Weyl Tensor Research Articles

Obtaining the multiple Debever null directions

Abstract The explicit expression of the multiple Debever null directions of an algebraically special spacetime are obtained in terms of the electric and magnetic parts of the Weyl tensor. An algorithm for the determination of the Petrov-Bel type and the algorithm to obtain the multiple Debever null directions are implemented as functions of a new package of xAct, a Mathematica suite of packages for tensor manipulations. These functions are applied to two examples.

On generalized pseudo conformally symmetric manifolds

In this paper, a type of Riemannian manifold, namely generalized pseudo conformally symmetric manifold is studied. Several geometric properties of such spaces are studied. By imposing different restrictions on the conformal curvature tensor, we have obtained several properties. If the conformal curvature tensor is harmonic, then the form of the scalar curvature is obtained. Also, the relations among the 1-forms under various conditions are obtained.

Electric-magnetic duality of dyonic Kerr-Newman-NUT-AdS spacetimes

We study the (anti-)self-duality conditions under which the electric and magnetic parts of the conserved charges of the dyonic Kerr-Newman-NUT-anti-de Sitter solution become equivalent. Within a holographic framework, the stress tensor and the boundary Cotton tensor are computed from the electric/magnetic content of the Weyl tensor. The holographic stress tensor/Cotton tensor duality is recovered along the (anti-)self-dual curve in parameter space. We show that the latter not only implies a duality relation for the mass but also for the angular momentum. The partition function is computed to first order in the saddle-point approximation and a Bogomol’nyi-Prasad-Sommerfield bound is obtained. The ground state of the theory is enlarged to all the (anti-)self-dual configurations when the SO(4) and U(1) Pontryagin densities are introduced. We demonstrate this at the level of the action and variations thereof. Published by the American Physical Society 2024

General relativistic gravitational induction and causal temperatures

Abstract In this paper, we describe the process of general relativistic gravitational induction in spherically symmetric spacetimes by defining an energy momentum tensor for the induction process, which is divergence-free and hence conserved. The aforementioned tensor explicitly describes how the matter-free gravity, as measured by the geometrical Weyl curvature, interacts with the matter. This tensor is clearly different from the energy momentum tensor of the standard matter and we transparently show that in spherical symmetry, the Bianchi identities reduce to the conservation laws for these two such energy momentum tensors. Working with a semitetrad covariant formalism in spherically symmetric spacetimes, we then demonstrate the process of constructing a consistent causal thermodynamical picture for the free gravity and matter interaction via the general non-truncated Israel-Stewart heat transport equation. As an illustrative example, we consider the Lemaitre-Tolman-Bondi spacetime to highlight the relationship between the shear and the Weyl curvature in determining the inductive heat flux.

Some canonical metrics via Aubin's local deformations

English: In this paper, using special metric deformations introduced by Aubin, we construct Riemannian metrics satisfying non-vanishing conditions concerning the Weyl tensor, on every compact manifold. In particular, in dimension four, we show that there are no topological obstructions for the existence of metrics with non-vanishing Bach tensor.

Self-dual cosmology

We construct cosmological spacetimes with a self-dual Weyl tensor whose dynamics are described by conformally coupled scalars with only cubic self-interactions. Similar to the previously discovered cases in flat and (Anti) de Sitter backgrounds, the interactions are characterized by a bracket that encodes a kinematic algebra. We discuss how the color-kinematics duality and double copy are realized in these cosmological backgrounds. If we further impose that the Ricci scalar is that of an FLRW spacetime, we find two new self-dual metrics corresponding to radiation-dominated and coasting (non-accelerating) FLRW backgrounds. Relaxing this requirement, we find an infinite family of solutions given by three different conformal classes of cosmological self-dual metrics. These solutions approximate those of FLRW as long as we impose a simple additional constraint on the scalar theory.

Compact locally homogeneous manifolds with parallel Weyl tensor

Abstract We construct new examples of compact ECS manifolds, that is, of pseudo-Riemannian manifolds with parallel Weyl tensor that are neither conformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, the rank being the dimension of a distinguished null parallel distribution discovered by Olszak. Previously known examples of compact ECS manifolds, in every dimension greater than 4, were all of rank 1, geodesically complete, and none of them was locally homogeneous. In contrast, our new examples — all of them geodesically incomplete — realize all odd dimensions starting from 5 and are of rank 2, as well as locally homogeneous.

Analytical solutions of planar geometry with zero complexity in extended gravity

In this paper, we discuss various analytical solutions of the planner geometry with anisotropic fluid as well as vanishing complexity factor condition in the framework of [Formula: see text] theory. The matching conditions are employed at the interior and exterior hypersurfaces. The connection between the matter parameters and the Weyl tensor has been constructed. For dissipative and non-dissipative structures, we examine four different conformal Killing vector possibilities, while some possibilities adhere to the [Formula: see text] condition along with the matching conditions, where [Formula: see text] indicates the trace free part of the electric component of the Riemann tensor. Several theoretical models are provided to show how a non-static system evolves.

Off-shell invariants of linearized 4D,N=2 supergravity in the harmonic approach

Using the harmonic superspace approach, we construct, at the linearized level, N=2 supersymmetric curvatures generalizing scalar curvature, Ricci curvature and Weyl tensor. These supercurvatures are the building blocks of various linearized 4D,N=2 Einstein supergravity invariants. The supercurvatures involving the scalar and Ricci curvatures are analytic harmonic N=2 superfields, while the Weyl supertensor is a chiral N=2 superfield. As the basic distinguished feature of our construction, all these objects are expressed through the fundamental analytic gauge prepotentials h++M,M=(αα˙,α+,α˙+,5). The related characteristic features are the heavy use of harmonic derivatives and harmonic zero-curvature equations. On a number of instructive examples, we describe the component reduction of the superfield invariants constructed. Published by the American Physical Society 2024

Generalization of conformal Hamada operators

The six-derivative conformal scalar operator was originally found by Hamada in its critical dimension of spacetime, d=6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=6$$\\end{document}. We generalize this construction to arbitrary dimensions d by adding new terms cubic in gravitational curvatures and by changing its coefficients of expansion in various curvature terms. The consequences of global scale-invariance and of infinitesimal local conformal transformations are derived for the form of this generalized operator. The system of linear equations for coefficients is solved giving explicitly the conformal Hamada operator in any d. Some singularities in construction for dimensions d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} and d=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=4$$\\end{document} are noticed. We also prove a general theorem that a scalar conformal operator with n derivatives in d=n-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=n-2$$\\end{document} dimensions is impossible to construct. Finally, we compare our explicit construction with the one that uses conformal covariant derivatives and conformal curvature tensors. We present new results for operators built with different orders of conformal covariant derivatives.

Observable strong field effects of extra spacetime dimension in the braneworld black hole

Inspired by the string theory, the braneworld picture introduces extra dimensions beyond the four that may have observable non-trivial effects in short distance (strong field) gravity experiments. A case in point is the Randall–Sundrum braneworld picture that projects the 5d bulk Weyl tensor onto the 3d brane providing a stress tensor in the effective Einstein field equations on the brane. Dadhich, Maartens, Papadopoulos and Rezania (DMPR) derived an exact braneworld black hole solution of the brane vacuum field equations. The solution formally resembles that of Reissner–Nordström but is physically different from it since the ”tidal charge” Υ in the solution is not the electric charge but an imprint from the fifth dimension allowing both signs in the power law modification ±Υ2r2 to the Schwarzschild metric Υ=0. The corresponding black holes are designated as DMPR±. We study here the effect of Υ on strong field lensing observables and compare in the eikonal limit the ring down quasinormal mode (QNM) frequencies of DMPR− with those of DMPR＋ , the two variants of tidal charge modified Schwarzschild black hole (Υ=0). It turns out that the tidal charge can significantly modify the Schwarzschild lensing observables and QNM frequencies. In particular, we find that the Pretorius–Khurana critical exponent γ of circular null orbits in the DMPR− black hole has a lower value than that for the Schwarzschild black hole, which indicates a stronger Lyapunov instability suggesting that the accretion disks of DMPR− black holes would appear brighter. The case of the SgrA* black hole is considered for a possible constraint on Υ from the EHT observation of its shadow size.

Gravitational waves on Kerr black holes: I. Reconstruction of linearized metric perturbations

Abstract The gravitational perturbations of a rotating Kerr black hole are notoriously complicated, even at the linear level. In 1973, Teukolsky showed that their physical degrees of freedom are encoded in two gauge-invariant Weyl curvature scalars that obey a separable wave equation. Determining these scalars is sufficient for many purposes, such as the computation of energy fluxes. However, some applications -- such as second-order perturbation theory -- require the reconstruction of metric perturbations. In principle, this problem was solved long ago, but in practice, the solution has never been worked out explicitly. Here, we do so by writing down the metric perturbation (in either ingoing or outgoing radiation gauge) that corresponds to a given mode of either Weyl scalar. Our formulas make no reference to the Hertz potential (an intermediate quantity that plays no fundamental role) and involve only the radial and angular Kerr modes, but not their derivatives, which can be altogether eliminated using the Teukolsky--Starobinsky identities. We expect these analytic results to prove useful in numerical studies and for extending black hole perturbation theory beyond the linear regime.

Renormalization of conformal infinity as a stretched horizon

Abstract In this paper, we provide a comprehensive study of asymptotically flat spacetime in even dimensions d ⩾ 4 . We analyze the most general boundary condition and asymptotic symmetry compatible with Penrose’s definition of asymptotic null infinity I through conformal compactification. Following Penrose’s prescription and using a minimal version of the Bondi–Sachs gauge, we show that I is naturally equipped with a Carrollian stress tensor whose radial derivative defines the asymptotic Weyl tensor. This analysis describes asymptotic infinity as a stretched horizon in the conformally compactified spacetime. We establish that charge aspects conservation can be written as Carrollian Bianchi identities for the asymptotic Weyl tensor. We then provide a covariant renormalization for the asymptotic symplectic potential, which results in a finite symplectic flux and asymptotic charges. The renormalization scheme works even in the presence of logarithmic anomalies.

On pseudo-𝒦-symmetric semi-Riemannian manifolds and relativistic applications

In this paper, we present a novel curvature tensor called the [Formula: see text]-curvature tensor, which is formed by combining the Weyl tensor and the [Formula: see text]-curvature tensor. It is proved that a semi-Riemannian manifold with traceless [Formula: see text]-curvature tensor is Einstein, while a [Formula: see text]-curvature flat semi-Riemannian manifold exhibits constant sectional curvature. We show that a space-time with traceless [Formula: see text]-curvature tensor or [Formula: see text]-curvature flat represents dark energy era. A semi-Riemannian manifold with divergence-free [Formula: see text]-curvature tensor is Weyl harmonic. In a space-time where the [Formula: see text]-curvature tensor is of divergence-free, both the isotropic pressure and energy density are constants. Under certain conditions, it is shown that a pseudo-[Formula: see text]-symmetric semi-Riemannian manifold reduces to a pseudo-Riemannian manifold. It is demonstrated that pseudo-[Formula: see text]-symmetric space-times can be regarded as generalized Robertson–Walker (GRW) space-times. Finally, a concrete example of pseudo-[Formula: see text]-symmetric semi-Riemannian manifolds is presented.

Effectiveness of Weyl gravity in probing quantum corrections to AdS black holes

Computing leading higher curvature contributions to thermodynamic quantities of AdS black hole is drastically simplified once the higher curvature terms are expressed in terms of powers of Weyl tensor by applying proper field redefinitions, avoiding the usual complications caused by higher derivative Gibbons-Hawking-York term or surface counterterms. We establish the method by computing the Euclidean action of general rotating anti–de Sitter (AdS) black holes in five-dimensional quadratic curvature theories with or without supersymmetry and verifying the results numerically. Our result is the state of the art for charged rotating AdS black holes in five-dimensional minimal gauged supergravity including corrections from all three supersymmetric curvature squared terms. Our approach facilitates precision tests in the AdS/CFT correspondence and should be applicable in diverse dimensions. Published by the American Physical Society 2024

Shear-free inhomogeneous energy density in 4D Einstein-Gauss-Bonnet spherical systems

We explore the inhomogeneity factors for the initially regular relativistic spheres in 4D-Einstein-Gauss-Bonnet (EGB) theory. The corresponding equations of motion are derived once the generic expressions for the kinematical variables are obtained for spherically symmetric self-gravitating system. By using the non-zero divergence of the stress-energy tensor, the independent components of Bianchi identities are constructed. To enable a thorough explanation of the inhomogeneity of the particular shear free matter distribution, we computed two distinct components of evolution equations employing the Weyl tensor. We then investigate the requisite variables for the irregularity by looking at particular scenarios in both the adiabatic and non-adiabatic domains. These instances demonstrate how, in addition to other factors, the Gauss-Bonnet terms contribute to the regularity requirements of the collapsing fluid.

Quasi-Bach flow and quasi-Bach solitons on Riemannian manifolds

In this article, we introduce quasi-Bach tensor and correspondingly introduce almost quasi-Bach solitons, thereby generalizing the existing notion of Bach tensor and almost Bach solitons. We explore some properties of gradient quasi Bach solitons with harmonic Weyl curvature tensor. We study the relationships between Weyl tensor, Cotton tensor, tensor D introduced by Cao, and quasi Bach tensor. We also find the evolution of volume, Einstein metric, Ricci curvature and scalar curvature, under the quasi Bach flow. Our results obtained here extends the results of Bach solitons and Bach flow. Finally, we obtain the characterization of gradient quasi-Bach soliton of type I, a particular quasi Bach soliton, on the product manifolds S2×H2 and R2×H2. Our exploration generalizes gradient Bach soliton on R2×H2 obtained by P. T. Ho, while the gradient soliton on S2×H2 is a novel one and is complementary to the results obtained by P. T. Ho.

Charges and topology in linearised gravity

Covariant conserved 2-form currents for linearised gravity are constructed by contracting the linearised curvature with conformal Killing-Yano tensors. The corresponding conserved charges were originally introduced by Penrose and have recently been interpreted as the generators of generalised symmetries of the graviton. We introduce an off-shell refinement of these charges and find the relation between these improved Penrose charges and the linearised version of the ADM momentum and angular momentum. If the graviton field is globally well-defined on a background Minkowski space then some of the Penrose charges give the momentum and angular momentum while the remainder vanish. We consider the generalisation in which the graviton has Dirac string singularities or is defined locally in patches, in which case the conventional ADM expressions are not invariant under the graviton gauge symmetry in general. We modify them to render them gauge-invariant and show that the Penrose charges give these modified charges plus certain magnetic gravitational charges. We discuss properties of the Penrose charges, generalise to toroidal Kaluza-Klein compactifications and check our results in a number of examples.

Alternate computation of gravitational effects from a single loop of inflationary scalars

We present a new computation of the renormalized graviton self-energy induced by a loop of massless, minimally coupled scalars on de Sitter background. Our result takes account of the need to include a finite renormalization of the cosmological constant, which was not included in the first analysis. We also avoid preconceptions concerning structure functions and instead express the result as a linear combination of 21 tensor differential operators. By using our result to quantum-correct the linearized effective field equation we derive logarithmic corrections to both the electric components of the Weyl tensor for gravitational radiation and to the two potentials which quantify the gravitational response to a static point mass.

Static and spherically symmetric vacuum spacetimes with non-expanding principal null directions in f(R) gravity

In this work we characterize all the static and spherically symmetric vacuum solutions in f(R) gravity when the principal null directions of the Weyl tensor are non-expanding. In contrast to General Relativity, we show that the Nariai spacetime is not the only solution of this type when general f(R) theories are considered. In particular, we find four different solutions for the non-constant Ricci scalar case, all of them corresponding to the same theory, given by f(R)=r0-1R-3/r021/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R) = r_0^{-1}\\left|R-3/r_0^2\\right|^{1/2}$$\\end{document}, where r0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0$$\\end{document} is a non-null constant. Finally, we briefly present some geometric properties of these solutions.