equations. In the past, the maste…
An algebraic-hyperbolic method for solving the Hamiltonian and momentum constraints has recently been shown to be well posed for general nonlinear perturbations of the initial data for a Schwarzschild black hole. This is a new approach to solving the constraints of Einstein's equations which does not involve elliptic equations and has potential importance for the construction of binary black hole data. In order to shed light on the underpinnings of this approach, we consider its application to obtain solutions of the constraints for linearized perturbations of Minkowski space. In that case, we…
The black-hole evaporation implies that the quantum-field propagators in a local Minkowski frame acquire a correction, which gives rise to this process. The modification of the propagators causes, in turn, non-trivial local effects due to the radiative/loop diagrams in non-linear QFTs. In particular, there should be imprints of the evaporation in QED, if one goes beyond the tree-level approximation. Of special interest in this respect is the region near the black-hole horizon, which, already at tree level, appears to show highly non-classical features, e.g., negative energy density and energy …
Gravity is not only able to be mimicked in flat spacetimes, but also in curved spacetimes. We study analogue gravity models in curved spacetime by considering the relativistic Gross-Pitaevskii theory and Yang-Mills theory in the fixed background spacetime geometry. The results show that acoustic metrics can be emergent from curved spacetimes yielding a Hadamard product of a real metric-tensor and an analogue metric-tensor. Taking \emph{quantum vortices} as \emph{test particles}, we evaluate their released energy ratio during the "gravitational binding". The $2+1$-dimensional flat Minkowski met…
It has recently been demonstrated (Class. Quantum Grav. 31, 085010, 2014) that the conformally invariant wave equation on a Minkowski background can be solved with a fully pseudospectral numerical method. In particular, it is possible to include spacelike infinity into the numerical domain, which is appropriately represented as a cylinder, and highly accurate numerical solutions can be obtained with a moderate number of gridpoints. In this paper, we generalise these considerations to the spherically-symmetric wave equation on a Schwarzschild background. In the Minkowski case, a logarithmic sin…
We conjecture that (when the notion of Hadamard state is suitably adapted) there is no isometry-invariant Hadamard state for the massive or massless covariant Klein-Gordon equation defined on the region of the Kruskal spacetime to the left of a surface of constant Schwarzschild radius in the right Schwarzschild wedge when Dirichlet boundary conditions are put on that surface. We also prove that, with a suitable definition for 'boost-invariant Hadamard state' (which we call 'strongly boost-invariant globally-Hadamard') which takes into account both the existence of the timelike boundary and the…
In [7] Klainerman introduced the hyperboloidal method to prove the global existence results for nonlinear Klein-Gordon equations by using commuting vector fields. In this paper, we extend the hyperboloidal method from Minkowski space to Lorentzian spacetimes. This approach is developed in [14] for proving, under the maximal foliation gauge, the global nonlinear stability of Minkowski space for Einstein equations with massive scalar fields, which states that, the sufficiently small data in a compact domain, surrounded by a Schwarzschild metric, leads to a unique, globally hyperbolic, smooth and…
Killing-Yano (KY) two and three forms of a class of spherically symmetric space-times that includes the well-known Minkowski, Schwarzschild, Reissner-Nordstrom, Robertson-Walker and six different forms of de Sitter space-times as special cases are derived in a unified and exhaustive manner. It is directly proved that while the Schwarzschild and Reissner-Nordstrom space-times do not accept any KY 3-form and they accept only one 2-form, the Robertson-Walker space-time admits four KY 2-forms and only one KY 3-form. Maximal number of KY-forms are obtained for Minkowski and all known forms of de Si…
We consider the critical gravity theory with a scalar field in four dimensions. We find that this theory has the solution corresponding to the de Sitter (dS), anti-de Sitter (AdS), and Minkowski background depending on whether the action includes the cosmological term or not. The Minkowski background is the solution which cannot be obtained in the model without a scalar field. At the critical point, we show that the Abbott-Deser (AD) mass of the Schwarzschild-de Sitter (SdS) black hole and the energy for the massless graviton vanish, whose situation is not changed from the model without a scal…
On gravitational collapse and integrable singularities
Università di Bologna（博洛尼亚大学）；I.N.F.N., Sezione di Bologna（意大利国家核物理研究所博洛尼亚分部）；Alma Mater Research Center on Applied Mathematics (AM)（阿尔特玛母应用数学研究中心）；Universidad de Tarapacá（塔拉帕卡大学）
We calculate the probability amplitude for tree-level elastic electron-muon scattering in Minkowski spacetime with carefully prepared initial and final wave packets. The obtained nonzero amplitude implies a nonvanishing probability for detecting a recoil electron outside the light-cone of the initial muon. Transposing this Minkowski-spacetime scattering result to a near-horizon spacetime region of a massive Schwarzschild black hole and referring to a previously proposed Gedankenexperiment, we conclude that, in principle, it is possible to have information transfer from inside the black-hole ho…
The Klein-Gordon equation in the Rindler space-time is studied carefully. It is shown that the stable properties depend on using what time coordinate to define the initial time. If we use the Rindler time, the scalar field is stable. Alternatively, if we use the Minkowski time, the scalar field may be regarded unstable to some extent. Furthermore, the complete extension of the Rindler space time is the Minkowski space time, we could also study the stable problem of the Rindler space time by the Klein-Gordon equation completely in the Minkowski coordinates system. The results support that the R…
We formulate a new method to calculate the gravitational reaction force on a particle of mass $μ$ orbiting a massive black hole of mass $M$. In this formalism, the tail part of the retarded Green function, which is responsible for the reaction force, is calculated at the level of the Teukolsky equation. Our method naturally allows a systematic post-Minkowskian (PM) expansion of the tail part at short distances. As a first step, we consider the case of a Schwarzschild black hole and explicitly calculate the first post-Newtonian (1PN) tail part of the Green function. There are, however, a couple…
We extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino-Schoen/Chrusciel-Delay. We combine conformal techniques and vector field methods: a naive adaptation of the ``Morawetz vector field'' to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino-Schoen/Chrusciel-Delay space-times. A good classification of solutions that peel arises from …
In terms of spin-charge separated variables, the Minkowski space Yang-Mills BPST instanton describes a locally conformally flat doubly-wrapped cigar manifold that can be viewed as a Euclidean quantum black hole. An ensemble of instantons then corresponds to a ``spacetime foam'' that creates a locally conformally flat spacetime from ``nothing'' as a quantum fluctuation.
Recently Abramowicz and Bajtlik [ArXiv: 0905.2428 (2009)] have studied the twin paradox in Schwarzschild spacetime. Considering circular motion they showed that the twin with a non-vanishing 4-acceleration is older than his brother at the reunion and argued that in spaces that are asymptotically Minkowskian there exists an absolute standard of rest determining which twin is oldest at the reunion. Here we show that with vertical motion in Schwarzschild spacetime the result is opposite: The twin with a non-vanishing 4-acceleration is younger. We also deduce the existence of a new relativistic ti…
We simulate the satellite constellations of two Global Navigation Satellite Systems: Galileo (EU) and GPS (USA). Satellite motions are described in the Schwarzschild space-time produced by an idealized spherically symmetric non rotating Earth. The trajectories are then circumferences centered at the same point as Earth. Photon motions are described in Minkowski space-time, where there is a well known relation, Coll, Ferrando & Morales-Lladosa (2010), between the emission and inertial coordinates of any event. Here, this relation is implemented in a numerical code, which is tested and applied. …
\emph{Resonance Casimir-Polder interaction(RCPI)} occurs in nature when one or more atoms are in their excited states and the exchange of real photon is involved between them due to vacuum fluctuations of the quantum fields. In recent times, many attempts have been made to show that the curved spacetime such as the \emph{de-Sitter spacetime} can be separated from a \emph{thermal Minkowski spacetime} using \emph{RCPI}. Motivated from these ideas, here we study the \emph{RCPI} between two atoms that interact with a massive scalar field in Schwarzschild spacetime provided the atoms are placed in …
We perturbatively calculate classical radiation in Yang-Mills theory and dilaton gravity, to next-to-leading order in couplings. The radiation is sourced by the scattering of two relativistic massive scalar sources with the dynamical effect taken into account, corresponding to the post-Minkowskian regime in gravity. We show how to arrange the Yang-Mills radiation such that the duality between colors and kinematics is manifest, including the three-term Jacobi identity. The search for duality-satisfying expressions exploits an auxiliary bi-adjoint scalar theory as a guide for locality. The doubl…
The 2-surface characterization of special classical radiative Higgs-, Yang-Mills and linear zero-rest-mass (l.z.r.m) fields with any spin is investigated. We determine all the zero quasi-local mass Higgs- and Yang-Mills field configurations with compact semisimple gauge groups, and show that they are plane waves (provided the Higgs field is massless and linear) and appropriate generalizations of plane waves (`Yang-Mills pp-waves'), respectively. A tensor field (generalizing the energy-momentum tensor for the Maxwell field and of the Bel-Robinson tensor for the linearized gravitational field) i…
