Conformally Flat Research Articles

On Conformally Flat Manifolds with Semi-parallel Ricci Tensor and Applications to the Study of Affine Hyperspheres

On Conformally Flat Manifolds with Semi-parallel Ricci Tensor and Applications to the Study of Affine Hyperspheres

Conformally flat almost cosymplectic 3-manifolds

In this paper, we consider an almost cosymplectic 3-manifold M such that its scalar curvature is invariant along the Reeb vector field. We prove that if the Reeb vector field of M is harmonic, then M is conformally flat if and only if it is locally isometric to the product ℝ × N 2(c), where N 2(c) is a Kähler surface of constant sectional curvature c. Almost cosymplectic 3-manifolds on which the Reeb vector field satisfies the h-a condition together with conformal flatness are also classified.

A comparative study of embedding class one, conformally flat, vanishing complexity factor and conformally symmetric solutions and their impacts on compact star structure

For the first time, we have presented in detail the comparative studies of embedding class one (CO), conformally flat (CF), vanishing complexity factor (COM), and conformally symmetric (CS) solutions, which are the easiest way of exploring new solutions of the field equations. All these solutions simplify the two metric potentials problem to one metric potential through specific bridge equations. To compare on the same footing, we have assumed the same grr metric function and solved the gtt metric functions via the bridge equations. We then compare all the physical quantities like density, pressure, anisotropy, adiabatic index, etc. To check which type of matter is appropriate for these solutions, we have plotted the equation of states (EoSs) and found the best-fitted functions. We have found that CO solution obeys quadratic EoS, the COM solution best fits with linear EoS, CF solution contains normal and exotic matters, while the CS solution follows a cubic polynomial, and also includes normal, stiff, and exotic matters. In the end, we have plotted the M−R curves and fitted them with observed masses of a few neutron stars to predict their radii.

Characterizations of Kerr-de Sitter in arbitrary dimension from null infinity.

The Kerr and Kerr-de Sitter metrics share remarkable local geometric properties in four dimensions. Gibbons et al. found a generalization of the Kerr-de Sitter metric to higher dimensions, to which the local characterization above cannot be applied. One viable approach to characterize this family is to understand the behaviour of these metrics at future null infinity. We review Friedrich's and Fefferman-Graham formalisms to discuss the asymptotic initial value problem of ([Formula: see text])-vacuum spacetimes in arbitrary dimensions and study their properties: geometric identification and conformal equivalence of data, Killing initial data and conformal equivalence of boundary conformal Killing vectors (CKV). These results are used to review a recent characterization of Kerr-de Sitter in terms of its asymptotic data, namely conformal flatness at [Formula: see text] together with a canonical TT tensor constructed from specific CKV at [Formula: see text]. Allowing for arbitrary CKV defines the (larger) Kerr-de Sitter-like class. All these metrics can be obtained explicitly as limits or analytic extensions of Kerr-de Sitter. The Kerr-de Sitter-like class is also characterized by the property of being Kerr-Schild and fulfilling a certain falloff condition. In addition, in five dimensions, this class corresponds to all algebraically special metrics with non-degenerate optical matrix. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

General relativistic moving-mesh hydrodynamic simulations with arepo and applications to neutron star mergers

ABSTRACT We implement general relativistic hydrodynamics in the moving-mesh code arepo. We also couple a solver for the Einstein field equations employing the conformal flatness approximation. The implementation is validated by evolving isolated static neutron stars using a fixed metric or a dynamical space–time. In both tests, the frequencies of the radial oscillation mode match those of independent calculations. We run the first moving-mesh simulation of a neutron star merger. The simulation includes a scheme to adaptively refine or derefine cells and thereby adjusting the local resolution dynamically. The general dynamics are in agreement with independent smoothed particle hydrodynamics and static-mesh simulations of neutron star mergers. Coarsely comparing, we find that dynamical features like the post-merger double-core structure or the quasi-radial oscillation mode persist on longer time scales, possibly reflecting a low numerical diffusivity of our method. Similarly, the post-merger gravitational wave emission shows the same features as observed in simulations with other codes. In particular, the main frequency of the post-merger phase is found to be in good agreement with independent results for the same binary system, while, in comparison, the amplitude of the post-merger gravitational wave signal falls off slower, i.e. the post-merger oscillations are less damped. The successful implementation of general relativistic hydrodynamics in the moving-mesh arepo code, including a dynamical space–time evolution, provides a fundamentally new tool to simulate general relativistic problems in astrophysics.

Conformal flatness of compact three-dimensional Cotton-parallel manifolds

A three-dimensional pseudo-Riemannian manifold is called essentially conformally symmetric (ECS) if its Cotton tensor is parallel but nowhere-vanishing. In this note we prove that three-dimensional ECS manifolds must be noncompact or, equivalently, that every compact three-dimensional Cotton-parallel pseudo-Riemannian manifold must be conformally flat.

Carrollian Conformal Fields and Flat Holography

The null conformal boundary I of Minkowski spacetime M plays a special role in scattering theory, as it is the locus where massless particle states are most naturally defined. We construct quantum fields on I, which create these massless states from the vacuum and transform covariantly under Poincaré symmetries. Because the latter symmetries act as Carrollian conformal isometries of I, these quantum fields are Carrollian conformal fields. This group theoretic construction is intrinsic to I by contrast to existing treatments in the literature. However, we also show that the standard relativistic massless quantum fields in M, when pulled back to I, provide a realisation of these Carrollian conformal fields. This correspondence between bulk and boundary fields should constitute a basic entry in the dictionary of flat holography. Finally, we show that I provides a natural parametrisation of the massless particles as described by irreducible representations of the Poincaré group and that in an appropriate conjugate basis, they indeed transform like Carrollian conformal fields.

The Global Property of Generic Conformally Flat Hypersurfaces in R4

A conformally flat hypersurface f:M3→R4 in the four-dimensional Euclidean space R4 is said to be generic if the hypersurface has three distinct principal curvatures everywhere. In this paper, we study the generic conformally flat hypersurfaces in R4 using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with a vanishing Möbius form under the Möbius transformation group of R4. Second, we investigate the global behavior of the compact generic conformally flat hypersurfaces and give some integral formulas about the Möbius invariant of these hypersurfaces.

Charged anisotropic spherical collapse in f(R,T,Q) gravity

This paper discusses the gravitational collapse of dynamical self-gravitating fluid distribution in f ( R , T , Q ) gravity, where Q = R φ ϑ T φ ϑ . In this regard, we assume a charged anisotropic spherical geometry involving dissipation flux and adopt standard model of the form R + Φ T + Ψ Q , where Φ and Ψ symbolize real-valued coupling parameters. The Misner–Sharp as well as Müler–Israel Stewart mechanisms are employed to formulate the corresponding dynamical and transport equations. We then interlink these evolution equations which help to study the impact of state variables, heat dissipation, modified corrections and charge on the collapse rate. The Weyl scalar is further expressed in terms of the modified field equations. The necessary and sufficient condition of conformal flatness of the considered configuration and homogeneous energy density is obtained by applying some constraints on the model along with disappearing charge and anisotropy. Finally, we discuss different cases to investigate how the spherical matter source is affected by the charge and modified corrections. • We discuss the gravitational collapse of dynamical self-gravitating fluid distribution. • We assume a charged anisotropic spherical geometry involving dissipation flux. • The necessary and sufficient condition of conformal flatness and homogeneous energy density is obtained. • We discuss different cases how the spherical matter source is affected by the charge and modified corrections.

Conformal Flatness and Conformal Vector Fields on Umbilically Synchronized Space-times

Conformal Flatness and Conformal Vector Fields on Umbilically Synchronized Space-times

Analysis of hyperbolically symmetric fluid configurations in modified Gauss–Bonnet gravity

This paper studies in detail the general physical properties inherent to the static fluid configurations possessing hyperbolic symmetry through the mechanism of modified Gauss–Bonnet gravity (GBG). The energy density of the anisotropic fluid configuration seems to be essentially negative, which indicates that any possible application of these fluid distributions needs severe physical constraints wherein quantum processes become significant. Several exact hyperbolically symmetric solutions with their corresponding generating functions are presented by imposing certain conditions (i.e., conformal flatness, vanishing complexity factor and stiff equation of state) on the fluid variables in the presence of higher-curvature GBG-corrections. Few of the provided stellar solutions exhibit the evolution of hyperbolically symmetric matter configurations whose center of symmetry is surrounded by a vacuum cavity.

Lie symmetry approach to the time-dependent Karmarkar condition

We obtain solutions of the time-dependent Einstein Field Equations which satisfy the Karmarkar condition via the method of Lie symmetries. Spherically symmetric spacetime metrics are used with metric functions set to impose conformal flatness, Weyl-free collapse and shear-free collapse. In particular, a solution was found which satisfies the heat-flux boundary condition of Santos, and a radiating stellar model was then obtained and investigated. Solutions obtained which do not allow for the application of the junction conditions at a boundary surface may lend themselves to cosmological models. This is a first attempt in generating solutions satisfying the Karmarkar condition via the method of Lie symmetries and our example of a radiating model highlights the viability of this method.

Classification of static black holes in Einstein phantom-dilaton Maxwell–anti-Maxwell gravity systems

The uniqueness theorem for static, spherically symmetric, asymptotically flat, higher dimensional phantom black holes, with non-degenerate event horizon , being the solutions of Einstein phantom/dilaton Maxwell/anti-Maxwell gravity systems is considered. Conformal positive energy theorem and conformal transformations authorize the crucial tools for exploiting the boundary conditions and conformal flatness.

Hyperbolically symmetric sources in [formula omitted] gravity

This paper studies the static fluid distributions admitting the hyperbolically symmetric structure by extending the work of Herrera et al. (2021) in f(R,T) gravity. The unavailability of the fluid distribution in the region which is close to the center of symmetry has been observed. A vacuum cavity can be used to depict this region around the center. The effective matter density emerges inevitably to be negative and this implies that the conceivable use of such fluids needs transcendental physical conditions when the significant quantum effects are observed. Furthermore, the Tolman mass, as well as an appropriate formulation of the mass function, are particularly computed. Along with this, we evaluated the structure scalars through the orthogonal splitting of the Riemann tensor in this modified gravity. Imposing the conditions of conformal flatness and stiff equation of state, we constructed several models for the less-complex systems. We also evaluated the generating functions to study the fluid distribution in this theory. A general hypothesis to attain precise solutions are also conferred through matter-curvature coupling.

A Spherically Symmetric Gravitational Solution of Nearly Conformally Flat Metric Measure Space

A Spherically Symmetric Gravitational Solution of Nearly Conformally Flat Metric Measure Space

A complete study of conformally flat pseudo-symmetric spacetimes in the theory of [formula omitted]-gravity

The motive in this article is twofold. First we investigate the geometrical structures of a pseudo-symmetric spacetime (PS)4 with a timelike vector under the condition of conformal flatness. We classify it into two possible types: constant Ricci scalar and closed velocity vector. Then we further study (PS)4 type spacetime solutions of F(R)-gravity theory and demonstrate that the pressure and energy density of the effective cosmological perfect fluid separately possess certain typical relations with the geometry and the gravity sector of the theory. Based on this result, some observational and cosmological analyzes are done for the different F(R)-gravity models and the energy conditions are investigated accordingly, supporting the accelerated expansion of the universe.

Free data at spacelike {mathscr {I}} and characterization of Kerr-de Sitter in all dimensions

We study the free data in the Fefferman–Graham expansion of asymptotically Einstein (n+1)-dimensional metrics with non-zero cosmological constant. We analyze the relation between the electric part of the rescaled Weyl tensor at {mathscr {I}}, D, and the free data at {mathscr {I}}, namely a certain traceless and transverse part of the n-th order coefficient of the expansion mathring{g}_{(n)}. In the case Lambda <0 and Lorentzian signature, it was known [23] that conformal flatness at {mathscr {I}} is sufficient for D and mathring{g}_{(n)} to agree up to a universal constant. We recover and extend this result to general signature and any sign of non-zero Lambda . We then explore whether conformal flatness of {mathscr {I}} is also neceesary and link this to the validity of long-standing open conjecture that no non-trivial purely magnetic Lambda -vacuum spacetimes exist. In the case of {mathscr {I}} non-conformally flat we determine a quantity constructed from an auxiliary metric which can be used to retrieve mathring{g}_{(n)} from the (now singular) electric part of the Weyl tensor. We then concentrate in the Lambda >0 case where the Cauchy problem at {mathscr {I}} of the Einstein vacuum field equations is known to be well-posed when the data at {mathscr {I}} are analytic or when the spacetime has even dimension. We establish a necessary and sufficient condition for analytic data at {mathscr {I}} to generate spacetimes with symmetries in all dimensions. These results are used to find a geometric characterization of the Kerr-de Sitter metrics in all dimensions in terms of its geometric data at null infinity.

Conformal transformation with multiple scalar fields and geometric property of field space with Einstein-like solutions

Multiple scalar fields appear in vast modern particle physics and gravity models. When they couple to gravity non-minimally, conformal transformation is utilized to bring the theory into Einstein frame. However, the kinetic terms of scalar fields are usually not canonical, which makes analytic treatment difficult. Here we investigate under what conditions the theories can be transformed to the quasi-canonical form, in which case the effective metric tensor in field space is conformally flat. We solve the relevant nonlinear partial differential equations for arbitrary number of scalar fields and present several solutions that may be useful for future phenomenological model building, including the $\sigma$-model with a particular non-minimal coupling. We also find conformal flatness can always be achieved in some modified gravity theories, for example, Starobinsky model.

Conformal Flat Metrics with Prescribed Mean Curvature on the Boundary

Conformal Flat Metrics with Prescribed Mean Curvature on the Boundary

Dynamical Analysis of Charged Dissipative Cylindrical Collapse in Energy-Momentum Squared Gravity

This paper investigates the dynamics of charged cylindrical collapse with the dissipative matter configuration in f(R,TαβTαβ) theory. This newly formulated theory resolves the primordial singularity and provides feasible cosmological results in the early universe.Moreover, its implications occur in high curvature regime where the deviations of energy-momentum squared gravity from general relativity is confirmed. We establish dynamical and transport equations through the Misner–Sharp and Mu¨ler–Israel Stewart techniques, respectively. We then couple these equations to examine the impact of effective fluid parameters and correction terms on the collapsing phenomenon. A connection between the modified terms, matter parameters, and Weyl tensor is also developed. To obtain conformal flatness, we choose a particular model of this theory and assume that dust matter with zero charge leads to conformal flatness and homogenous energy density. We found that the modified terms, dissipative matter, and electromagnetic field reduce the collapsing phenomenon.