We study static spherically-symmetric solutions of non-Abelian gauge theory coupled to Conformal Gravity. We find solutions for the self-gravitating pure Yang-Mills case as well as monopole-like solutions of the Higgs system. The former are localized enough to have finite mass and approach asymptotically the vacuum geometry of Conformal Gravity, while the latter do not decay fast enough to have analogous properties.
Cosmic string solutions of the abelian Higgs model with conformal coupling to gravity are shown to exist. The main characteristics of the solutions are presented and the differences with respect to the minimally coupled case are studied. An important difference is the absence of Bogomolnyi cosmic string solutions for conformal coupling. Several new features of the abelian Higgs cosmic strings of both types are discussed. The most interesting is perhaps a relation between the angular deficit and the central magnetic field which is bounded by a critical value.
Motivated by the Higgs Inflation scenario, we study static spherically-symmetric solutions of the non-Abelian Higgs model coupled non-minimally to Gravity. We find solutions for the self-gravitating sphaleron as well as monopole-like solutions and study the impact of the non-minimal coupling on their properties. Finally we discuss shortly the possibility that these solutions interact gravitationally with star-like objects like boson stars.
We study 4-dimensional non-charged and charged spherically symmetric spacetimes in conformal teleparallel equivalent of general relativity. We apply the gravitational field equations in non-charged and charged spacetimes to the diagonal and non-diagonal vierbeins and derive their sets of the non-linear differential equations. It is demonstrated that the Schwarzschild, for the non-charged case, and the Reissner-Nordström, for the charged case, are the only black hole solutions for the spherically symmetric case in the framework of the conformal teleparallel equivalent of general relativity theo…
Static spherically symmetric solutions for conformal gravity in three dimensions are found. Black holes and wormholes are included within this class. Asymptotically the black holes are spacetimes of arbitrary constant curvature, and they are conformally related to the matching of different solutions of constant curvature by means of an improper conformal transformation. The wormholes can be constructed from suitable identifications of a static universe of negative spatial curvature, and it is shown that they correspond to the conformal matching of two black hole solutions with the same mass.
Three different methods to quantize the spherically symmetric sector of electromagnetism are presented: First, it is shown that this sector is equivalent to Abelian BF-theory in four spacetime dimensions with suitable boundary conditions. This theory, in turn, is quantized by both a reduced phase space quantization and a spin network quantization. Finally, the outcome is compared with the results obtained in the recently proposed general quantum symmetry reduction scheme. In the magnetically uncharged sector, where all three approaches apply, they all lead to the same quantum theory.
In canonical gravity, general covariance is implemented by hypersurface-deformation symmetries on phase space. The different versions of hypersurface deformations required for full covariance have complicated interplays with one another, governed by non-Abelian brackets with structure functions. For spherically symmetric space-times, it is possible to identify a certain Abelian substructure within general hypersurface deformations, which suggests a simplified realization as a Lie algebra. The generators of this substructure can be quantized more easily than full hypersurface deformations, but …
Conformal invariance can ameliorate or eliminate the singularities residing in the black holes, and may still exist in the strong gravity regimes close to these black holes. In this paper, we try to probe this conformal invariance by looking into the wave absorption and scattering by the nonsingular static spherical black holes. The partial and total absorption cross section, as well as the differential scattering cross section, are presented for black holes with different choices of conformal parameters. Although the photon trajectories are unchanged from the Schwarzschild case since the spac…
We study spherically-symmetric structures in Conformal Gravity and in a scalar-tensor extension and gain some more insight about these gravitational theories. In both cases we analyze solutions in two systems: perfect fluid solutions and boson stars of a self-interacting complex scalar field. In the purely tensorial (original) theory we find in a certain domain of parameter space finite mass solutions with a linear gravitational potential but without a Newtonian contribution. The scalar-tensor theory exhibits a very rich structure of solutions whose main properties are discussed. Among them, s…
We present the explicit form for all the four dimensional, static, spherically symmetric solutions in $(4+n)$-d Abelian Kaluza-Klein theory by performing a subset of $SO(2,n)$ transformations corresponding to four $SO(1,1)$ boosts on the Schwarzschild solution, supplemented by $SO(n)/SO(n-2)$ transformations. The solutions are parameterized by the mass $M$, Taub-Nut charge $a$, $n$ electric $\vec{\cal Q}$ and $n$ magnetic $\vec{\cal P}$ charges. Non-extreme black holes (with zero Taub-NUT charge) have either the Reissner-Nordstr\" om or Schwarzschild global space-time. Supersymmetric extreme b…
We construct a scalar polynomial curvature invariant that transforms covariantly under a conformal transformation from any spherically symmetric metric. This invariant has the additional property that it vanishes on the event horizon of any black hole that is conformal to a static spherical metric.
A study of proper teleparallel conformal vector field in spherically symmetric static space-times is given using the direct integration technique and diagonal tetrads. In this study we show that the above space-times do not admit proper teleparallel conformal vector fields.
We consider dynamical spherically symmetric spacetimes, which are conformal to the static spherically symmetric metrics, and find new solutions of Einstein equations by symmetry considerations. Our study help us classify various conformal Black Holes that are embedded within a dynamic background into the one class of solutions with the same conformal symmetry. In addition, Thermodynamics, mathematical and gravitational properties are addressed. These solutions point to have a better resolution of the meaning of the Black Holes in the dynamic background.
Conformally flat spherically symmetric cosmological models representing a charged perfect fluid as well as a bulk viscous fluid distribution have been obtained. The cosmological constant Λis found positive and is a decreasing function of time which is consistent with the recent supernovae observations. The physical and geometrical properties of the models are discussed.
We present an algorithm for the construction of conformal coordinates in the interior of a spherically symmetric, collapsing matter cloud in general relativity. This algorithm is based on the numerical integration of radial null geodesics. As an application we generate conformal diagrams for collapsing spherical dust clouds and analyze the causal structure of the resulting spacetimes.
The boundary energy - momentum tensors for a static observer in the conformally flat Rindler geometry are considered. We found the surface energy is positive far form the Planck world but the transversal pressures are negative. The kinematical parameters associated to a nongeodesic congruence of static observers are computed. The entropy $S$ corresponding to the degrees of freedom on the two surface of constant $ρ$ and $t$ equals the horizon entropy of a black hole with a time dependent mass and the Padmanabhan expression $E = 2 S T$ is obeyed. The two surface shear tensor is vanishing but the…
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.
A contribution linear in r to the gravitational potential can be created by a suitable conformal duality transformation: the conformal factor is 1/(1+r)^2 and r will be replaced by r/(1+r), where r is the Schwarzschild radial coordinate. Thus, every spherically symmetric solution of conformal Weyl gravity is conformally related to an Einstein space. This result finally resolves a long controversy about this topic. As a byproduct, we present an example of a spherically symmetric Einstein space which is a limit of a sequence of Schwarzschild-de Sitter space-times but which fails to be expressabl…
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…
A nonstationary spherically symmetric problem for conformal geometrodynamics equations is considered and general exact solutions in quadratures are obtained. Involvement of Weyl degrees of freedom allows us to consider the problem with arbitrary initial data, as for the conformal geometrodynamics equations the Cauchy problem is set up without connections to initial data. The results of this paper are not confined with the framework of the perturbation theory and open up new avenues for study of the process of space-time singularity evolution in time.
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…
All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat representations of these geometries. We find all solutions for the conformal factor in 3, 4 and 6 dimensions. We write them in closed form, either in terms of elliptic or elementary functions. We are particularly interested in 3-dimensional spaces because of the link to General Relativity. In particular, all 3-dimensional constant negative scalar curvature spheri…
We present an algorithm for the construction of conformal coordinates in the interior of a spherically symmetric, collapsing matter cloud in general relativity. This algorithm is based on the numerical integration of the radial null geodesics and a local analysis of their behavior close to the singularity. As an application, we consider a collapsing spherical dust cloud, generate the corresponding conformal diagram and analyze the structure of the resulting singularity. A new bound on the initial data which guarantees that the singularity is visible from future null infinity is also obtained.
We find a new property in $W^2$-conformal gravity in spherical symmetry. We demonstrate that the charge of the electromagnetic field varies with respect to the partial scaling symmetry (conformal transformations in subspaces of a spacetime) in the conformal gravity. We find that the electric and magnetic charges can vanish or appear via partial conformal transformations for the on-shell configurations in the conformal gravity. We find a solution of a Yang-Mills SU(2)-charged black hole in the conformal gravity, and further demonstrate that the SU(2) charge has similar properties with electroma…
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…
Set of analytic solutions of the geodesic equation in a spherical conformal spacetime is presented. Solutions of this geodesics can be expressed in terms of the Weierstrass {\wp} function and the Kleinian σ function. Using conserved energy and angular momentum we can characterize the different orbits. Also, considering parametric diagrams and effective potentials, we plot some possible orbits. Moreover, with the help of analytical solutions, we investigate the light deflection for such an escape orbit.
This paper has been removed by arXiv administrators because it plagiarizes astro-ph/0611537, astro-ph/0506256, astro-ph/0203033, astro-ph/0311128, gr-qc/0505144, astro-ph/0611460, and astro-ph/0610840.
We prove the theorem: The necessary and sufficient condition for a spherically symmetric spacetime to represent an isothermal perfect fluid (barotropic equation of state with density falling off as inverse square of the curvature radius) distribution without boundary is that it is conformal to the ``minimally'' curved (gravitation only manifesting in tidal acceleration and being absent in particle trajectory) spacetime.
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 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…
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.
In this manuscript, we study the nearly flat approximation of a conformally invariant gravitational theory in metric measure space (MMS). In addition, we investigate the transformation of the energy-momentum tensor in this context and obtain the vacuum solution of MMS and its weak field limit in the spherically symmetric coordinates. We show that while it is already a vacuum solution, it can simulate dark matter when restricted to the framework of general relativity, i.e., a symmetry-broken conformal frame. This is done by employing a density function which is an essential part of MMS. We deri…
Spherically symmetric models of loop quantum gravity have been studied recently by different methods that aim to deal with structure functions in the usual constraint algebra of gravitational systems. As noticed by Gambini and Pullin, a linear redefinition of the constraints (with phase-space dependent coefficients) can be used to eliminate structure functions, even Abelianizing the more-difficult part of the constraint algebra. The Abelianized constraints can then easily be quantized or modified by putative quantum effects. As pointed out here, however, the method does not automatically provi…
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 present a new class of solutions for static spherically symmetric wormhole spacetimes in conformal gravity and outline a detailed method for their construction. As an explicit example, we construct a class of traversable and non-traversable wormholes that are locally conformal to Schwarzschild-(anti) de Sitter spacetimes. These wormhole spacetimes are exact vacuum solutions in, but not being limited to, Weyl gravity and conformal scalar-tensor theories. Importantly, the method implies that every conformal theory of gravity with static spherically symmetric solutions will trivially contain w…
We construct a conformal scattering theory for Dirac fields in the interior of a Reissner-Nordström-like black hole, between the black hole event horizon and the Cauchy horizon. The main result is a resolution of the characteristic Cauchy problem for the Dirac equation on the horizons by solving a system of wave equations and re-interpreting its solution as the components of the required Dirac solution.
The general structure of the spherically symmetric solutions in the Weyl conformal gravity is described. The corresponding Bach equations are derived for the special type of metrics, which can be considered as the representative of the general class. The complete set of the pure vacuum solutions is found. It consists of two classes. The first one contains the solutions with constant two-dimensional curvature scalar of our specific metrics, and the representatives are the famous Robertson--Walker metrics. One of them we called the "gravitational bubbles", which is compact and with zero Weyl ten…
We study in spherical symmetry the conformal compactification for hyperboloidal foliations with nonvanishing constant mean curvature. The conformal factor and the coordinates are chosen such that null infinity is at a fixed radial coordinate location.
A calculation of the entropy of static, electrically charged, black holes with spherical, toroidal, and hyperbolic compact and oriented horizons, in D spacetime dimensions, is performed. These black holes live in an anti-de Sitter spacetime, i.e., a spacetime with negative cosmological constant. To find the entropy, the approach developed by Solodukhin is followed. The method consists in a redefinition of the variables in the metric, by considering the radial coordinate as a scalar field. Then one performs a 2+(D-2) dimensional reduction, where the (D-2) dimensions are in the angular coordinat…
We adopt a reference-metric approach to generalize a covariant and conformal version of the Z4 system of the Einstein equations. We refer to the resulting system as ``fully covariant and conformal", or fCCZ4 for short, since it is well suited for curvilinear as well as Cartesian coordinates. We implement this fCCZ4 formalism in spherical polar coordinates under the assumption of spherical symmetry using a partially-implicit Runge-Kutta (PIRK) method and show that our code can evolve both vacuum and non-vacuum spacetimes without encountering instabilities. Our method does not require regulariza…
The near horizon aspects (and beyond) of a black hole metric, which belongs to a large class of static spherically symmetric black holes, are considered here. It has been realized recently that an atom falling into a black hole leads to the generation of acceleration radiation through virtual transitions. In recent studies, it has been argued that this acceleration radiation can be understood from the near horizon physics of the black hole. The near horizon approximation leads to conformal symmetry in the problem. We go beyond the near horizon approximation in our analysis. This breaks the con…
The spherically symmetric thin shells are the nearest generalizations of the point-like particles. Moreover, they serve as the simple sources of the gravitational fields both in General Relativity and much more complex quadratic gravity theories. We are interested in the special and physically important case when all the quadratic in curvature tensor (Riemann tensor) and its contractions (Ricci tensor and scalar curvature) terms are present in the form of the square of Weyl tensor. By definition, the energy-momentum tensor of the thin shell is proportional to Dirac delta-function. We construct…
We present numerical solutions of the hyperboloidal initial value problem for a self-gravitating scalar field in spherical symmetry, using a variety of standard hyperbolic slicing and shift conditions that we adapt to our hyperboloidal setup. We work in the framework of conformal compactification, and study both evolutions that employ the preferred conformal gauge, which simplifies the formal singularities of our equations at null infinity, and evolutions without this simplification. In previous work we have used a staggered grid, which excludes null infinity, while now we include the option o…
We study the stability of static, spherically symmetric, traversable wormholes existing due to conformal continuations in a class of scalar-tensor theories with zero scalar field potential (so that Fisher's well-known scalar-vacuum solution holds in the Einstein conformal frame). Specific examples of such wormholes are those with nonminimally (e.g., conformally) coupled scalar fields. All boundary conditions for scalar and metric perturbations are taken into account. All such wormholes are shown to be unstable under spherically symmetric perturbations. The instability is proved analytically wi…
The general solution for non-rotating perfect-fluid spacetimes admitting one Killing vector and two conformal (non-isometric) Killing vectors spanning an abelian three-dimensional conformal algebra (C_3) acting on spacelike hypersurfaces is presented. It is of Petrov type D; some properties of the family such as matter contents are given. This family turns out to be an extension of a solution recently given in \cite{SeS} using completely different methods. The family contains Friedman-Lema\^ıtre-Robertson-Walker particular cases and could be useful as a test for the different FLRW perturbation…
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.
We continue our study in 4-dimension to derive non-charged and charged (Anti)-de Sitter black hole solutions in conformal teleparallel equivalent of general relativity. The non-charged and charged equations of motion are applied to a spherically symmetric tetrad and the non-linear differential equations are derived. It is shown that the output solutions of the two cases are identical to those obtained in teleparallel equivalent theory to general relativity. As a result, it is found that in the conformal teleparallel equivalent theory to general relativity, the scalar field cannot influence on …
We address the problem of finding static and spherically symmetric anisotropic compact stars in general relativity that admit conformal motions. The study is framed in the language of f(R) gravity theory in order to expose opportunity for further study in the more general theory. Exact solutions of compact stars are found under the assumption that spherically symmetric spacetimes which admits conformal motion with matter distribution anisotropic in nature. In this work, two cases have been studied for the existence of such solutions: first, we consider the model given by f(R) = R and then f(R)…
We consider conformal gravity as a gauge natural theory. We study its conservation laws and superpotentials. We also consider the Mannheim and Kazanas spherically symmetric vacuum solution and discuss conserved quantities associated to conformal and diffeomorphism symmetries.
We consider the general procedure for proving no-hair theorems for static, spherically symmetric black holes. We apply this method to the abelian Higgs model and find a proof of the no-hair conjecture that circumvents the objections raised against the original proof due to Adler and Pearson.
We study the formation and the evaporation of a spherically symmetric black hole in conformal gravity. From the collapse of a spherically symmetric thin shell of radiation, we find a singularity-free non-rotating black hole. This black hole has the same Hawking temperature as a Schwarzschild black hole with the same mass, and it completely evaporates either in a finite or in an infinite time, depending on the ensemble. We consider the analysis both in the canonical and in the micro-canonical statistical ensembles. Last, we discuss the corresponding Penrose diagram of this physical process.
We present new soliton and hairy black hole solutions of Einstein-non-Abelian-Proca theory in asymptotically anti-de Sitter space-time with gauge group ${\mathfrak {su}}(2)$. For static, spherically symmetric configurations, we show that the gauge field must be purely magnetic, and solve the resulting field equations numerically. The equilibrium gauge field is described by a single function $ω(r)$, which must have at least one zero. The solitons and hairy black holes share many properties with the corresponding solutions in asymptotically flat space-time. In particular, all the solutions we st…
We review basic features of cosmological models with large-scale classical non-Abelian Yang-Mills (YM) condensates. There exists a unique SU(2) YM configuration (generalizable to larger gauge groups) compatible with homogeneity and isotropy of the three-space which is parameterized by a single scalar field. In the past various aspects of Einstein-Yang-Mills (EYM) cosmology were discussed in the context of the Early Universe. Due to conformal invariance, solvable EYM FRW models exist both on the classical and quantum levels. To develop the YM model for dark energy one has to find mechanisms of …
