Weyl Tensor Research Articles

First BGG operators on homogeneous conformal geometries

We study first BGG operators and their solutions on homogeneous conformal geometries. We focus on conformal Killing tensors, conformal Killing–Yano forms and twistor spinors in particular. We develop an invariant calculus that allows us to find solutions explicitly using only algebraic computations. We also discuss applications to holonomy reductions and conserved quantities of conformal circles. We demonstrate our result on examples of homogeneous conformal geometries coming mostly from general relativity.

Curvature Dynamics in General Relativity

The equations of general relativity are recast in the form of a wave equation for the Weyl tensor. This allows reformulation of gravitational wave theory in terms of curvature waves, rather than metric waves. The existence of two transverse polarization states for curvature waves is proven and in the linearized approximation the quadrupole formula is rederived. A perturbative scheme to extend the linearized result to the non-linear regime is outlined.

Gravitational-gauge vector interaction in the Hořava–Lifshitz framework

An anisotropic model describing gravity-vector gauge coupling at all energy scales is presented. The starting point is the 4+1 dimensional non–projectable Hořava–Lifshitz gravity theory subject to a geometrical restriction. Renormalizability arguments require all possible interactions in the potential up to terms with z = 4 spatial derivatives on the geometrical tensor fields: the Riemann and Weyl tensors. The latter being necessary on a 4+1 dimensional formulation. The dimensional reduction to 3+1 dimensions gives rise to a model invariant under foliation-preserving diffeomorphisms (FDiff) and U(1) symmetry groups. The reduced theory on the kinetic conformal point (), propagates the same spectrum of the Einstein–Maxwell theory. Moreover, at low energies, on the IR point α = 0, β = 1, its field equations are exactly the Einstein–Maxwell ones in a particular gauge condition. The Minkowski ground state is stable provided several restrictions on the coupling parameters are satisfied, they are explicitly obtained. The quantum propagators of the physical degrees of freedom are obtained and after an analysis of the first and second class constraints the renormalizability by power counting is proved, provided that the aforementioned restrictions on the coupling parameters are satisfied.

On conformally flat centroaffine hypersurfaces with semi-parallel cubic form

In this paper, we study locally strongly convex centroaffine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of centroaffine metric. As the main result, we classify such hypersurfaces being not of flat centroaffine metric. In particular, 2,3-dimensional locally strongly convex centroaffine hypersurfaces with semi-parallel cubic form are completely determined.

On Sultana–Dyer spacetime: Curvatures and geometric structures

Sultana–Dyer black hole, obtained by a time-dependent conformal transformation of Schwarzschild black hole, is an exact solution of general relativity in spherical symmetry. This paper provides the investigation of geometrical properties of the Sultana–Dyer spacetime by means of covariant derivative(s) of the geometric quantity “curvature”. It is shown that such a spacetime is [Formula: see text]-quasi-Einstein, Einstein spacetime of level [Formula: see text] and fulfills the generalized Roter type condition. The spacetime admits pseudosymmetric Weyl curvature as well as pseudosymmetric conharmonic curvature. Also, Weyl compatibility and Riemann compatibility of the Ricci tensor are shown. Finally, a comparison is drawn between Schwarzschild and Sultana–Dyer spacetimes with respect to their curvature restricted geometric properties.

On the volume functional of compact manifolds with harmonic Weyl tensor

AbstractThe main aim of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold M with boundary and under the harmonic Weyl tensor condition. In particular, we prove that a critical metric with a harmonic Weyl tensor on a simply connected compact manifold with the boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form , , and . To this end, we first conclude the classification of such critical metrics under the Bach‐flat assumption and then we prove that both geometric conditions are equivalent in this situation.

Regular evaporating black holes with stable cores

A feature shared by many regular black hole spacetimes is the occurrence of a Cauchy horizon. It is then commonly believed that this renders the geometry unstable against perturbations through the mass-inflation effect. In this work, we perform the first dynamical study of this effect taking into account the mass-loss of the black hole due to Hawking radiation. It is shown that the time-dependence of the background leads to two novel types of late-time behavior whose properties are entirely determined by the Hawking flux. The first class of attractor-behavior is operative for regular black holes of the Hayward and renormalization group improved type and characterized by the square of the Weyl curvature growing as $v^6$ at asymptotically late times. This singularity is inaccessible to a radially free-falling observer though. The second class is realized by Reissner-Nordstr{\"o}m black holes and regular black holes of the Bardeen type. In this case the curvature scalars remain finite as $v\rightarrow\infty$. Thus the Hawking flux has a profound effect on the mass-inflation instability, either weakening the effect significantly or even expelling it entirely.

Characterizations of a spacetime admitting ψ-conformal curvature tensor

In this paper, we introduce ?-conformal curvature tensor, a new tensor that generalizes the conformal curvature tensor. At first, we deduce a few fundamental geometrical properties of ?-conformal curvature tensor and pseudo ?-conharmonically symmetric manifolds and produce some interesting outcomes. Moreover, we study ?-conformally flat perfect fluid spacetimes. As a consequence, we establish a number of significant theorems about Minkowski spacetime, GRW-spacetime, projective collineation. Moreover, we show that if a?-conformally flat spacetime admits a Ricci bi-conformal vector field, then it is either conformally flat or of Petrov type N. At last, we consider pseudo?conformally symmetric spacetime admitting harmonic ?-conformal curvature tensor and prove that the semi-symmetric energy momentum tensor and Ricci semi-symmetry are equivalent and also, the Ricci collineation and matter collineation are equivalent.

Liftings from a para-Sasakian manifold to its tangent bundles

The purpose of the present paper is to study the liftings of a quarter symmetric non-metric connection from a para-Sasakian manifold to its tangent bundles. By liftings, some results of the curvature tensor, projective curvature tensor, concircular curvature tensor and conformal curvature tensor wrt a quarter symmetric non-metric connection in a P-Sasakian manifold to its tangent bundles are obtained.

A study of different class of almost Para-Sasakian Riemannian manifold

In this work, the author thought about and looked into the Nijenhuis tensor, the Affine Connection, and the conformal curve tensor on Nearly Para Sasakian different class-Riemannian differential Manifold. We developed and proved the different results for above mentioned class. We also established new condition of Geodesic for different class-Riemannian differential Manifold.

Some results on pseudosymmetric normal paracontact metric manifolds

TIn this article, the M-projective and Weyl curvature tensors on a normal paracontact metric manifold are discussed. For normal paracontact metric manifolds, pseudosymmetric cases are investigated and some interesting results are obtained. We show that a semisymmetric normal paracontact manifold is of constant sectional curvature. We also obtain that a pseudosymmetric normal paracontact metric manifold is an $\eta$-Einstein manifold. Finally, we support our topic with an example.

The Investigation of Some Tensor Conditions for α-Kenmotsu Pseudo-Metric Structures

This paper aims to study some semi-symmetric and curvature tensor conditions on α-Kenmotsu pseudo-metric manifolds. Some conditions of semi-symmetric, locally symmetric, and the Ricci semi-symmetric are considered on such manifolds. Also, the relationships between the M-projective curvature tensor and conformal curvature tensor, concircularly curvature tensor, and conharmonic curvature tensor are investigated. Finally, an example of α-Kenmotsu pseudo-metric structure is given.

Closed Vacuum Static Spaces with Zero Radial Weyl Curvature

Closed Vacuum Static Spaces with Zero Radial Weyl Curvature

Self-gravitating spherically symmetric systems with Q h condition in chameleonic Brans–Dicke gravity

Abstract This paper is asserted to explore the self-gravitating spherically symmetric anisotropic fluids in Chameleonic Brans–Dicke theory as dark energy matter. The dissipative and non-dissipative cases for the evolution of the system are discussed evidently satisfying the quasi-homologous condition with vanishing complexity (Y TF ) factor, which is identified in the trace free part of the electric Riemann tensor in splitting of the curvature tensor. We formulate different equations through conformal tensor, mass function, shear stress tensor, scalar field to govern self-gravitating systems. A few models describe center filled fluid distribution whereas some of them have cavities surrounding the center by means of matching conditions on the boundary as well as on inner surfaces. The temperature of the respective models is also discussed here. Finally, we conclude the work by comparing it with GR.

TCFHs, hidden symmetries and M-theory backgrounds

We present the twisted covariant form hierarchy (TCFH) of 11-dimensional supergravity and so demonstrate that the form bilinears of supersymmetric solutions satisfy a generalisation of the conformal Killing-Yano equation with resepct to the TCFH connection. We also compute the Killing-Stäckel, KY and closed conformal Killing-Yano tensors of all spherically symmetric M-branes that include the M2-brane, M5-brane, KK-monopole and pp-wave and demonstrate that their geodesic flows are completely integrable by giving all independent conserved charges in involution. We then find that all form bilinears of pp-wave and KK-monopole solutions generate (hidden) symmetries for spinning particle probes propagating on these backgrounds. Moreover, there are Killing spinors such that some of the 1-, 2- and 3-form bilinears of the M2-brane solution generate symmetries for spinning particle probes. We also explore the question on whether the form bilinears are sufficient to prove the integrability of particle probe dynamics on 11-dimensional supersymmetric backgrounds.

The effective theory of gravity and dynamical vacuum energy

Gravity and general relativity are considered as an Effective Field Theory (EFT) at low energies and macroscopic distances. The effective action of the conformal anomaly of light or massless quantum fields has significant effects on macroscopic scales, due to associated light cone singularities that are not captured by an expansion in local curvature invariants. A compact local form for the Wess-Zumino effective action of the conformal anomaly and stress tensor is given, requiring the introduction of a new light scalar field, which it is argued should be included in the low energy effective action for gravity. This scalar conformalon couples to the conformal part of the spacetime metric and allows the effective value of the vacuum energy, described as a condensate of an exact 4-form abelian gauge field strength F = dA, to change in space and time. This is achieved by the identification of the torsion dependent part of the Chern-Simons 3-form of the Euler class with the gauge potential A, which enters the effective action of the conformal anomaly as a J · A interaction analogous to electromagnetism. The conserved 3-current J describes the worldtube of 2-surfaces that separate regions of differing vacuum energy. The resulting EFT thus replaces the fixed constant Λ of classical gravity, and its apparently unnaturally large sensitivity to UV physics, with a dynamical condensate whose ground state value in empty flat space is Λeff = 0 identically. By allowing Λeff to vary rapidly near the 2-surface of a black hole horizon, the proposed EFT of dynamical vacuum energy provides an effective Lagrangian framework for gravitational condensate stars, as the final state of complete gravitational collapse consistent with quantum theory. The possible consequences of dynamical vacuum dark energy for cosmology, the cosmic coincidence problem, and the role of conformal invariance for other fine tuning issues in the Standard Model are discussed.

Power-law inflation satisfies Penrose’s Weyl curvature hypothesis

Based on entropy considerations and the arrow of time Penrose argued that the universe must have started in a special initial singularity with vanishing Weyl curvature. This is often interpreted to be at odds with inflation. Here we argue just the opposite, that Penrose's persuasions are in fact consistent with inflation. Using the example of power law inflation, we show that inflation begins with a past null singularity, where Weyl tensor vanishes when the metric is initially exactly conformally flat. This initial state precisely obeys Penrose's conditions. The initial null singularity breaks $T$-reversal spontaneously and picks the arrow of time. It can be regulated and interpreted as a creation of a universe from nothing, initially fitting in a bubble of Planckian size when it materializes. Penrose's initial conditions are favored by the initial $O(4)$ symmetry of the bubble, selected by extremality of the regulated Euclidean action. The predicted observables are marginally in tension with the data, but they can fit if small corrections to power law inflation kick in during the last 60 efolds.

Bach equation and the matching of spacetimes in conformal cyclic cosmology models

We consider the problem of matching two spacetimes, the previous and present aeons, in the Conformal Cyclic Cosmology model. The common boundary between them inherits two sets of constraints -- one for each solution of the Einstein field equations extended to the conformal boundaries. The previous aeon is assumed to be an asymptotically de Sitter spacetime, so the standard conformal formulation of the Einstein field equations suffice to derive the constraints on the future null infinity. For the future aeon, which is supposed to evolve from an initial singularity, they are obtained with the use of the Bach equation. This equation is regular at the past conformal infinity for conformally flat and conformally Einstein spacetimes, so we will mostly focus on them here. An example of the electrovacuum spacetime which does not fall into this class and has regular conformal Bach tensor will be discussed in the appendix.

On Kähler structures of Taub-Nut and kerr spaces

In this paper, we study the Kählerian nature of Taub-NUT and Kerr spaces which are gravitational instanton and black hole solutions in general relativity. We show that Euclidean Taub-NUT metric is hyper-Kähler with respect to the usual almost complex structures by employing an alternative explicit coframe, and Euclidean Kerr metric is globally conformally Kähler. We also show that conformally scaled Euclidean Kerr space admits a Kähler structure by applying a conformal scaling factor stemming from the Lee-form of the original metric or alternatively a factor coming from self-dual part of the Weyl tensor [Formula: see text].

Peeling in generalized harmonic gauge

It is shown that a large class of systems of non-linear wave equations, based on the good–bad–ugly model, admit formal solutions with polyhomogeneous expansions near null infinity. A particular set of variables is introduced which allows us to write the Einstein field equations in generalized harmonic gauge (GHG) as a good–bad–ugly system and the functional form of the first few orders in such an expansion is found by applying the aforementioned result. Exploiting these formal expansions of the metric components, the peeling property of the Weyl tensor is revisited. The question addressed is whether or not the use of GHG, by itself, causes a violation of peeling. Working in harmonic gauge, it is found that log-terms that prevent the Weyl tensor from peeling do appear. The impact of gauge source functions and constraint additions on the peeling property is then considered. Finally, the special interplay between gauge and constraint addition, as well as its influence on the asymptotic system and the decay of each of the metric components, is exploited to find a particular gauge which suppresses this specific type of log-term to arbitrarily high order.