We study static spherically symmetric solutions to the vacuum field equations of quadratic gravity in the presence of a cosmological constant $Λ$. Motivated by the trace no-hair theorem, we assume the Ricci scalar to be constant throughout a spacetime. Furthermore, we employ the conformal-to-Kundt metric ansatz that is valid for all static spherically symmetric spacetimes and leads to a considerable simplification of the field equations. We arrive at a set of two ordinary differential equations and study its solutions using the Frobenius-like approach of (infinite) power series expansions. While the indicial equations considerably restrict the set of possible leading powers, careful analysis of higher-order terms is necessary to establish the existence of the corresponding classes of solutions. We thus obtain various non-Einstein generalizations of the Schwarzschild, (anti-)de Sitter [or (A)dS for short], Nariai, and Plebański-Hacyan spacetimes. Interestingly, some classes of solutions allow for an arbitrary value of $Λ$, while other classes admit only discrete values of $Λ$. For most of these classes, we give recurrent formulas for all series coefficients. We determine which classes contain the Schwarzschild-(A)dS black hole as a special case and briefly discuss the physical interpretation of the spacetimes. In the discussion of physical properties, we naturally focus on the generalization of the Schwarzschild-(A)dS black hole, namely the Schwarzschild-Bach-(A)dS black hole, which possesses one additional Bach parameter. We also study its basic thermodynamical properties and observable effects on test particles caused by the presence of the Bach tensor. This work is a considerable extension of our letter [Phys. Rev. Lett., 121, 231104, 2018].
We categorize the global structure of spherically symmetric static solutions of Einstein SU(2) Yang Mills equations with positive cosmological constant that are smooth at the center of spherical symmetry.
We consider static spherically symmetric solutions of the Einstein equations with cosmological constant coupled to the SU(2) Yang Mills equations. We prove that most solutions can be continued back to the origin of spherical symmetry and that the qualitative behavior of solutions near this origin does not depend on the value of the cosmological constant.
The paper deals with static traversable wormhole having shape function a polynomial of the radial coordinate of degree 2. Embedding of the wormholes in space-time is examined both analytically and graphically. Also both timelike and null geodesics are studied in this wormhole geometry.
We consider static spherically symmetric solutions of Einstein's equations coupled to an SU(2) Yang Mills field that are smooth at the center of spherical symmetry. We prove that with small cosmological constant there exist solutions that possess a coordinate singularity at some r that is not maximum. The singularity can be removed with a Kruskal-like change of coordinates.
The junction conditions between static and non-static space-times are studied for analyzing gravitational collapse in the presence of a cosmological constant. We have discussed about the apparent horizon and their physical significance. We also show the effect of cosmological constant in the collapse and it has been shown that cosmological constant slows down the collapse of matter.
We propose to describe the dynamics of a cosmological term in the spherically symmetric case by an r-dependent second rank symmetric tensor invariant under boosts in the radial direction. This proposal is based on the Petrov classification scheme and Einstein field equations in the spherically symmetric case. The inflationary equation of state p=-ρis satisfied by the radial pressure. The tangential pressure is calculated from the conservation equation (the contracted Bianchi identity).
We analyse the evolution of the rotational type cosmological perturbation in a gravity with general quadratic order gravitational coupling terms. The result is expressed independently of the generalized nature of the gravity theory, and is simply interpreted as a conservation of the angular momentum.
We present the first numerically stable nonlinear evolution for the leading-order gravitational effective field theory (Quadratic Gravity) in the spherically-symmetric sector. The formulation relies on (i) harmonic gauge to cast the evolution system into quasi-linear form (ii) the Cartoon method to reduce to spherical symmetry in keeping with harmonic gauge, and (iii) order-reduction to 1st-order (in time) by means of introducing auxiliary variables. Well-posedness of the respective initial-value problem is numerically confirmed by evolving randomly perturbed flat-space and black-hole initial …
A class of exact solutions for the Einstein-Maxwell field equations are obtained by assuming the erstwhile cosmological constant $ Λ$ to be a space-variable scalar, viz., $ Λ=Λ(r) $. The source considered here is static, spherically symmetric and anisotropic charged fluid. The solutions obtained are matched continuously to the exterior Reissner-Nordström solution and each of the four solutions represents an electromagnetic mass model.
Interior solutions of Einstein's equations with a non-zero cosmological constant are given for static and spherically symmetric configurations of uniform density. The metric tensor and pressure are determined for both positive and negative values of the cosmological constant. Limits on the outer radius of the interior solutions are established. It is shown that, contrary to the cases of the limits on the interior Schwarzschild and Reissner--Nordström solutions with a zero cosmological constant, these limits do not fully coincide with the conditions of embeddability of the optical reference geo…
The purpose of this study is to describe a perfect fluid matter distribution that leads to a constant curvature region, thanks to the effect of a non-minimal coupling. This distribution exhibits a density profile within the range found in the interstellar medium and an adequate matching of the metric components at its boundary. By identifying this constant curvature with the value of the cosmological constant, and superimposing the spherical distributions arising from different matter sources throughout the universe, one is able to mimic a large-scale homogeneous cosmological constant solution…
In this work a class of interior solution for Einstein field equations corresponding to a spherically symmetric anisotropic fluid sphere has been obtained under the assumption that the cosmological constant is spatially variable. The solution obtained has the characteristics that the pressure and the cosmological parameter vanish at the centre and at the boundary with a maximum value somewhere inside the body. It has been argued that a variable $ Λ$ is as much important physically in Astrophysics as in Cosmology.
Exact solutions of the Einstein's field equations describing a spherically symmetric cosmological model without a big bang or any other kind of singularity recently obtained by Dadhich and Patel (2000) are revisited. The matter content of the model is a shear-free perfect fluid with isotropic pressure and a radial heat flux. Three different exact solutions are obtained for both perfect fluid and fluid with bulk viscosity. It turns out that the cosmological rerm $Λ(t)$ is a decreasing function of time, which is consistent with recent observations of type Ia supernovae.
We review recent results concerning the spherically symmetric Einstein-scalar field system with positive cosmological constant. We do so by comparing with the classical results of Christodoulou concerning the asymptotically flat case (vanishing cosmological constant) and by discussing some of the issues which have emerged since the publication of our main results. Concerning the positive cosmological constant case, we also sketch an alternative proof of global in (Bondi) time existence, based on energy estimates, which is presumably more flexible and, consequently, amenable to generalizations;…
We extend here the canonical treatment of spherically symmetric (quantum) gravity to the most simple matter coupling, namely spherically symmetric Maxwell theory with or without a cosmological constant. The quantization is based on the reduced phase space which is coordinatized by the mass and the electric charge as well as their canonically conjugate momenta, whose geometrical interpretation is explored. The dimension of the reduced phase space depends on the topology chosen, quite similar to the case of pure (2+1) gravity. We investigate several conceptual and technical details that might be…
Einstein's field equations with cosmological constant are analysed for a static, spherically symmetric perfect fluid having constant density. Five new global solutions are described. One of these solutions has the Nariai solution joined on as an exterior field. Another solution describes a decreasing pressure model with exterior Schwarzschild-de Sitter spacetime having decreasing group orbits at the boundary. Two further types generalise the Einstein static universe. The other new solution is unphysical, it is an increasing pressure model with a geometric singularity.
Conditions for smooth cosmological models are set out and applied to inhomogeneous spherically symmetric models constructed by matching together different Lemaitre-Tolman-Bondi solutions to the Einstein field equations. As an illustration the methods are applied to a collapsing dust sphere in a curved background. This describes a region which expands and then collapses to form a black hole in an Einstein de Sitter background. We show that in all such models if there is no vacuum region then the singularity must go on accreting matter for an infinite LTB time.
A exact de Sitter-like cosmological solution of quadratic gravitation with torsion has been found. In the limit of constant energy and pressure, it becomes a exact de Sitter spacetime. It exists in a wide class of quadratic gravity theories and is the same in vacuum for all the models in this class, no matter how the coefficients of the quadratic terms in the Lagrangian are. It describes an accelerating universe and gives a cosmological constant which is of the order of magnitude of the observed value. In vacuum the universe is a de Sitter spacetime without torsion. When matter presents, howev…
We give a complete classification of all static, spherically symmetric solutions of the SU(2) Einstein-Yang-Mills theory with a positive cosmological constant. Our classification proceeds in two steps. We first extend solutions of the radial field equations to their maximal interval of existence. In a second step we determine the Carter-Penrose diagrams of all 4-dimensional space-times constructible from such radial pieces. Based on numerical studies we sketch a complete phase space picture of all solutions with a regular origin.
The Vlasov-Einstein system describes a self-gravitating, collisionless gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with spherical, plane, or hyperbolic symmetry and prove that for small initial data solutions exist up to a spacetime singularity which is a curvature and a crushing singularity. An important tool in the analysis is a local existence result with a continuation criterion saying that solutions can be extended as long as the momenta in the support of the phase-space distribution of the matter remain bounded.
We investigate spherical, isothermal and polytropic steady accretion models in the presence of the cosmological constant. Exact solutions are found for three classes of isothermal fluids, assuming the test gas approximation. The cosmological constant damps the mass accretion rate and - above certain limit - completely stops the steady accretion onto black holes. A "homoclinic-type" accretion flow of polytropic gas has been discovered in AdS spacetimes in the test-gas limit. These results can have cosmological connotation, through the Einstein--Straus vacuole model of embedding local structures…
We present here the canonical treatment of spherically symmetric (quantum) gravity coupled to spherically symmetric Maxwell theory with or without a cosmological constant. The quantization is based on the reduced phase space which is coordinatized by the mass and the electric charge as well as their canonically conjugate momenta, whose geometrical interpretation is explored. The dimension of the reduced phase space depends on the topology chosen, quite similar to the case of pure (2+1) gravity. We also compare the reduced phase space quantization to the algebraic quantization. Altogether, we o…
A spherically symmetric charged ideal fluid solution of Einstein field equation is given in the presence of the cosmological constant and two well known example of this type of solution is presented. If the matter is confined in a region, the exterior spacetime is considered as RN-de Sitter (Reissner-Nordstrom de Sitter) and to complete solution matching conditions are examined. We show that the function which is related to the dynamics of the system will determine the fate of the system: expansion, contraction or bouncing situations may occur for different configurations. The initial conditio…
In this work we investigate analytic static and spherically symmetric solutions of a generalized theory of gravity in the Einstein-Cartan formalism. The main goal consists in analyzing the behavior of the solutions under the influence of a quadratic curvature term in the presence of cosmological constant and no torsion. In the first incursion we found an exact de Sitter-like solution. This solution is obtained by imposing vanishing torsion in the field equations. On the other hand, by imposing vanishing torsion directly in the action, we are able to find a perturbative solution around the Schw…
We present two classes of inhomogeneous, spherically symmetric solutions of the Einstein-Maxwell-Perfect Fluid field equations with cosmological constant generalizing the Vaidya-Shah solution. Some special limits of our solution reduce to the known inhomogeneous charged perfect fluid solutions of the Einstein field equations and under some other limits we obtain new charged and uncharged solutions with cosmological constant. Uncharged solutions in particular represent cosmological models where the universe may undergo a topology change and in between is a mixture of two different Friedmann-Rob…
The quantum cosmological wavefunction for a quadratic gravity theory derived from the heterotic string effective action is obtained near the inflationary epoch and during the initial Planck era. Neglecting derivatives with respect to the scalar field, the wavefunction would satisfy a third-order differential equation near the inflationary epoch which has a solution that is singular in the scale factor limit $a(t)\to 0$. When scalar field derivatives are included, a sixth-order differential equation is obtained for the wavefunction and the solution by Mellin transform is regular in the $a\to 0$…
We study static spherically and hyperbolically symmetric solutions of the Einstein equations in the presence of a conformally coupled scalar field and compare them with those in the space filled with a minimally coupled scalar field. We then study the Kantowski-Sachs cosmological solutions, which are connected with the static solutions by the duality relations. The main ingredient of these relations is an exchange of roles between the radial and the temporal coordinates, combined with the exchange between the spherical and hyperbolical two-dimensional geometries. A brief discussion of question…
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…
Observational manifestations of accelerated expansion of the universe, in particular, recent data for Type Ia supernovae, baryon acoustic oscillations, for the Hubble parameter $H(z)$ and cosmic microwave background constraints are described with different cosmological models. We compare the $Λ$CDM, the models with generalized and modified Chaplygin gas and the model with quadratic equation of state. For these models we estimate optimal model parameters and their permissible errors with different approaches to calculation of sound horizon scale $r_s(z_d)$. Among the considered models the best …
We present a set of equations describing the cosmological gravitational wave in a gravity theory with quadratic order gravitational coupling terms which naturally arise in quantum correction procedures. It is known that the gravitational wave equation in the gravity theories with a general $f(R)$ term in the action leads to a second order differential equation with the only correction factor appearing in the damping term. The case for a $R^{ab} R_{ab}$ term is completely different. The gravitational wave is described by a fourth order differential equation both in time and space. However, curi…
Homogeneous isotropic spatial flat cosmological models with two torsion functions in vacuum are built and investigated in the framework of de Sitter gauge theory of gravity. It is shown that by certain choices of parameters of gravitational Lagrangian the cosmological equations have some exact constant solutions that turn out to be stable de Sitter critical points of dynamical systems and can explain observable acceleration of cosmological expansion. The role of the space-time torsion provoking the acceleration of cosmological expansion is shown.
We study the evolution of a self interacting scalar field in Einstein-Gauss-Bonnet theory in four dimension where the scalar field couples non minimally with the Gauss-Bonnet term. Considering a polynomial coupling of the scalar field with the Gauss-Bonnet term, a self-interaction potential and an additional perfect fluid distribution alongwith the scalar field, we investigate different possibilities regarding the outcome of the collapsing scalar field. The strength of the coupling and choice of the self-interaction potential serves as the pivotal initial conditions of the models presented. Th…
We investigate motion of test particles in exact spacetimes with an expanding impulsive gravitational wave which propagates in Minkowski, de Sitter or anti-de Sitter universe. Using the continuous form of these metrics we derive explicit junction conditions and simple refraction formulae for null, timelike and spacelike geodesics crossing a general impulse of this type. In particular, we present a detailed geometrical description of the motion of test particles in a special class of axially symmetric spacetimes in which the impulse is generated by a snapped cosmic string.
We use Boulware's Hamiltonian formalism of quadratic gravity theories in order to analyze the classical behaviour of Bianchi cosmological models for a Lagrangian density containing quadratic terms in the curvature. For this purpose we define a canonical transformation which leads to a clear distinction between two main variants of the quadratic theory, namely the R-squared or conformal Lagrangian densities. In this paper we restrict the study to the first variant. For Bianchi type I and IX models we give the explicit forms of the super-Hamiltonian constraint, of the ADM Hamiltonian density and…
Given spherically symmetric characteristic initial data for the Einstein-scalar field system with a positive cosmological constant, we provide a criterion, in terms of the dimensionless size and dimensionless renormalized mass content of an annular region of the data, for the formation of a future trapped surface. This corresponds to an extension of Christodoulou's classical criterion by the inclusion of the cosmological term.
Extending the study of spherically symmetric metrics satisfying the dominant energy condition and exhibiting singularities of power-law type initiated in SI93, we identify two classes of peculiar interest: focusing timelike singularity solutions with the stress-energy tensor of a radiative perfect fluid (equation of state: $p={1\over 3} ρ$) and a set of null singularity classes verifying identical properties. We consider two important applications of these results: to cosmology, as regards the possibility of solving the horizon problem with no need to resort to any inflationary scenario, and t…
We consider the generalized set of theories of gravitation whose Lagrangians contain the term $R^{2}$ : $L=\sqrt{-g}(R+βR^{2})$. Inserting the RW metric with an imposed non-singular and inflationary behaviour of the scale factor $a(t)$, and using a arbitrary perfect fluid, we study the properties of $ρ$ and $p$ in this context. By requiring the positivity of the energy density, as well as real and finite velocity of sound, we can obtain the range of values of $β$ that ensure the inflationary behaviour and absence of singularity.
We examine a large class of inhomogeneous spherically symmetric spacetimes that generalize the Lemaitre-Tolman-Bondi dust solutions to nonzero pressure ("LTB spacetimes"). Local covariant LTB objects can be expressed as perturbations of covariant quasi-local (QL) scalars that satisfy evolution equations of equivalent Friedman-Lemaitre-Robertson-Walker (FLRW) scalars. Thus, the dynamics of these spacetimes can be rigorously described as non-linear, gauge invariant and covariant perturbations on a formal FLRW background given by the QL scalars. Since LTB spacetimes are compatible with a wide var…
We study different manifestations of the speed of light in theories of gravity where metric and connection are regarded as independent fields. We find that for a generic gravity theory in a frame with locally vanishing affine connection, the usual degeneracy between different manifestations of the speed of light is broken. In particular, the space-time causal structure constant ($c_{ST}$) may become variable in that local frame. For theories of the form $f(R,R^{μν} R_{μν})$, this variation in $c_{ST}$ has an impact on the definition of the luminosity distance (and distance modulus), which can …
Using the framework for solving the spherically symmetric field equations in observational coordinates given in Araujo and Stoeger (1999), their formulation and solution in the perturbed FLRW sperically symmetric case with observational data representing galaxy redshifts, number counts and observer area distances, both as functions of redshift on our past light cone, are presented. The importance of the central conditions, those which must hold on our world line C, is emphasized. In detailing the solution for these perturbations, we discuss the gauge problem and its resolution in this context,…
Using a recently developed perturbation formalism based on curvature quantities, we investigate the linear stability of black holes and solitons with Yang-Mills hair and a negative cosmological constant. We show that those solutions which have no linear instabilities under odd- and even- parity spherically symmetric perturbations remain stable under odd-parity, linear, non-spherically symmetric perturbations.
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…
Using the principle of least action, the motion equations for a singular hypersurface of arbitrary type in quadratic gravity are derived. Equations containing the "external pressure" and the "external flow" components of the surface energy-momentum tensor together with the Lichnerowicz conditions serve to find the hypersurface itself, while the remaining ones define arbitrary functions that arise due to the implicit presence of the delta function derivative. It turns out that neither double layers nor thin shells exist for the quadratic Gauss-Bonnet term. It is shown that there is no "external…
In this paper I examine black hole and cosmological space-times in Born-Infeld-Einstein theory with electric and magnetic charges. The field equations are derived and written in the form $G_{μν}=-κT_{μν}$ for spherically symmetric space-times. The energy-momentum tensor is not the Born-Infeld energy-momentum tensor, but can be obtained from Born-Infeld theory by letting $a\to ia$, where $a$ is the Born-Infeld parameter. It is shown that there is a curvature singularity in spherically symmetric space-times at a nonzero radial coordinate and that, as in Reissner-Nordstrom space-times, there are …
We present a set of equations describing the evolution of the scalar-type cosmological perturbation in a gravity with general quadratic order curvature coupling terms. Equations are presented in a gauge ready form, thus are ready to implement various temporal gauge conditions depending on the problems. The Ricci-curvature square term leads to a fourth-order differential equation for describing the spacetime fluctuations in a spatially homogeneous and isotropic cosmological background.
We consider conditions for existence and stability of a static cosmological solution in quadratic gravity. It appears that such a solution for a Universe filled by only one type of perfect fluid is possible in a wide range of the equation of state parameter $w$ and for both positively and negatively spatially curved Universe. We show that the static solution for the negative curvature is always unstable if we require positive energy density of the matter content. On the other hand, a static solution with positive spatial curvature can be stable under certain restrictions. Stability of this sol…
We propose a method for calculating vacuum fluctuations on the background of a spherical impulsive gravitational wave which results in a finite expression for the vacuum expectation value of the stress-energy tensor. The method is based on first including a cosmological constant as an auxiliary constant. We show that the result for the vacuum expectation value of the stress-energy tensor in second-order perturbation theory is finite if both the cosmological constant and the infrared parameter tend to zero at the same rate.
Within the framework of the minimum quadratic Poincare gauge theory of gravity in the Riemann-Cartan spacetime the dynamics of homogeneous anisotropic Bianchi types I-IX spinning-fluid cosmological models is investigated. A basic equation set for these models is obtained and analyzed. In particular, exact solutions for the Bianchi type-I spinning-fluid and Bianchi type-V perfect-fluid models are found in analytic form.
