We provide in this paper a first step to obtain the conformal scattering theory for the linearized gravity fields on the Schwarzschild spacetime by using the conformal geometric approach. We will show that the existing decay results for the solutions of the Regge-Wheeler and Zerilli equations obtained recently by L. Anderson, P. Blue and J. Wang \cite{ABlu} is sufficient to obtain the conformal scattering.
The equations governing the perturbations of the Schwarzschild metric satisfy the Regge-Wheeler-Zerilli-Moncrief system. Applying the technique introduced in [2], we prove an integrated local energy decay estimate for both the Regge-Wheeler and Zerilli equations. In these proofs, we use some constants that are computed numerically. Furthermore, we make use of the $r^p$ hierarchy estimates [13, 32] to prove that both the Regge-Wheeler and Zerilli variables decay as $t^{-\frac{3}{2}}$ in fixed regions of $r$.
We prove Birkhoff-type results showing that $L^2$ solutions of the linearized Einstein equations around Riemannian Kottler ("Schwarzschild-anti de Sitter") metrics in arbitrary dimension and horizon topology, which are not controlled by "master functions" are pure gauge. Together with earlier results this implies that the $TT$-gauge-fixed linearized Einstein operator for these metrics is non-degenerate for open ranges of the mass parameter.
We show that existing decay results for scalar fields on the Schwarzschild metric are sufficient to obtain a conformal scattering theory. Then we re-interpret this as an analytic scattering theory defined in terms of wave operators, with an explicit comparison dynamics associated with the principal null geodesic congruences. The case of the Kerr metric is also discussed.
In this paper, we address the issue of linear stability of Schwarzschild space- time subject to certain axisymmetric perturbations. In particular, we prove that associ- ated solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, decay to a linearized Kerr metric. Our method employs a complex line bundle interpretation applied to a connection-level object, allow- ing for direct analysis of this connection-level object by the linearized Einstein equations, in contrast with the recent breakthrough of Dafermos-Holzegel-Rodnians…
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…
We discuss how the assumption of variable equation of state (EoS) allows the elimination of the instability at equilibrium throat radius $a_0=3M$ featured by previous Schwarzschild thin-shell wormhole models. Unobstructed stability regions are found for three choices of variable EoS. Two of these EoS entail linear stability at every equilibrium radius. Particularly, the thin-shell remains stable as $a_0$ approaches the Schwarzschild radius $2M$. A perturbative analysis of the wormhole equation of motion is carried out in the case of variable Chaplygin EoS. The squared proper angular frequency …
A new class of conformal invariants for a given spacetime $M$ is introduced exploiting the conformal geometry of any light ray $Γ$. Each congruence of light rays passing through a given point $p$ defines the sky $S(p)$ of such point. The new conformal invariants are defined on the bundle of skies of the spacetime $M$, being called sky invariants accordingly. The natural conformal covariant derivative defined on a light ray and its associated covariant calculus allows us show the existence of a natural conformal invariant differential of arc that, together with the restriction of the curvature …
The coupled equations for the scalar modes of the linearized Einstein equations around Schwarzschild's spacetime were reduced by Zerilli to a 1+1 wave equation with a potential $V$, on a field $Ψ_z$. For smooth metric perturbations $Ψ_z$ is singular at $r_s=-6M/(\ell-1)(\ell+2)$, $\ell$ the mode harmonic number, and $V$ has a second order pole at $r_s$. This is irrelevant to the black hole exterior stability problem, where $r>2M>0$, and $r_s <0$, but it introduces a non trivial problem in the naked singular case where $M<0$, then $r_s>0$, and the singularity appears in the relevant range of $r…
The generic null geodesic of the Schwarzschild--Kruskal--Szekeres geometry has a natural complexification, an elliptic curve with a cusp at the singularity. To realize that complexification as a Riemann surface without a cusp, and also to ensure conservation of energy at the singularity, requires a branched cover of the space-time over the singularity, with the geodesic being doubled as well to obtain a genus two hyperelliptic curve with an extra involution. Furthermore, the resulting space-time obtained from this branch cover has a Hamiltonian that is null geodesically complete. The full comp…
In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, remain uniformly bounded and decay to a linearized Kerr metric on the exterior region. We employ Hodge decomposition to split the solution into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature. For the co-cl…
Among the coordinates used to construct a conformal compactification of the Schwarzschild spacetime, none of them simultaneously extend smoothly both through an event horizon and beyond null infinity.To construct such coordinates, instead of starting with the Kruskal-Szekeres coordinates we assume direct analytic transformation between Schwarzschild and compactified coordinates and determine their behavior on the event horizon and at null infinity. We then propose an example of such coordinates and illustrate the way they cover the conformally extended Schwarzschild spacetime as well as their …
A gauge-invariant treatment of the monopole- ($l=0$) and dipole ($l=1$) modes in linear perturbations of the Schwarzschild background spacetime is proposed. Through this gauge-invariant treatment, we derived the solutions to the linearized Einstein equation for these modes with a generic matter field. In the vacuum case, these solutions include the Kerr parameter perturbations in the $l=1$ odd modes and the additional mass parameter perturbations of the Schwarzschild mass in the $l=0$ even modes. The linearized version of Birkhoff's theorem is also confirmed in a gauge-invariant manner. In thi…
We show that the hidden conformal symmetry of the Schwarzschild black hole is realized from the AdS$_2$ sector of the AdS$_2\times S^2$, but not from the Rindler spacetime which is the genuine near-horizon geometry of the Schwarzschild black hole. This implies that purely imaginary quasinormal frequencies obtained using the hidden conformal symmetry is not suitable for describing the largely damped modes around the Schwarzschild black hole.
We study the stability of Schwarzschild-Tangherlini (ST) black holes in fourth-order gravity which provides a higher dimensional linearized massive equation. The linearized-Ricci tensor perturbation is employed to exhibit unstable modes featuring the Gregory-Laflamme (GL) instability of higher dimensional black strings, in comparison to the stable ST black holes in Einstein gravity. It turns out that the GL instability of the ST black holes in the fourth-order gravity originates from the massiveness, but not a nature of fourth-order derivative theories giving ghost states.
In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild met- rics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively cor- respond to the closed and co-closed, or polar and axial, solutions in the case of four spacetime dimensions, while the two-tensor portion is a new feature in the higher-dimensional setting. Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge decomposition, we produce decoupled gauge-invariant master qu…
The analytic expression obtained in the preceding project for the massless conformal scalar propagator in the Hartle-Hawking vacuum state for small values of the Schwarzschild radial coordinate above $ r = 2M$ is analytically extended into the interior of the Schwarzschild black hole. The result of the analytical extension coincides with the exact propagator for a small range of values of the Schwarzschild radial coordinate below $ r = 2M$ and is an analytic expression which manifestly features its dependence on the background space-time geometry. This feature as well as the absence of any ass…
We investigate gravitational radiation memory and its corresponding effect on the asymptotic symmetries of a body whose exterior is a boosted Schwarzschild spacetime. First, in the context of linearized theory, we consider such a Schwarzschild body which is initially at rest, then goes through a radiative stage and finally emerges as a boosted Schwarzschild body. We show that the proper retarded solution of the exterior Schwarzschild spacetime for this process can be described in terms of the ingoing Kerr-Schild form of the Schwarzschild metric for both the initial and final states. An outgoin…
The conformal structure of the Schwarzschild-de Sitter spacetime is analysed using the extended conformal Einstein field equations. To this end, initial data for an asymptotic initial value problem for the Schwarzschild-de Sitter spacetime is obtained. This initial data allows to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild-de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild-de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear pertur…
An exact solution is obtained for the gravitational bending of light in static, spherically symmetric metrics which includes the Schwarzschild-de Sitter spacetime and also the Mannheim-Kazanas metric of conformal Weyl gravity. From the exact solution, we obtain a small bending-angle-approximation for a lens system where the source, lens and observer are co-aligned. This expansion improves previous calculations where we systematically avoid parameter ranges that correspond to non-existent null trajectories. The linear coefficient $γ$ characteristic to conformal gravity is shown to contribute en…
We study the equations of conformal gravity, as given by Mannheim, in the weak field limit, so that a linear approximation is adequate. Specializing to static fields with spherical symmetry, we obtain a second-order equation for one of the metric functions. We obtain the Green function for this equation, and represent the metric function in the form of integrals over the source. Near a compact source such as the Sun the solution no longer has Schwarzschild form. We conclude that a solution of Mannheim type cannot exist for these field equations.
The emergence of the quantum gravitational effects in a very high energy regime necessitates some corrections to the thermodynamics of Black holes. In this letter, we investigate a possible modification to the thermodynamics of Schwarzschild anti de Sitter (SAdS) black holes due to rainbow gravity model. Using the correspondence between a $(d+1)-$dimensional SAdS black hole and a conformal filed theory in $d$-dimensional spacetime, one may find the corrections to the Cardy-Verlinde formula from the modified thermodynamics of the black hole. Furthermore, we show that the corrected Cardy-Verlind…
An analysis of conformal geodesics in the Schwarzschild-de Sitter and Schwarzschild-anti de Sitter families of spacetimes is given. For both families of spacetimes we show that initial data on a spacelike hypersurface can be given such that the congruence of conformal geodesics arising from this data cover the whole maximal extension of canonical conformal representations of the spacetimes without forming caustic points. For the Schwarzschild-de Sitter family, the resulting congruence can be used to obtain global conformal Gaussian systems of coordinates of the conformal representation. In the…
On the basis of the C-metric, we investigate the conformal Schwarzschild - deSitter spacetime and compute the source stress tensor and study its properties, including the energy conditions. Then we study its extremal version ($b^{2} = 27m^{2}$, where $b$ is the deS radius and $m$ is the source mass), when the metric is nonstatic. The weak-field version is analyzed in several frames, and the metric becomes flat with the special choice $b = 1/a$, $a$ being the constant acceleration of the Schwarzschild-like mass or black hole. This form is Rindler's geometry in disguise and is also conformal to …
A Lorentz and conformally invariant `Schrödinger-like' equation for a massless complex scalar function $ψ$ is derived from an invariant action, and it is shown how the same $ψ$ can be used to calculate both the gravitational field $h_{μν}$ of linearized Einstein gravity in the TT gauge and the electromagnetic field $F_{μν}$ in the Lorenz gauge, and that the main difference between classical and quantum aspects of such fields lies in a certain condition that the underlying scalar wave must satisfy to keep it nondispersive. It is also shown how the existence of gravitons can be inferred from the…
Within linearized perturbation theory, black holes decay to their final stationary state through the well-known spectrum of quasinormal modes. Here we numerically study whether nonlinearities change this picture. For that purpose we study the ringdown frequencies of gauge-invariant second-order gravitational perturbations induced by self-coupling of linearized perturbations of Schwarzschild black holes. We do so through high-accuracy simulations in the time domain of first and second-order Regge-Wheeler-Zerilli type equations, for a variety of initial data sets. We consider first-order even-pa…
We show that the scalar wave equation at low frequencies in the Schwarzschild geometry enjoys a hidden SL(2,R) invariance, which is not inherited from an underlying symmetry of the spacetime itself. Contrary to what happens for Kerr black holes, the vector fields generating the SL(2,R) are globally defined. Furthermore, it turns out that under an SU(2,1) Kinnersley transformation, which maps the Schwarzschild solution into the near horizon limit AdS_2 x S^2 of the extremal Reissner-Nordstr"om black hole (with the same entropy), the Schwarzschild hidden symmetry generators become exactly the is…
A point of view is presented, according to which, the well known picture with the Schwarzschild black hole in canonical general relativity is one in a whole class of conformal representations of the same physical situation, that are physically equivalent. Symmetry arguments are presented that favour a conformal picture without singularity instead of the Schwarzschild one.
We analyze the behaviour of geodesic motion of test particles in the spacetime of a specific class of axially symmetric static vacuum solutions to the Einstein equations, hereafter referred to as linearized multipole solution (LM). We discuss about its suitability to describe a quasi-spherical spacetime. The existence of an ISCO (innermost stable circular orbit) very close to the (singular) horizon of the source, is established. The existence of such stable orbit, inner than the one of the Schwarzschild metric, as well as the appearance of a splitting in the admissible region of circular orbit…
The monopole solution of the Einstein vacuum field equations (Schwarzschild`s solution) in Weyl coordinates involves a metric function that can be interpreted as the gravitational potential of a bar of length $2m$ with constant linear density. The question addressed in this work is whether similar representations can be constructed for Weyl solutions other than the spherically symmetric one. A new family of static solutions of the axisymmetric vacuum field equations generalizing the M-Q$^{(1)}$ solution is developed. These represent slight deviations from spherical symmetry in terms of the rel…
Noise Kernel and Stress Energy Bi-Tensor of Quantum Fields in Conformally-Optical Metrics: Schwarzschild Black Holes
在论文II [N. G. Phillips和B. L. Hu，previous abstract]我们给出了光时空中量子标量场的噪声核的正则化的细节，采用改进的点分离方案，并对格林函数进行高斯近似.给出了热平坦空间和光学史瓦西度规的正则化噪声核。本文研究Schwarzschild黑洞中标量场的噪声核。点分离方法中的大部分工作是确定发散部分如何共形变换。对于Schwarzschild度规，我们发现在远场区Hawking通量的应力张量的涨落与Campos和Hu早先给出的分析结果相一致[A. Campos和B. L. Hu，Phys. Rev. D { \bf 58}（1998）125021; Int.J.Theor. { \bf 38}（1999）1253]。我们还验证了Page的结果[D. N. Page，Phys. Rev.{ \bf D25}，1499（1982）]的应力张量，其中，虽然经常使用，仍然缺乏严格的证明，因为在他的原始工作的保形变换的直接使用是规避。然而，在光学的情况下，我们表明，高斯近似适用于格林函数产生显着的错误，在史瓦西地平线上的噪声内核。如前所述，我们确定故障发生在第四协变导数阶。
We show that the Schwarzschild-(A)dS black hole mechanics possesses a hidden symmetry under the three-dimensional Poincaré group. This symmetry shows up after having gauge-fixed the diffeomorphism invariance in the symmetry-reduced homogeneous Einstein-$Λ$ model and stands as a physical symmetry of the system. It dictates the geometry both in the black hole interior and exterior regions, as well as beyond the cosmological horizon in the Schwarzschild-dS case. It follows that one can associate a set of non-trivial conserved charges to the Schwarzschild-(A)dS black hole which act in each causall…
We revisit the classical stability of Schwarzschild black hole in fourth-order theories of gravity. On the contrary of the stability of this black hole, it turns out that the linearized perturbations exhibit unstable modes featuring the Gregory-Laflamme instability of five-dimensional black string. This shows clearly an instability of black hole in fourth-order gravity with $α=-β/3$.
Geometric scattering for nonlinear wave equations on the Schwarzschild metric
Thang Long Institute of Mathematics and Applied Sciences (TIMAS), Thang Long University（长龙数学与应用科学研究所（TIMAS），长龙大学）；Faculty of Mathematics and Informatics, Hanoi University of Science and Technology（河内科技大学数学与信息学院）
Thermodynamics and information recovery of Schwarzschild AdS black holes in conformal Killing gravity
Mohammed I University（穆罕默德一世大学）；Astrophysical and Cosmological Center（天体物理与宇宙学中心）；Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa（里斯本大学科学学院天体物理学与空间科学研究所）；Departamento de Física, Faculdade de Ciências da Universidade de Lisboa（里斯本大学科学学院物理系）
For conformally invariant gravity theories defined on Riemannian spacetime and having the Schwarzschild--de-Sitter (SdS) metric as a solution in the Einstein gauge, we consider whether one may conformally rescale this solution to obtain flat rotation curves, such as those observed in galaxies, without the need for dark matter. Contrary to recent claims in the literature, we show that if one works in terms of quantities that can be physically measured, then in any conformal frame the trajectories followed by `ordinary' matter particles are merely the timelike geodesics of the SdS metric, as one…
The linearized Kerr metric is considered and put in some Gauss coordinates which are further {\em intrinsic} ones. The linear and angular 4-momenta of this metric are calculated in these coordinates and the resulting value is just zero. Thus, the global vanishing previously found for the Schwarzschild metric remains linearly stable under slow rotational perturbations of this metric.
We study the gravity action built from two gauge fields corresponding to the generators of the conformal group. Starting with the action from which one can obtain Einstein gravity and conformal gravity upon imposing suitable constraints, we keep two independent gauge fields and integrate out the field corresponding to the generator of Lorentz transformations. We identify the two gauge fields with two vierbeins and perturb them around an Anti--de Sitter space. This gives the linearized equations that differ from both, Einstein gravity and conformal gravity linearized equations. We also study th…
