We investigate photon surfaces and their stability in a less symmetric spacetime, a general static warped product with a warping function acting on a Riemannian submanifold of codimension two. We find a one-dimensional pseudopotential that gives photon surfaces as its extrema regardless of the spatial symmetry of the submanifold. The maxima and minima correspond to unstable and stable photon surfaces, respectively. It is analogous to the potential giving null circular orbits in a spherically symmetric spacetime. We also see that photon surfaces indeed exist for the spacetimes which are solutions to the Einstein equation. The parameter values for which the photon surfaces exist are specified. As we show finally, the pseudopotential arises due to the separability of the null geodesic equation, and the separability comes from the existence of a Killing tensor in the spacetime. The result leads to the conclusion that photon surfaces may exist even in a less symmetric spacetime if the spacetime admits a Killing tensor.
Using gyroscopes we generalize results, obtained for the gravitomagnetic clock effect in the particular case when the exterior spacetime is produced by a rotating dust cylinder, to the case when the vacuum spacetime is described by the general cylindrically symmetric Lewis spacetime. Results are contrasted with those obtained for the Kerr spacetime.
We derive matter collineations for some static spherically symmetric spacetimes and compare the results with Killing, Ricci and Curvature symmetries. We conclude that matter and Ricci collineations are not, in general, the same.
Using Lie symmetry methods for differential equations we have investigated the symmetries of a Lagrangian for a plane symmetric static spacetime. Perturbing this Lagrangian we explore its approximate symmetries. It has a non-trivial first-order approximate symmetry.
We prove strong cosmic censorship for T^2-symmetric cosmological spacetimes (with spatial topology T^3 and vanishing cosmological constant Lambda) with collisionless matter. Gowdy symmetric spacetimes constitute a special case. The formulation of the conjecture is in terms of generic C^2-inextendibility of the metric. Our argument exploits a rigidity property of Cauchy horizons, inherited from Killing fields.
Self-similar spacetimes are of importance to cosmology and to gravitational collapse problems. We show that self-similarity or the existence of a homothetic Killing vector field for spherically symmetric spacetimes implies the separability of the spacetime metric in terms of the co-moving coordinates and that the metric is, uniquely, the one recently reported in [cqg1]. The spacetime, in general, has non-vanishing energy-flux and shear. The spacetime admits matter with any equation of state.
Boost-rotation symmetric spacetimes are the only locally asymptotically flat axially symmetric electrovacuum spacetimes with a further symmetry that are radiative. They are realized by uniformly accelerated particles of various kinds or black holes. Their general properties are summarized. Several examples of boost-rotation symmetric solutions of the Maxwell and Einstein equations are studied: uniformly accelerated electric and magnetic multipoles, the Bonnor-Swaminarayan solutions, the C-metric and the spinning C-metric.
Boost-rotation symmetric vacuum spacetimes with spinning sources which correspond to gravitational field of uniformly accelerated spinning "particles" are studied. Regularity conditions and asymptotic properties are analyzed. News functions are derived by transforming the general spinning boost-rotation symmetric vacuum metric to Bondi-Sachs coordinates.
We present an axially symmetric spacetime which contains closed timelike curves, and hence violates the causality condition. The metric belongs to type III in the Petrov classification scheme with vanishing expansion, shear and twist. The matter-energy represents a pure radiation field with a negative cosmological constant. The spacetime is asymptotically anti-de Sitter space in the radial direction.
Dirac's constraint analysis and the symplectic structure of geodesic equations are obtained for the general cylindrically symmetric stationary spacetime. For this metric, using the obtained first order Lagrangian, the geodesic equations of motion are integrated, and found some solutions for Lewis, Levi-Civita, and Van Stockum spacetimes.
A complete classification of locally spherically symmetric four-dimensional Lorentzian spacetimes is given in terms of their local conformal symmetries. The general solution is given in terms of canonical metric types and the associated conformal Lie algebras. The analysis is based upon the local conformal decomposition into 2+2 reducible spacetimes and the Petrov type. A variety of physically meaningful example spacetimes are discussed.
This work deals with intersection points: conjugate points and cut points, of timelike geodesics emanating from a common initial point in special spacetimes. The paper contains three results. First, it is shown that radial timelike geodesics in static spherically symmetric spacetimes are globally maximal (have no cut points) in adequate domains. Second, in one of ultrastatic spherically symmetric spacetimes, Morris--Thorne wormhole, it is found which geodesics have cut points (and these must coincide with conjugate points) and which ones are globally maximal on their entire segments. This resu…
An analitical approximation of $<ϕ^2>$ for a massive scalar field in a zero temperature vacuum state in static spherically symmetric spacetimes is obtained. The calculations are based on the method for computing vacuum expectations values for scalar fields in general static spherically symmetric spacetimes derived by Anderson, Hiscock and Samuel [Phys. Rev. D {\bf 51}, 4337 (1995)]. The analitical approximation is used to compute $<ϕ^2>$ in Schwarzschild and wormhole spacetimes.
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.
We prove a global foliation result, using areal time, for T^2 symmetric spacetimes with a positive cosmological constant. We then find a class of solutions that exhibit AVTD behavior near the singularity.
Sonic point/photon sphere (SP/PS) correspondence is a theoretical phenomenon which appears in fluid dynamics on curved spacetime and its existence has been recently proved in quite wide situations as theorems. The theorems state that a sonic point (SP) of radiation fluid flow must be on an unstable photon sphere (PS) when the fluid flows radially or rotationally on an equatorial plane in spherically symmetric spacetime of arbitrary dimensions. In this paper, we investigate SP/PS correspondence in spherically, planarly and hyperbolically symmetric spacetime. As the corresponding objects of phot…
The crease set of an event horizon is studied in a spacetime with discrete or continuous symmetry. It determines possible topologies on spatial sections of an event horizon. We thereby investigate the classification of stable topological structure of the crease sets in a spacetime with any symmetry. In practice, we show the classification of the crease set in axially symmetric spacetime. By that we realize the topological structure of axially symmetric event horizons. We will finc that many new topological structures become stable which is not stable without symmetry.
We expound in detail a method frequently used to reduce the Dirac equation in D-dimensional (D >= 4) spherically symmetric spacetimes to a pair of coupled partial differential equations in two variables. As a simple application of these results we exactly calculate the quasinormal frequencies of the uncharged Dirac field propagating in the D-dimensional Nariai spacetime.
We dispose of some objections raised by Manko et al. (gr-qc/0604091) on a recently published paper on the role of Poynting vector in the ocurrence of vorticity in electrovaccum spacetimes (2006, Class. Quantum Grav. 23, 2395)
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 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].
It is known that spherically symmetric spacetimes admit flat spacelike foliations. We point out a simple method of seeing this result via the Hamiltonian constraints of general relativity. The method yields explicit formulas for the extrinsic curvatures of the slicings.
Noether symmetries from geodetic Lagrangian for time-conformal plane symmetric spacetime are presented. Here, time conformal factor is used to find the approximate Noether symmetries. This is a generalization of the idea discussed by I. Hussain and A. Noether symmetries from geodetic Lagrangian for time-conformal plane symmetric spacetime are presented. Here, time conformal factor is used to find the approximate Noether symmetries. This is a generalization of the idea discussed by I. Hussain and A. Qadir [3,4], where they obtained approximate Noether symmetries from Lagrangian for a particular…
We study properties of resonant solutions to the scalar wave equation in several axially symmetric spacetimes. We prove that non-axial resonant modes do not exist neither in the Lanczos dust cylinder, the $(2+1)$ extreme BTZ spacetime nor in a class of simple rotating wormhole solutions. Moreover, we find unstable solutions to the wave equation in the Lanczos dust cylinder and in the $r^2 <0$ region of the extreme $(2+1)$ BTZ spacetime, two solutions that possess closed timelike curves. Similarities with previous results obtained for the Kerr spacetime are explored.
We summarize the consequences of the twin assumptions of (discrete) self-similarity and spherical symmetry for the global structure of a spacetime. All such spacetimes can be constructed from two building blocks, the "fan" and "splash". Each building block contains one radial null geodesic that is invariant under the self-similarity (self-similarity horizon).
We give a definition and derive the equations of motion for the center of mass and angular momentum of an axially symmetric, isolated system that emits gravitational and electromagnetic radiation. A central feature of this formulation is the use of Newman-Unti cuts at null infinity that are generated by worldlines of the spacetime. We analyze some consequences of the results and comment on the generalization of this work to general asymptotically flat spacetimes.
Boost-rotation symmetric spacetimes are exceptional as they are the only exact asymptotically flat solutions to the Einstein equations describing spatially bounded sources ("point-like" particles, black holes) undergoing non-trivial motion ("uniform acceleration") with radiation. We construct the Newtonian limit of these spacetimes: it yields fields of uniformly accelerated sources in classical mechanics. We also study the special-relativistic limit of the charged rotating C-metric and so find accelerating electromagnetic magic field.
We investigate matter collineations of plane symmetric spacetimes when the energy-momentum tensor is degenerate. There exists three interesting cases where the group of matter collineations is finite-dimensional. The matter collineations in these cases are either four, six or ten in which four are isometries and the rest are proper.
The objective of the present paper is to study 4-dimensional almost pseudo Ricci symmetric perfect fluid spacetimes $(APRS)_4$. We show that a Robertson-Walker spacetime is $(APRS)_4$ and vice versa under certain condition imposed on its scale factor. Some popular toy models of $F(R)$-gravity are also studied under the current setting and various energy conditions are investigated.
We present a covariant decomposition of Einstein's Field Equations which is particularly suitable for perturbations of spherically symmetric -- and general locally rotationally symmetric -- spacetimes. Based upon the utility of the 1+3 covariant approach to perturbation theory in cosmology, the semi-tetrad, 1+1+2 approach presented here should be useful for analysing perturbations of a variety of systems in a covariant and gauge-invariant manner. Such applications range from stellar objects to cosmological models such as the spherically symmetric Lemaitre-Tolman-Bondi solutions or the class of…
In this paper, the exact vacuum solution of a general plane symmetric spacetime is investigated in metric f(R) gravity with the assumption of constant Ricci scalar. For this solution, we have studied the generalized Landau-Lifshitz energy-momentum complex in this theory to determine the energy distribution expressions for some specific f(R) models. Also, we show that these models satisfy the constant curvature condition.
Analytic Solutions for Geodesic Motion in Static Axially Symmetric Spacetime
Coordenação de Astronomia e Astrofísica, Observatório Nacional（天文与天体物理协调局，国家天文台）；Departamento de Física Teórica, Universidade do Estado do Rio de Janeiro (UERJ)（理论物理系，里约热内卢州立大学）；Sorbonne Université, UPMC Université Paris 06, LERMA, Observatoire de Paris-Meudon（索邦大学，巴黎第六大学，天体物理学研究实验室，巴黎-默东天文台）
We study scalar field and electromagnetic perturbations on Locally Rotationally Symmetric (LRS) class II spacetimes, exploiting a recently developed covariant and gauge-invariant perturbation formalism. From the Klein-Gordon equation and Maxwell's equations, respectively, we derive covariant and gauge-invariant wave equations for the perturbation variables and thereby find the generalised Regge-Wheeler equations for these LRS class II spacetime perturbations. As illustrative examples, the results are discussed in detail for the Schwarzschild and Vaidya spacetime, and briefly for some classes o…
This paper investigates the global properties of a class of spherically symmetric spacetimes. The class contains the maximal development of asymptotically flat spherically symmetric initial data for a wide variety of coupled Einstein-matter systems. For this class, it is proven here that the existence of a single trapped surface or marginally trapped surface implies the completeness of future null infinity and the formation of an event horizon whose area radius is bounded by twice the final Bondi mass.
Some new aspects of axially symmetric spacetimes are discussed. These results open the door for future interplay between analytical and numerical studies. The new developments are based on the role of the total mass in axial symmetry. Finally, a list of relevant open problems is presented. These problems can be hopefully solved with an interaction between numerical and analytical insights.
In Schwarzschild spacetime the value $r=3m$ of the radius coordinate is characterized by three different properties: (a) there is a ``light sphere'', (b) there is ``centrifugal force reversal'', (c) it is the upper limiting radius for a non-transparent Schwarschild source to act as a gravitational lens that produces infinitely many images. In this paper we prove a theorem to the effect that these three properties are intimately related in {\em any} spherically symmetric static spacetime. We illustrate the general results with some examples including black-hole spacetimes and Morris-Thorne worm…
This paper is devoted to the study of matter collineations of plane symmetric spacetimes (for a particular class of spacetimes) when the energy-momentum tensor is non-degenerate. There exists many interesting cases where we obtain proper matter collineations. The matter collineations in these cases are {\it four}, \emph{five}, {\it six}, \emph{seven} and {\it ten} with some constraints on the energy-momentum tensor. We have solved some of these constraints to obtain solutions of the Einstein field equations.
In this talk we show that any spherically symmetric spacetime admits locally a maximal spacelike slicing. The above condition is reduced to solve a decoupled system of first order quasi-linear partial differential equations. The solution may be accomplished analytical or numerically. We provide a general procedure to construct such maximal slicings.
Some future global properties of cosmological solutions for the Einstein-Vlasov-Maxwell system with surface symmetry are presented. Global existence is proved, the homogeneous spacetimes are future complete for causal trajectories, and the same is true for inhomogeneous plane-symmetric solutions with small initial data. In the latter case some decay properties are also obtained at late times. Similar but slightly weaker results hold for hyperbolic symmetry.
We construct conformastat spherically symmetric spacetimes representing anisotropic fluid matter distributions from given solutions of the Poisson's equation of Newtonian gravity and its corresponding circular speed profile. As simple examples, we present three families of spherically symmetric spacetimes which we apply in constructing new models of relativistic anisotropic thick spherical shells, and of relativistic galaxy models composite by a central spherical bulge, the thick disk and the dark matter halo, writing in this case the metric in cylindrical coordinates. Moreover, the geodesic m…
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…
It is shown that only the maximally-symmetric spacetimes can be expressed in both the Robertson-Walker form and in static form - there are no other static forms of the Robertson-Walker spacetimes. All possible static forms of the metric of the maximally-symmetric spacetimes are presented as a table. The findings are generalized to apply to functionally more general spacetimes: it is shown that the maximally symmetric spacetimes are also the only spacetimes that can be written in both orthogonal-time isotropic form and in static form.
Marginally trapped surfaces (MTSs) are commonly used in numerical relativity to locate black holes. For dynamical black holes, it is not known generally if this procedure is sufficiently reliable. Even for Schwarzschild black holes, Wald and Iyer constructed foliations which come arbitrarily close to the singularity but do not contain any MTSs. In this paper, we review the Wald-Iyer construction, discuss some implications for numerical relativity, and generalize to the well known Vaidya spacetime describing spherically symmetric collapse of null dust. In the Vaidya spacetime, we numerically lo…
We construct a new class of plane-symmetric solutions possessing a curvature singularity which is null and weak, like the spacetime singularity at the Cauchy horizon of spinning (or charged) black holes. We then analyse the stability of this singularity using a rigorous non-perturbative method. We find that within the framework of (linearly-polarized) plane-symmetric spacetimes this type of null weak singularity is locally stable. Generically, the singularity is also scalar-curvature. These observations support the new picture of the null weak singularity inside spinning (or charged) black hol…
We apply the Dirac-Bergmann algorithm for the analysis of constraints to gauge theories defined on spherically symmetric black hole backgrounds. We find that the constraints for a given theory are modified on such spacetimes through the presence of additional contributions from the horizon. As a concrete example, we consider the Maxwell field on a black hole background, and determine the role of the horizon contributions on the dynamics of the theory.
In this paper, the gauge choices in general spherically symmetric spacetimes have been explored. We construct the gauge invariant variables and the master equations for both the Detweiler easy gauge and the Regge-Wheeler gauge, respectively. The particular cases for $l=0,1$ are also been investigated. Our results provide analytical calculations of metric perturbation in general spherically symmetric spacetimes, which can be applied to various cases, including the Effective-One-Body problem. A simple example is presented to show how the metric perturbation components are related to the source p…
We describe a method to analyze causal geodesics in static and spherically symmetric spacetimes of Kerr-Schild form which, in particular, allows for a detailed study of the geodesics in the vicinity of the central singularity by means of a regularization procedure based on a generalization of the McGehee regularization for the motion of Newtonian point particles moving in a power-law potential. The McGehee regularization was used by Belbruno and Pretorius to perform a dynamical system regularization of the central singularity of the motion of massless test particles in the Schwarzschild spacet…
We give the necessary and sufficient (local) conditions for a metric tensor to be a non conformally flat spherically symmetric solution. These conditions exclusively involve explicit concomitants of the Riemann tensor. As a direct application we obtain the {\em ideal} labeling of the Schwarzschild, Reissner-Nordström and Lemaître-Tolman-Bondi solutions.
In this paper we derive the inverse spatial Laplacian for static, spherically symmetric backgrounds by solving Poisson's equation for a point source. This is different from the electrostatic Green function, which is defined on the four dimensional static spacetime, while the equation we consider is defined on the spatial hypersurface of such spacetimes. This Green function is relevant in the Hamiltonian dynamics of theories defined on spherically symmetric backgrounds, and closed form expressions for the solutions we find are absent in the literature. We derive an expression in terms of elemen…
We present calculations of gyroscope precession in spacetimes described by Levi-Civita and Lewis metrics, under different circumstances. By doing so we are able to establish a link between the parameters of the metrics and observable quantities, providing thereby a physical interpretation for those parameters, without specifying the source of the field.
We examine potential deformations of inner black hole and cosmological horizons in Reissner-Nordström de-Sitter spacetimes. While the rigidity of the outer black hole horizon is guaranteed by theorem, that theorem applies to neither the inner black hole nor past cosmological horizon. Further for pure deSitter spacetime it is clear that the cosmological horizon can be deformed (by translation). For specific parameter choices, it is shown that both inner black hole and cosmological horizons can be infinitesimally deformed. However these do not extend to finite deformations. The corresponding res…
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 prove that all spherically symmetric static spacetimes which are both regular at r=0 and satisfying the single energy condition rho + p_r + p_t >= 0 cannot contain any black hole region (equivalently, they must satisfy 2m/r <= 1 everywhere). This result holds even when the spacetime is allowed to contain a finite number of matching hypersurfaces. This theorem generalizes a result by Baumgarte and Rendall when the matter contents of the space-time is a perfect fluid and also complements their results in the general non-isotropic case.
In this paper we consider a static, cylindrically symmetric spacetime with coincident f(Q) gravity. Since the field equation of this spacetime in symmetric teleparallel gravity is suitable for choosing the function of f(Q) in the form of power series and exponential forms, perfect fluid solutions of these forms are discussed. Energy densities, directional pressures and energy conditions are plotted and analysed for a few different metric potentials. Although cosmic strings violate all energy conditions in both f(Q) functions, the Levi-Civita solution violates all energy conditions in the power…
The conformal field equations are used to discuss the local existence of spherically symmetric solutions to the Einstein-Yang-Mills system which behave asymptotically like the anti-de Sitter spacetime. By using a gauge based on conformally privileged curves we obtain a formulation of the problem in terms of an initial boundary value problem on which a general class of maximally dissipative boundary conditions can be discussed. The relation between these boundary conditions and the notion of mass on asymptotically anti-de Sitter spacetimes is analysed.
