Exact Solutions In General Relativity Research Articles

Self-similar collapse in Painlevé–Gullstrand coordinates

We report a family of self-similar exact solutions in General Relativity. The solutions are found in a Painleve-Gullstrand coordinate system but can also be transformed smoothly into a diagonal form. The solutions represent a gravitational collapse leading to three possible outcomes, depending on the parameter space: (i) a collapse followed by a bounce and dispersal of the clustered matter distribution, (ii) a rapid collapse followed by a bounce and an eventual re-collapse, and (iii) a standard collapse leading to zero proper volume. Profiles of the energy conditions are studied for all of the scenarios, and it is noted that a bounce is usually associated with a violation of the Null Energy Condition. It is found that more than one null surfaces (apparent horizons) can develop during the collapse. We also discuss that for a general metric tensor having a conformal symmetry, some regions of the parameter space allows a formation of null throat, much like a wormhole. Matching the metric with a Schwarzschild metric in Painleve–Gullstrand form leads to the geodesic equation for a zero energy falling particle in the exterior.

An axially symmetric spacetime with causality violation in Ricci-inverse gravity

In this paper, Ricci-inverse gravity is investigated. It is an alternative theory of gravity that introduces into the Einstein–Hilbert action an anti-curvature scalar that is obtained from the anti-curvature tensor which is the inverse of the Ricci tensor. An axially symmetric spacetime with causality violation is studied. Two classes of the model are discussed. Different sources of matter are considered. Then a direct relation between the content of matter and causality violation is shown. Our results confirm that Ricci-inverse gravity allows the existence of Closed Time-like Curves (CTCs) that lead to the violation of causality. Furthermore, a comparison is made between the results of general relativity and Ricci-inverse gravity. Other spacetimes, such as Gödel and Gödel-type universes, which are exact solutions of general relativity and allow for causality violations, are also explored in Ricci-inverse gravity framework.

Black bounces, wormholes, and partly phantom scalar fields

Simpson and Visser recently proposed a phenomenological way to avoid some kinds of space-time singularities by replacing a parameter whose zero value corresponds to a singularity (say, $r$) with the manifestly nonzero expression $r(u) = \sqrt{u^2 + b^2}$, where $u$ is a new coordinate, and $b =$ \const $>0$. This trick, generically leading to a regular minimum of $r$ beyond a black hole horizon (called a "black bounce"), may hopefully mimic some expected results of quantum gravity, and was previously applied to regularize the Schwarzschild, Reissner-Nordstr\"om, Kerr and some other metrics. In this paper it is applied to regularize the Fisher solution with a massless canonical scalar field in general relativity (resulting in a traversable wormhole) and a family of static, spherically symmetric dilatonic black holes (resulting in regular black holes and wormholes). These new regular metrics represent exact solutions of general relativity with a sum of stress-energy tensors of a scalar field with a nonzero self-interaction potential and a magnetic field in the framework of nonlinear electrodynamics with a Lagrangian function $L(F)$, $F = F_{\mu\nu} F^{\mu\nu}$. A novel feature in the present study is that the scalar fields involved have "trapped ghost" properties, that is, are phantom in a strong-field region and canonical outside it, with a smooth transition between the regions. It is also shown that any static, spherically symmetric metric can be obtained as an exact solution to the Einstein equations with the stress-energy tensor of the above field combination.

Black bounces as magnetically charged phantom regular black holes in Einstein-nonlinear electrodynamics gravity coupled to a self-interacting scalar field

The "black-bounce" spacetime geometries, were recently proposed in [A. Simpson, M. Visser, JCAP 02 (2019) 042] as regular black holes that bouncing into a future incarnation of the universe. In this work we will present several black-bounce exact solutions of General Relativity. Among them is a novel type of black-bounce solutions, which in contrast of the Simpson-Visser type [A. Simpson, M. Visser, JCAP 02 (2019) 042], do not have the Ellis wormhole metric as a particular case. The source of these solutions are linear superposition of phantom scalar fields and nonlinear electromagnetic fields.

Non-asymptotically flat Einstein-Scalar black hole

Exact solutions in general relativity and other theories in physics are important. One of such theories is the Einstein-Scalar theory. We present an exact class of solutions in the Einstein-Scalar theory with a specific nonlinear self-interacting potential. The potential admits a local minimum and a local maximum in general configuration. The solutions include AdS and dS black hole and non-black hole spacetimes. The scalar field is regular on and outside the horizon and asymptotically vanishes when We also study the thermal stability of the black hole solution and it is shown that the black hole is thermally stable for a certain interval of the radius of the horizon.

Note on the asymptotic structure of Kerr-Schild form

The Kerr-Schild form provides a natural way of realizing the classical double copy that relates exact solutions in general relativity to exact solutions in gauge theory. In this paper, we examine the asymptotic structure of Kerr-Schild form. In Newman-Unti gauge, we find a generic solution space satisfying the Kerr-Schild form in series expansion around null infinity. The news function in the solution space is chiral and can not lead to a mass loss formula. A class of asymptotically flat complex pp-wave solutions in closed form is obtained from the solution space.

Sources in the Weyl Double Copy.

The Weyl double copy relates exact solutions in general relativity to exact solutions in gauge theory, formulated in the spinorial language. To date, the Weyl double copy is understood and employed only for vacuum spacetimes, and hence only to vacuum gauge theories. In this Letter, we propose an extension to the Weyl double copy that provides a systematic procedure for treating gravitational sources. We show that this extended Weyl double copy gives a new perspective to the Kerr-Newman black hole and the general class of Petrov type D electrovac spacetimes.

The Kerr-Schild double copy in Lifshitz spacetime

The Kerr-Schild double copy is a map between exact solutions of general relativity and Maxwell’s theory, where the nonlinear nature of general relativity is circumvented by considering solutions in the Kerr-Schild form. In this paper, we give a general formulation, where no simplifying assumption about the background metric is made, and show that the gauge theory source is affected by a curvature term that characterizes the deviation of the background spacetime from a constant curvature spacetime. We demonstrate this effect explicitly by studying gravitational solutions with non-zero cosmological constant. We show that, when the background is flat, the constant charge density filling all space in the gauge theory that has been observed in previous works is a consequence of this curvature term. As an example of a solution with a curved background, we study the Lifshitz black hole with two different matter couplings. The curvature of the background, i.e., the Lifshitz spacetime, again yields a constant charge density; however, unlike the previous examples, it is canceled by the contribution from the matter fields. For one of the matter couplings, there remains no additional non-localized source term, providing an example for a non-vacuum gravity solution corresponding to a vacuum gauge theory solution in arbitrary dimensions.

Exact plane-symmetric cosmological solutions for a massless scalar field and cosmological constant using the Riccati equation

The Einstein–Klein–Gordon equations have been solved exactly for the case of a minimally coupled, plane-symmetric static massless scalar field with cosmological constant. Here, using the Riccati equation, we find additional, overlooked solutions, as well as anisotropic cosmologies derived by analytic continuation. Setting an integration constant equal to zero, the latter solutions become equivalent to known homogeneous, isotropic cosmologies featuring stiff matter. The interplay between the scalar field stress energy and cosmological constant allows several outcomes for universal evolution, including a singularity-free bounce cosmology. The Riccati equation appears to be particularly well suited for finding certain exact solutions in general relativity.

Kerr-Schild double field theory and classical double copy

The Kerr-Schild (KS) formalism is a powerful tool for constructing exact solutions in general relativity. In this paper, we present a generalization of the conventional KS formalism to double field theory (DFT) and supergravities. We introduce a generalized KS ansatz for the generalized metric in terms of a pair of null vectors. Applying this ansatz to the equations of motion of DFT, we construct the generalized KS field equation. While the generalized KS equations are quadratic in the fields, we show that it is possible to find solutions by considering linear equations only. Furthermore, we construct a Killing spinor equation under the generalized KS ansatz. Based on this formalism, we show that the classical double copy structure, which represents solutions of the Einstein equation in terms of solutions of the Maxwell equation, can be extended to the entire massless string NS-NS sector. We propose a supersymmetric classical double copy which shows that solutions of the Killing spinor equation can be realized in terms of solutions of the BPS equation of the supersymmetric Maxwell theory.

On relativistic generalization of Perelman\u2019s W-entropy and thermodynamic description of gravitational fields and cosmology

Using double 2+2 and 3+1 nonholonomic fibrations on Lorentz manifolds, we extend the concept of W-entropy for gravitational fields in general relativity (GR). Such F- and W-functionals were introduced in the Ricci flow theory of three dimensional (3-d) Riemannian metrics by Perelman (the entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159). Non-relativistic 3-d Ricci flows are characterized by associated statistical thermodynamical values determined by W-entropy. Generalizations for geometric flows of 4-d pseudo-Riemannian metrics are considered for models with local thermodynamical equilibrium and separation of dissipative and non-dissipative processes in relativistic hydrodynamics. The approach is elaborated in the framework of classical field theories (relativistic continuum and hydrodynamic models) without an underlying kinetic description, which will be elaborated in other work. The 3+1 splitting allows us to provide a general relativistic definition of gravitational entropy in the Lyapunov–Perelman sense. It increases monotonically as structure forms in the Universe. We can formulate a thermodynamic description of exact solutions in GR depending, in general, on all spacetime coordinates. A corresponding 2+2 splitting with nonholonomic deformation of linear connection and frame structures is necessary for generating in very general form various classes of exact solutions of the Einstein and general relativistic geometric flow equations. Finally, we speculate on physical macrostates and microstate interpretations of the W-entropy in GR, geometric flow theories and possible connections to string theory (a second unsolved problem also contained in Perelman’s work) in Polyakov’s approach.

Cosmological solutions off(T)gravity

In the cosmological scenario in $f\left( T\right) $ gravity, we find analytical solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe. The method that we apply is that of movable singularities of differential equations. For the isotropic universe, the solutions are expressed in terms of a Laurent expansion, while for the anisotropic universe we find a family of exact Kasner-like solutions in vacuum. Finally, we discuss when a nonlinear $f\left( T\right) $-gravity theory provides solutions for the teleparallel equivalence of general relativity and derive conditions for exact solutions of general relativity to solve the field equations of an $f(T)$ theory.

Exact solutions in modified massive gravity and off-diagonal wormhole deformations

There are explored off-diagonal deformations of "prime" metrics in Einstein gravity (for instance, for wormhole configurations) into "target" exact solutions in f(R,T)-modified and massive/ bi-metric gravity theories. The new classes of solutions may posses, or not, Killing symmetries and can be characterized by effective induced masses, anisotropic polarized interactions and cosmological constants. For nonholonomic deformations with (conformal) ellipsoid/ toroid and/or solitonic symmetries and, in particular, for small eccentricity rotoid configurations, we can generate wormholes like objects matching external black ellipsoid - de Sitter geometries. We conclude that there are nonholonomic transforms and/or non-trivial limits to exact solutions in general relativity when modified/ massive gravity effects are modeled by off-diagonal and/or nonholonomic parametric interactions.

Varieties of Parametric Classes of Exact Solutions in General Relativity Representing Static Fluid Balls

We have presented a method of obtaining parametric classes of spherically symmetric analytic solutions of the general relativistic field equations in canonical coordinates. A number of previously known classes of solutions have been rediscovered which describe perfect fluid balls with infinite central pressure and infinite central density though their ratio is positively finite and less than one. From the solution of one of the newly discovered classes, we have constructed a causal model in which outmarch of pressure and density is positive and monotonically decreasing, and pressure-density ratio is positive and less than one throughout within the balls. Corresponding to this model, we have maximized the Neutron star mass 2.40MΘ with the linear dimensions of 28.43 kms and surface red shift of 0.4142.

New parametric classes of exact solutions in general relativity and polytropic nature fluid ball with constant sound speed

In this paper we have presented a method of obtaining varieties of new parametric classes of spherically symmetric analytic solutions of the general relativistic field equations in canonical coordinates. A number of previously known classes of solutions has been rediscovered which describe perfect fluid balls with infinite central pressure and infinite central density though their ratio is positively finite and less then one. From the solutions of one of the class we have constructed a causal model of polytrope with constant sound speed Corresponding to the polytrope model we have maximized the Neutron star mass 3.26 M⊙ with the linear dimensions 32.27 kms with surface red shift 0.7355 and for other class we have constructed a causal model in which outmarch of pressure and density is monotonically decreasing and pressure–density ratio is positive and less than 1 throughout with in the ball. Corresponding to this model we have maximized the Neutron star mass 3.09 M⊙ with the linear dimensions 30.55 kms with surface red shifts 0.5811.

New class of regular and well behaved exact solutions in general relativity

We present a new class of spherically symmetric regular and well behaved solutions of the general relativistic field equations in isotropic coordinates. These solutions describe perfect fluid balls with positively finite central pressure and positively finite central density; their ratio is less than one and causality condition is obeyed at the centre. The solutions of this class, the outmarch of pressure, density pressure-density ratio and the ratio of sound speed to light is monotonically decreasing. Keeping in view of well behaved nature in terms of central red shift and surface red shift and by assuming the surface density ρb=2×1014 g/cm3, we constructed a Neutral star model for k=2, resulting into maximum mass ≈6.36MΘ, linear dimension ≈48.08 km, surface red shift ≈1.132 and central red shift ≈17.1314.

Nonholonomic Ricci Flows, Exact Solutions in Gravity, and Symmetric and Nonsymmetric Metrics

We provide a proof that nonholonomically constrained Ricci flows of (pseudo) Riemannian metrics positively result into nonsymmetric metrics (as explicit examples, we consider flows of some physically valuable exact solutions in general relativity). There are constructed and analyzed three classes of solutions of Ricci flow evolution equations defining nonholonomic deformations of Taub NUT, Schwarzschild, solitonic and pp--wave symmetric metrics into nonsymmetric ones.

Classifying geometries in general relativity:III. Classification in practice

This is the third in a series of papers concerning a project to set up acomputer database of exact solutions in general relativity which can beaccessed and updated by the user community. In this paper, we describehow the Cartan-Karlhede method for classifying a geometry isaccomplished in practice. We give as an example the classification ofthe Edgar-Lugwig conformally flat pure radiation metrics.

Classifying geometries in general relativity:II. Spinor tools

This is the second in a series of papers concerning a project to set up acomputer database of exact solutions in general relativity which can beaccessed and updated by the user community. In this paper, we discussthe choice of computer algebra platform and the general relativityapplication package. The derivative operators needed in the Cartan-Karlhede classification algorithm and the behaviour of symmetric spinors under frame rotations are then presented.

Classifying geometries in general relativity: I. Standard forms for symmetric spinors

This is the first in a series of papers concerning a project to set up a computer database of exact solutions in general relativity which can be accessed and updated by the user community. In this paper, we briefly discuss the Cartan-Karlhede invariant classification of geometries and the significance of the standard form of a spinor. We then present algorithms for putting the Weyl spinor, Ricci spinor and general spinors into standard form.