K. Saifullah, Shair-e-Yazdan
In this paper, conformal motions are studied in plane symmetric static spacetimes. The general solution of conformal Killing equations and the general form of the conformal Killing vector for these spacetimes are presented. All possibilities for the existence of conformal motions in these spacetimes are exhausted.
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Spacetimes which are conformally related to reducible 1+3 spacetimes are considered. We classify these spacetimes according to the conformal algebra of the underlying reducible spacetime, giving in each case canonical expressions for the metric and conformal Killing vectors, and provide physically meaningful examples.
Conformal Ricci collineations of static spherically symmetric spacetimes are studied. The general form of the vector fields generating conformal Ricci collineations is found when the Ricci tensor is non-degenerate, in which case the number of independent conformal Ricci collineations is \emph{fifteen}; the maximum number for 4-dimensional manifolds. In the degenerate case it is found that the static spherically symmetric spacetimes always have an infinite number of conformal Ricci collineations. Some examples are provided which admit non-trivial conformal Ricci collineations, and perfect fluid…
Conformal Killing vectors (CKVs) preserve the spacetime metric up to a factor. Homothetic vectors and Killing vectors are special cases of CKVs. Classification of some classes of spacetimes on the basis of CKVs give interesting results showing how homothetic and Killing vectors which form subsets of the set of CKVs can be recovered as a result of the above classification.
We investigate the conformal geometry of spherically symmetric spacetimes in general without specifying the form of the matter distribution. The general conformal Killing symmetry is obtained subject to a number of integrability conditions. Previous results relating to static spacetimes are shown to be a special case of our solution. The general inheriting conformal symmetry vector, which maps fluid flow lines conformally onto fluid flow lines, is generated and the integrability conditions are shown to be satisfied. We show that there exists a hypersurface orthogonal conformal Killing vector i…
We develop a new method in order to classify the Bianchi I spacetimes which admit conformal Killing vectors (CKV). The method is based on two propositions which relate the CKVs of 1+(n-1) decomposable Riemannian spaces with the CKVs of the (n-1) subspace and show that if 1+(n-1) space is conformally flat then the (n-1) spacetime is maximally symmetric. The method is used to study the conformal algebra of the Kasner spacetime and other less known Bianchi type I matter solutions of General Relativity.
We consider the future causal boundary as a tool to find obstructions to conformal extensions, the latter being a slight generalization to conformal compactifications.
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 …
Conformal matter collineations of the energy-momentum tensor for a general spherically symmetric static spacetime are studied. The general form of these collineations is found when the energy-momentum tensor is non-degenerate, and the maximum number of independent conformal matter collineations is \emph{fifteen}. In the degenerate case of the energy-momentum tensor it is found that these collineations have infinite degrees of freedom. In some subcases of degenerate energy-momentum, the Ricci tensor is non-degenerate, that is, there exist non-degenerate Ricci inheritance collineations.
We determine conformal symmetry classes for the pp-wave spacetimes. This refines the isometry classification scheme given by Sippel and Goenner (1986 {\it Gen. Rel. Grav.} {\bf 18} 1229). It is shown that every conformal Killing vector for the null fluid type $N$ pp-wave spacetimes is a conformal Ricci collineation. The maximum number of proper non-special conformal Killing vectors in a type $N$ pp-wave spacetime is shown to be three, and we determine the form of a particular set of type $N$ pp-wave spacetimes admitting such conformal Killing vectors. We determine the conformal symmetries of e…
In this paper, we present a complete classification of Locally Rotationally Symmetric (LRS) Bianchi type I spacetimes according to their Conformal Ricci Collineations (CRCs). When the Ricci tensor is non-degenerate, a general form of the vector field generating CRCs is found, subject to some integrability conditions. Solving the integrability conditions in different cases, it is found that the LRS Bianchi type I spacetimes admit 7, 10, 11 or 15- dimensional Lie algebra of CRCs for the choice of non-degenerate Ricci tensor. Moreover, it is found that these spacetimes admit infinite number of CR…
Conformal scattering of the wave equation in the Vaidya spacetime
该研究利用向量场方法构建Vaidya时空中标量波动方程的共形散射算子，借助Schwarzschild时空的标量场衰减结果得到能量估计，通过求解零无穷远上的Goursat问题证明迹算子值域稠密，最终得出散射算子为同构的结论。
On the conformal group of a globally hyperbolic spacetime
研究全局双曲时空的因果和共形自同构群，用序理论来回方法，在二维中得出有非紧致柯西曲面且有向的时空与闵可夫斯基平面\(\mathbb{M}^2\)因果同构，实现二维全局双曲时空相关自同构群部分分类及物理对称性描述。
A unified framework for graviton, "partially massless" graviton, and photon fields in de Sitter spacetime under conformal symmetry
本文基于Dirac六锥形式体系和Wigner群论分析，在德西特时空中建立共形不变框架，统一描述引力子、“部分无质量”引力子及光子场。
We discuss the uniqueness of the static spacetimes with non-trivial conformal scalar field. Then, we can show that the spacetime is unique to be the Bocharova-Bronnikov-Melnikov-Bekenstein solution outside the surface composed of the unstable circular orbit of photon(photon surface). In addition, we see that multi-photon surfaces having the same scalar field values do not exist.
Anisotropic Conformal Dark Gravity on the Lorentz Tangent Bundle Spacetime
本研究在洛伦兹切丛的广义拉格朗日几何框架下探究含暗引力的引力场各向异性共形结构，提出两类时空/宇宙学应用，推导的广义类弗里德曼方程可在特定条件下退化为已有研究及广义相对论的对应结果。
We prove that semiclassical gravity in conformally static, globally hyperbolic spacetimes with a massless, conformally coupled Klein-Gordon field is well posed, when viewed as a coupled theory for the dynamical conformal factor of the metric and the Klein-Gordon theory. Namely, it admits unique and stable solutions whenever constrained fourth-order initial data for the conformal factor and suitably defined Hadamard initial data for the Klein-Gordon state are provided on a spacelike Cauchy surface. As no spacetime symmetries are imposed on the conformal factor, the present result implies that, …
Conformal transformations of spacetimes without observer horizons
本文研究无观察者视界因果时空中的共形变换，将其划分为逃逸与非逃逸两类，并应用于 Einstein 静态宇宙及本质性判定。
On the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime
本文利用扩展共形爱因斯坦场方程和共形高斯规范，证明了亚极端Schwarzschild-de Sitter时空宇宙学区域在共形边界附近的非线性稳定性，为宇宙学视界上的微扰稳定性论证奠定基础。
Conformal geodesics and the evolution of spacetimes with positive Cosmological constant
本文提出基于共形测地线的共形高斯规范方法，将正宇宙学常数下的爱因斯坦场方程化为对称双曲系统，并用于研究类德西特时空及施瓦西-德西特时空在共形边界附近的演化性质。
Conformally compactified (3+1)-dimensional Minkowski spacetime may be identified with the projective light cone in (4+2)-dimensional spacetime. In the latter spacetime the special conformal group acts via rotations and boosts, and conformal inversion acts via reflection in a single coordinate. Hexaspherical coordinates facilitate dimensional reduction of Maxwell electromagnetic field strength tensors to (3+1) from (4 + 2) dimensions. Here we focus on the operation of conformal inversion in different coordinatizations, and write some useful equations. We then write a conformal invariant and a p…
Considering the degenerate and non-degenerate cases, we provide a complete classification of static plane symmetric spacetimes according to conformal Ricci collineations (CRCs) and conformal matter collineations (CMCs). In case of non-degenerate Ricci tensor, a general form of vector field generating CRCs is found in terms of unknown functions of t and x, subject to some integrability conditions. The integrability conditions are then solved in different cases depending upon the nature of Ricci tensor and it is concluded that static plane symmetric spacetimes possess 7, 10 or 15-dimensional Lie…
Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime
该研究利用共形几何方法，基于已有的Regge-Wheeler和Zerilli方程解的衰减结果，为施瓦西时空上线性化引力场的共形散射理论构建了初步基础。
Ricci-flat metrics of the ultrahyperbolic signature which enjoy the l-conformal Galilei symmetry are constructed. They involve the AdS_2-metric in a way similar to the near horizon black hole geometries. The associated geodesic equations are shown to describe a second order dynamical system for which the acceleration generators are functionally independent.
An outstanding question lying at the core of the AdS/CFT correspondence in string theory is the holographic prescription problem for Einstein metrics, which asserts that one can slightly perturb the conformal geometry at infinity of the anti-de Sitter space and still obtain an asymptotically anti-de Sitter spacetime that satisfies the Einstein equations with a negative cosmological constant. The purpose of this paper is to address this question by providing a precise quantitative statement of the real-time holographic principle for Einstein spacetimes, to outline its proof and to discuss its p…
Conformal scattering theory for the Dirac field on Kerr spacetime
本研究借助共形几何方法，在Dirac场逐点衰减假设下构建Kerr时空上自旋1/2无质量Dirac方程的共形散射理论，且该构建在逐点衰减已证实的Schwarzschild及极慢转动Kerr黑洞外部区域有效。
We construct a complete conformal scattering theory for Maxwell fields in the static exterior region of a Reissner-Nordstrøm-de Sitter black bole spacetime. This is done using uniform energy decay results that we obtain in a separate paper, to show that the trace operators are injective and have closed ranges. We then solve the Goursat problem (characteristic Cauchy problem) for Maxwell fields on the null boundaries showing that the trace operators are also surjective.
We explicitly prove that a class of finite quantum gravitational theories (in odd as well as in even dimension) is actually a range of anomaly-free conformally invariant theories in the spontaneously broken phase of the conformal Weyl symmetry. At classical level we show how the Weyl conformal invariance is likely able to tame the spacetime singularities that plague not only Einstein gravity, but also local and weakly non-local higher derivative theories. This latter statement is rigorously proved by a singularity theorem that applies to a large class of weakly non-local theories. Following th…
We find all Ricci semi-symmetric as well as all conformally semi-symmetric spacetimes. Neither of these properties implies the other. We verify that only conformally flat spacetimes can be Ricci semi-symmetric without being conformally semi-symmetric and show that only vacuum spacetimes and spacetimes with just a $Λ$-term can be Ricci semi-symmetric without being conformally semi-symmetric.
Generalized Birkhoff theorems and 2+2 direct pruduct spacetimes in Weyl conformal gravity
本文研究了Weyl conformal引力中2+2直接积时空的广义布里科夫定理，并探讨了其与爱因斯坦引力中已知解的共形关系。
We provide a simple proof that conformally semi-symmetric spacetimes are actually semi-symmetric. We also present a complete refined classification of the semi-symmetric spacetimes.
A definition of quantum singularity for the case of static spacetimes has recently been extended to conformally static spacetimes. Here the theory behind quantum singularities in conformally static spacetimes is reviewed, and then applied to a class of spherically symmetric, conformally static spacetimes, including as special cases those studied by Roberts, by Fonarev, and by Husain, Martinez, and Núñez. We use solutions of the generally coupled, massless Klein-Gordon equation as test fields. In this way we find the ranges of metric parameters and coupling coefficients for which classical time…
A class of curves with special conformal properties (conformal curves) is studied on the Reissner-Nordström spacetime. It is shown that initial data for the conformal curves can be prescribed so that the resulting congruence of curves extends smoothly to future and past null infinity. The formation of conjugate points on these congruences is examined. The results of this analysis are expected to be of relevance for the discussion of the Reissner-Nordström spacetime as a solution to the conformal field equations and for the global numerical evaluation of static black hole spacetimes.
The conformal algebra of a 1+3 decomposable spacetime can be computed from the conformal Killing vectors (CKV) of the 3-space. It is shown that the general form of such a 3-CKV is the sum of a gradient CKV and a Killing or homothetic 3-vector. It is proved that spaces of constant curvature always admit such conformal Killing vectors. As an example, the complete conformal algebra of a Gödel-type spacetime is computed. Finally it is shown that this method can be extended to compute the conformal algebra of more general non-decomposable spacetimes.
We show that a class of finite quantum non-local gravitational theories is conformally invariant at classical as well as at quantum level. This is actually a range of conformal anomaly-free theories in the spontaneously broken phase of the Weyl symmetry. At classical level we show how the Weyl conformal invariance is able to tame all the spacetime singularities that plague not only Einstein gravity, but also local and weakly non-local higher derivative theories. The latter statement is proved by a singularity theorem that applies to a large class of weakly non-local theories. Therefore, we are…
It is known that the extremal Reissner-Nordström black hole possesses a discrete conformal isometry that exchanges the black hole horizon with infinity. It is also known that the Reissner-Nordström-de Sitter spacetime posseses a similar discrete conformal isometry which exchanges the event horizon of the black hole with the cosmological horizon. In this short paper we will continue this line of thought and extend the conformal isometry of the latter spacetime and give an unphysical interpretation of the negative root.
We study the utilization of conformal compactification within the conformal approach to solving the constraints of general relativity for asymptotically flat initial data. After a general discussion of the framework, particular attention is paid to simplifications that arise when restricting to a class of initial data which have a certain $U(1)\times U(1)$ conformal symmetry.
All hypersurface homogeneous locally rotationally symmetric spacetimes which admit conformal symmetries are determined and the symmetry vectors are given explicitly. It is shown that these spacetimes must be considered in two sets. One set containing Ellis Class II and the other containing Ellis Class I, III LRS spacetimes. The determination of the conformal algebra in the first set is achieved by systematizing and completing results on the determination of CKVs in 2+2 decomposable spacetimes. In the second set new methods are developed. The results are applied to obtain the classification of …
We consider the problem of matching two spacetimes, the previous and present aeons, in the Conformal Cyclic Cosmology model. The common boundary between them inherits two sets of constraints -- one for each solution of the Einstein field equations extended to the conformal boundaries. The previous aeon is assumed to be an asymptotically de Sitter spacetime, so the standard conformal formulation of the Einstein field equations suffice to derive the constraints on the future null infinity. For the future aeon, which is supposed to evolve from an initial singularity, they are obtained with the us…
In this paper, we find all the Conformal Killing Vectors (CKVs) and their Lie Algebra for the recently reported [cqg1] spherically symmetric, shear-free separable metric spacetimes with non-vanishing energy or heat flux. We also solve the geodesic equatios of motion for the spacetime under consideration.
Conformal rescaling of conformal Yano--Killing tensors and relations between Yano and CYK tensors are discussed. Pullback of these objects to a submanifold is used to construct all solutions of a CYK equation in anti-de Sitter and de Sitter spacetimes. Properties of asymptotic conformal Yano--Killing tensors are examined for asymptotic anti-de Sitter spacetimes. Explicit asymptotic forms of them are derived. The results are used to construct asymptotic charges in asymptotic AdS spacetime. Well known examples like Schwarzschild-AdS, Kerr-AdS and NUT-AdS are examined carefully in the constructio…
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…
We revisit the invariance of the curved spacetime Maxwell equations under conformal transformations. Contrary to standard literature, we include the discussion of the four-current, the wave equations for the four-potential and the field, and the behaviour of gauge conditions under the conformal transformation.
The Robertson-Walker spacetimes are conformally flat and so are conformally invariant under the action of the Lie group SO(4,2), the conformal group of Minkowski spacetime. We find a local coordinate transformation allowing the Robertson-Walker metric to be written in a manifestly conformally flat form for all values of the curvature parameter k continuously and use this to obtain the conformal Killing vectors of the Robertson-Walker spacetimes directly from those of the Minkowski spacetime. The map between the Minkowski and Robertson-Walker spacetimes preserves the structure of the Lie algebr…
We discuss the uniqueness of asymptotically flat and static spacetimes in the $n$-dimensional Einstein-conformal scalar system. This theory potentially has a singular point in the field equations where the effective Newton constant diverges. We will show that the static spacetime with the conformal scalar field outside a certain surface $S_p$ associated with the singular point is unique.
A subclassification of stationary spacetimes, endowed with one timelike and one spacelike Killing vectors, i.e., Petrov $G{_2}I$ on $T_2$ spaces, is proposed. Special attention deserves the Collison's theorem [1] and the branch of metrics circularly cyclicly (axially) symmetric possessing additionally the conformal flatness property reported by Garcia and Campuzano [2].
We present covariant symmetry operators for the conformal wave equation in the (off-shell) Kerr-NUT-AdS spacetimes. These operators, that are constructed from the principal Killing-Yano tensor, its `symmetry descendants', and the curvature tensor, guarantee separability of the conformal wave equation in these spacetimes. We next discuss how these operators give rise to a full set of conformally invariant mutually commuting operators for the conformally rescaled spacetimes and underlie the $R$-separability of the conformal wave equation therein. Finally, by employing the WKB approximation we de…
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 …
This is a summary of how the definition of quantum singularity is extended from static space-times to conformally static space-times. Examples are given.
In this paper we characterize the source-free Einstein-Maxwell spacetimes which have a trace-free Chevreton tensor. We show that this is equivalent to the Chevreton tensor being of pure-radiation type and that it restricts the spacetimes to Petrov types \textbf{N} or \textbf{O}. We prove that the trace of the Chevreton tensor is related to the Bach tensor and use this to find all Einstein-Maxwell spacetimes with a zero cosmological constant that have a vanishing Bach tensor. Among these spacetimes we then look for those which are conformal to Einstein spaces. We find that the electromagnetic f…
This is the second of three papers on Conformal General Relativity (CGR). The conformal group is introduced here as the invariance group of the partial order of causal events in $n$D spacetime. Its general structure, discrete symmetries and field representations are described in detail. The spontaneous breakdown of conformal symmetry is then discussed and the role played by a ghost scalar field and a physical scalar field in 4D spacetime are evidenced. Kinematic--, conformal-- and proper--time hyperbolic coordinates are introduced in a negatively curved Milne spacetime for the purpose of provi…
We analyse various conformal properties of the extremal Reissner-Nordström spacetime. In particular, we obtain conformal representations of the neighbourhoods of spatial infinity, timelike infinity and the cylindrical end ---the so-called cylinders at spatial infinity and at the horizon, respectively--- which are regular with respect to the conformal Einstein field equations and their associated initial data sets. We discuss possible implications of these constructions for the propagation of test fields and non-linear perturbations of the gravitational field close to the horizon.
Connecting 2-Forms, Conformal Transformations, Curvature Invariants and Topological Classes in Einstein Spacetimes
本文利用四维洛伦兹群在2-形式上的运算结构，推广了伯克霍夫定理至施瓦西（反）德西特时空，并构造了共形变换的张量表示，进而将克雷奇曼标量与拓扑欧拉密度及霍金温度联系起来。
We provide a partial characterization of the conformal infinity of asymptotically de Sitter spacetimes by deriving constraints that relate the asymptotics of the stress-energy tensor with conformal geometric data. The latter is captured using recently defined objects, called higher conformal fundamental forms. For the boundary hypersurface, these generalize to higher order the trace-free part of the second form.
Embedding the stationary spacetimes into Brans-Dicke cosmology via conformal transformations
本文利用时间相关共形变换和 BD 场方程研究 FLRW 时空与共形 Kerr-Newman 黑洞，给出不同几何的膨胀解，并揭示原始真空或物质曲率条件对加速膨胀的影响。
In this paper we investigate conformal symmetries in Locally Rotationally Symmetric (LRS) spacetimes using a semitetrad covariant formalism. We demonstrate that a general LRS spacetime which rotates and spatially twists simultaneously has an inherent homothetic symmetry in the plane spanned by the fluid flow lines and the preferred spatial direction. We discuss the nature and consequence of this homothetic symmetry showing that a null Killing horizon arises when the heat flux has an extremal value. We also consider the special case of a perfect fluid and the restriction on the conformal geomet…
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…
In this paper we excavate, for the first time, the most general class of conformal Killing vectors, that lies in the two dimensional subspace described by the null and radial co-ordinates, that are admitted by the generalised Vaidya geometry. Subsequently we find the most general class of generalised Vaidya mass functions that give rise to such conformal symmetry. From our analysis it is clear that why some well known subclasses of generalised Vaidya geometry, like pure Vaidya or charged Vaidya solutions, admit only homothetic Killing vectors but no proper conformal Killing vectors with non co…
We investigate the necessary conditions for the two spacetimes, which are solutions to the Einstein field equations with an anisotropic matter source, to be related to each other by means of a conformal transformation. As a result, we obtain that if one of such spacetimes is a generalization of the Robertson-Walker solution with vanishing acceleration and vorticity, then the other one has to be in this class as well, i.e. the conformal factor will be a constant function on the hypersurfaces orthogonal to the fluid flow lines. The evolution equation for this function appears naturally as a dire…
In this paper we considered the most general form of non conformally flat cylindrically symmetric non-static space-times to study proper conformal motions using direct integration technique. We have shown that very special classes for cylindrically symmetric space-times admit proper conformal motions. This classification also covers non-static and static plane symmetric space-times. In [9] it was claimed that non conformally flat plane symmetric static space-times do not admit proper conformal motion. Here it is also shown that static and non static plane symmetric space-times admit proper con…
We investigate the matching, across cylindrical surfaces, of static cylindrically symmetric conformally flat spacetimes with a cosmological constant $Λ$, satisfying regularity conditions at the axis, to an exterior Linet-Tian spacetime. We prove that for $Λ\leq 0$ such matching is impossible. On the other hand, we show through simple examples that the matching is possible for $Λ>0$. We suggest a physical argument that might explain these results.
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