Weyl Tensor Research Articles

Lorentzian wormhole in the framework of loop quantum cosmology

In this paper, we construct a traversable static Lorentzian wormhole in the effective scenario of Loop Quantum Cosmology (LQC), where the field equations are modified due to the ultraviolet (UV) corrections introduced at large space-time curvatures. A stable wormhole can be constructed in the effective scenario without the violation of Null energy condition (NEC) by physical matter at the throat. The NEC is effectively violated due to the corrections in the field equations from LQC, resolving the Weyl curvature singularity at the throat. However, the physical matter does violate the Strong energy condition (SEC), suggesting the interesting possibility that dark energy can be harnessed into a wormhole. A possible explanation for this is the presence of inherent pressure isotropy in the UV-corrected field equations (discussed and compared to braneworld wormholes in the discussion). No additional exotic ingredient (violating NEC) is required, avoiding quantum instabilities. The tidal forces at the throat do not diverge and also the throat is found to be stable. The wormhole features an attractive geometry. LQC can resolve both types of curvature singularities appearing at the black hole centre and wormhole throat, without exotic matter.

Modified conformal extensions

We present a geometric construction and characterization of 2n-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson–Walker metric construction for n-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive a complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an explicit geometrically constructed Fefferman–Graham ambient metric and show the vanishing of the Q-curvature.

Weyl tensors, strongly regular graphs, multiplicative characters, and a quadratic matrix equation

We study solutions of a quadratic matrix equation arising in Riemannian geometry. Let S be a real symmetric n×n-matrix with zeros on the diagonal and let θ be a real number. We construct nonzero solutions (S,θ) of the set of quadratic equations∑kSi,k=0 and ∑kSi,kSk,j+Si,j2=θSi,j for i<j. Our solutions relate the equations to strongly regular graphs, to group rings, and to multiplicative characters of finite fields.

Traversable Lorentzian wormhole on the Shtanov-Sahni braneworld with matter obeying the energy conditions

In this paper we have explored the possibility of constructing a traversable wormhole on the Shtanov-Sahni braneworld with a timelike extra dimension. We find that the Weyl curvature singularity at the throat of the wormhole can be removed with physical matter satisfying the NEC ρ + p ≥ 0, even in the absence of any effective Λ-term or any type of charge source on the brane. (The NEC is however violated by the effective matter description on the brane arising due to effects of higher dimensional gravity.) Besides satisfying NEC the matter constituting the wormhole also satisfies the Strong Energy Condition (SEC), ρ + 3p ≥ 0, leading to the interesting possibility that normal matter on the brane may be harnessed into a wormhole. Incidentally, these conditions also need to be satisfied to realize a non-singular bounce and cyclic cosmology on the brane [1] where both past and future singularities can be averted. Thus, such a cyclic universe on the brane, constituted of normal matter can naturally contain wormholes. The wormhole shape function on the brane with a time-like extra dimension represents the tubular structure of the wormhole spreading out at large radial distances much better than in wormholes constructed in a braneworld with a spacelike extra dimension and have considerably lower mass resulting in minimization of the amount of matter required to construct a wormhole. Wormholes in the Shtanov-Sahni (SS) braneworld also have sufficiently low tidal forces, facilitating traversability. Additionally they are found to be stable and exhibit a repulsive geometry. We are left with the intriguing possibility that both types of curvature singularity can be resolved with the SS model, which we discuss at the end of the concluding section.

Emerging electric or magnetic parts of Weyl tensor

In this paper, we study the PNDP-manifold, which has shown to be effective in applications, particularly in the field of general relativity, by introducing a new topological approach to the emergent electric or magnetic parts associated to the Weyl tensor. The working hypothesis in this study is that the electric or magnetic parts of the Weyl tensor are the dimensions of the PNDP-manifold and the interaction that emerges as “virtual” point-like manifolds and as a manifold with positive/ negative dimensions results in the generation of the purely electric or purely magnetic parts of the Weyl tensor. The PNDP metric for the electric or magnetic parts of the Weyl tensor is also derived. The generalized Lorentz transformations on the PNDP manifold are introduced. These can give rise to novel forms of gauge theories if we consider local Lorentz transformations. Also we briefly show that principal fiber bundles with PNDP manifolds as base manifolds can be constructed.

Tensor Decompositions and Their Properties

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ), σ=1,2,3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.

The topology of compact rank-one ECS manifolds

AbstractPseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as essentially conformally symmetric (ECS) manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the rank of a certain distinguished null parallel distribution $\mathcal{D}$. All known examples of compact ECS manifolds are of rank one and have dimensions greater than 4. We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced when necessary by a twofold isometric covering, must be a bundle over the circle with leaves of $\mathcal{D}^\perp$ serving as the fibres. The same conclusion holds in the locally homogeneous case if one assumes that $\,\mathcal{D}^\perp$ has at least one compact leaf. We also show that in the pseudo-Riemannian universal covering space of any compact rank-one ECS manifold, the leaves of $\mathcal{D}^\perp$ are the factor manifolds of a global product decomposition.

Linearised conformal Einstein field equations

The linearisation of a second-order formulation of the conformal Einstein field equations (CEFEs) in generalised harmonic gauge (GHG), with trace-free matter is derived. The linearised equations are obtained for a general background and then particularised for the study linear perturbations around a flat background—the inversion (conformal) representation of the Minkowski spacetime—and the solutions discussed. We show that the generalised Lorenz gauge (defined as the linear analogue of the GHG-gauge) propagates. Moreover, the equation for the conformal factor can be trivialised with an appropriate choice for the gauge source functions; this permits a scri-fixing strategy using gauge source functions for the linearised wave-like CEFE-GHG, which can in principle be generalised to the nonlinear case. As a particular application of the linearised equations, the far-field and compact source approximation is employed to derive quadrupole-like formulae for various conformal fields such as the perturbation of the rescaled Weyl tensor.

Effective metric of spinless binaries with radiation-reaction effect up to fourth post-Minkowskian order in effective-one-body theory

By means of the scattering angles, we obtain an effective metric of spinless binaries with radiation-reaction effects up to fourth post-Minkowskian order, which is the foundation of the effective-one-body theory. We note that there are freedoms for the parameters of the effective metric because one equation corresponds to two parameters for each post-Minkowskian order. Accordingly, in order to construct a self-consistent effective-one-body theory in which the Hamiltonian, radiation-reaction forces and waveforms for the “plus” and “cross” modes of the gravitational wave should be based on the same physical model, we can fix these freedoms by requiring the null tetrad component of the gravitationally perturbed Weyl tensor Psi _4^B to be decoupled in the effective spacetime.

Born reciprocity and relativistic generalized uncertainty principle in Finsler structure: Fundamental tensor in discretized curved spacetime

The main findings of the generalized uncertainty principle (GUP), the phenomenological approach, for instance, the emergence of a minimal measurable length uncertainty, are obtained in various versions from theories of quantum gravity, such as string theory, loop quantum gravity, doubly special relativity and black hole physics. GUP counts for impacts of relativistic energies and finite gravitational fields on the fundamental theories of quantum mechanics (QM), the noncommutation and measurement uncertainty. Utilizing GUP in reconciling principles of general relativity (GR) and QM, thereby enables to draw convincing conclusions about quantum gravity. To resolve the shortcuts reported with the nonrelativistic three-dimensional GUP, namely, violation of Lorentz covariance, dependence on frame of reference, and violation of the linear additional law of momenta, we introduce relativistic four-dimensional generalized uncertainty principle (RGUP) to curved spacetime. To unify GR and QM, we apply the Born reciprocity principle (BRP), distance-momentum duality symmetry and RGUP to estimate the fundamental tensor in discretized curved spacetime. To this end, we generalize Riemann geometry. The Finsler geometry, which is characterized by manifold and Finsler structure, allows to directly apply RGUP to the Finsler structure of a free particle so that [Formula: see text] can be expressed as [Formula: see text], from which the metric tensor in discretized Riemann spacetime could be deduced. We conclude that [Formula: see text] is homogeneous with degree [Formula: see text] in [Formula: see text], while [Formula: see text] is [Formula: see text]-homogeneous resulting in [Formula: see text]. Despite, the astonishing similarity with the conformal transformation, know as Weyl tensor, this study suggests that principles of QMs could be unambiguously imposed on the resulting fundamental tensor. Also, we conclude that the features of Finsler geometry assumed in this study are likely the ones of the duel Hamilton geometry.

Cosmological dark matter from a bulk black hole

We study the cosmology of a three-brane in a specific five-dimensional scalar-gravity (i.e. soft-wall) background, known as the linear dilaton background. We discover that the Friedmann equation of the brane-world automatically contains a term mimicking pressureless matter. We propose to identify this term as dark matter. This dark matter arises as a projection of the bulk black hole on the brane, which contributes to the brane Friedmann equation via both the Weyl tensor and the scalar stress tensor. The nontrivial matter-like behavior is due to an exact cancellation between the Weyl and scalar pressures. We show that the Newtonian potential only receives a mild short-distance correction going as inverse distance squared, ensuring compatibility of the linear dilaton brane-world with observed 4D gravity. Our setup can be viewed as a consistent cosmological description of the holographic theories arising in the linear dilaton background. We also present more general scalar-gravity models where the brane cosmology features an effective energy density whose behavior smoothly interpolates between dark radiation, dark matter and dark energy depending on a model parameter.

Transport coefficients in AdS/CFT and quantum gravity corrections due to a functional measure

The presence of a functional measure is scrutinized on both sides of the dual gauge/gravity correspondence. Corrections to the transport coefficients in relativistic hydrodynamics are obtained using the linear response procedure. In particular, using first-order hydrodynamics, the shear viscosity, entropy density, diffusion constant, and speed of sound are shown not to acquire any corrections from the functional measure of gravity, for a Minkowski background metric. On the other hand, the energy density, the pressure, the relaxation time, the bulk viscosity, the decay rate of sound waves, and coefficients of conformal traceless tensor fields, are shown to carry significant quantum corrections due to the functional measure, even for a flat background. They all acquire an imaginary part that reflects the instability of the strongly-coupled fluids on the boundary CFT. This opens up the possibility of testing quantum gravity with the quark-gluon plasma.

EBWeyl: a code to invariantly characterize numerical spacetimes

Relativistic cosmology can be formulated covariantly, but in dealing with numerical relativity simulations a gauge choice is necessary. Although observables should be gauge-invariant, simulations do not necessarily focus on their computations, while it is useful to extract results invariantly. To this end, in order to invariantly characterize spacetimes resulting from cosmological simulations, we present two different methodologies to compute the electric and magnetic parts of the Weyl tensor, and , from which we construct scalar invariants and the Weyl scalars. The first method is geometrical, computing these tensors in full from the metric, and the second uses the 3 + 1 slicing formulation. We developed a code for each method and tested them on five analytic metrics, for which we derived and and the various scalars constructed from them with computer algebra software. We find excellent agreement between the analytic and numerical results. The slicing code outperforms the geometrical code for computational convenience and accuracy; on this basis we make it publicly available in github with the name EBWeyl. We emphasize that this post-processing code is applicable to any numerical spacetime in any gauge.

Self-consistent effective-one-body theory for spinning binaries based on post-Minkowskian approximation

This paper extends the research on the self-consistent effective-one-body theory of a real spinless two-body system based on the post-Minkowskian approximation (Science China, 65, 100411, (2022)) to the case of a binary system for the spinning black holes. An effective rotating metric and an improved Hamiltonian for the spinning black hole binaries were constructed. The decoupled equation for the null tetrad component of the gravitational perturbed Weyl tensor $\psi^B_{4}$ in the effective rotating spacetime is found with the help of the gauge transform characteristics of the Weyl tensors. The decoupled equation is then separated between radial and angular variables in the slowly rotating background spacetime, and a formal solution of $\psi^B_{4}$ is obtained. On this basis, the formal expressions of the radiation reaction force and the waveform for the ``plus'' and ``cross'' modes of the gravitational wave are presented. These results, obtained in the same effective spacetime, constitute a self-consistent effective-one-body theory for the spinning black hole binaries based on the post-Minkowskian approximation.

D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of Kähler–Einstein surfaces

D3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, H(textbf{y},bar{textbf{y}}) multiplying at power -1/2 the first summand, i.e., the Minkowski metric of the D3 brane world-sheet, and at power 1/2 the second summand, i.e., the Ricci-flat metric on a six-dimensional transverse space mathcal {M}_6, whose complex coordinates y are the arguments of the warp factor. Of particular interest is the case where mathcal {M}_6={text {tot}}[ Kleft[ left( mathcal {M}_Bright) right] is the total space of the canonical bundle over a complex Kähler surface mathcal {M}_B. This situation emerges in many cases while considering the resolution à la Kronheimer of singular manifolds of type mathcal {M}_6=mathbb {C}^3/Gamma , where Gamma subset mathrm {SU(3)} is a discrete subgroup. When Gamma = mathbb {Z}_4, the surface mathcal {M}_B is the second Hirzebruch surface endowed with a Kähler metric having mathrm {SU(2)times U(1)} isometry. There is an entire class {text {Met}}(mathcal{F}mathcal{V}) of such cohomogeneity one Kähler metrics parameterized by a single function mathcal{F}mathcal{K}(mathfrak {v}) that are best described in the Abreu–Martelli–Sparks–Yau (AMSY) symplectic formalism. We study in detail a two-parameter subclass {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}}subset {text {Met}}(mathcal{F}mathcal{V}) of Kähler–Einstein metrics of the aforementioned class, defined on manifolds that are homeomorphic to S^2times S^2, but are singular as complex manifolds. Actually, {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}}subset {text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}}subset {text {Met}}(mathcal{F}mathcal{V}) is a subset of a four parameter subclass {text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}} of cohomogeneity one extremal Kähler metrics originally introduced by Calabi in 1983 and translated by Abreu into the AMSY action-angle formalism.{text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}} contains also a two-parameter subclass {text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}mathbb {F}_2} disjoint from {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}} of extremal smooth metrics on the second Hirzebruch surface that we rederive using constraints on period integrals of the Ricci 2-form. The Kähler–Einstein nature of the metrics in {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}} allows the construction of the Ricci-flat metric on their canonical bundle via the Calabi Ansatz, which we recast in the AMSY formalism deriving some new elegant formulae. The metrics in {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}} are defined on the base manifolds of U(1) fibrations supporting the family of Sasaki–Einstein metrics textrm{SEmet}_5 introduced by Gauntlett et al. (Adv Theor Math Phys 8:711–734, 2004), and already appeared in Gibbons and Pope (Commun Math Phys 66:267–290, 1979). However, as we show in detail using Weyl tensor polynomial invariants, the six-dimensional Ricci-flat metric on the metric cone of {mathcal M}_5 in {text {Met}}(textrm{SE})_5 is different from the Ricci-flat metric on {text {tot}}[ Kleft[ left( mathcal {M}_{textrm{KE}}right) right] constructed via Calabi Ansatz. This opens new research perspectives. We also show the full integrability of the differential system of geodesics equations on mathcal {M}_B thanks to a certain conserved quantity which is similar to the Carter constant in the case of the Kerr metric.

On a class of Finsler metrics of quadratic Weyl curvature

The class of warped product manifolds plays an important role in differential geometry and physics. In this paper, we shall study product manifolds $\mathbb{R} \times M$ with Finsler metrics arising from warped product structure. We give an equivalent condition for those metrics to be of quadratic Weyl curvature.

Hadamard and boundary conditions for the Big Bang quantum vacuum

General relativity predicts final-type singularities inside black holes, as well as a cosmological initial-type singularity. Cosmic censorship protects external observers from black hole singularities, while Penrose’s Weyl curvature hypothesis protects the smoothness of the initial (Big Bang) singularity. We discuss a simple realization of the Weyl curvature hypothesis by assuming a very early radiation-dominated universe and analytically extending the expansion factor to negative values of conformal time. We impose time-reversal conditions at the Big Bang to characterize a natural set of preferred vacuum states for quantized matter fields. We implement the prescription of States of Low Energy constructed around the Big Bang obtaining Hadamard states. We also explore the physical implications of these vacua for cosmological dark matter production.

Extrinsic Paneitz operators and Q-curvatures for hypersurfaces

For any hypersurface M of a Riemannian manifold X, recent works introduced the notions of extrinsic conformal Laplacians and extrinsic Q-curvatures. Here we derive explicit formulas for the extrinsic version P4 of the Paneitz operator and the corresponding extrinsic fourth-order Q-curvature Q4 in general dimensions. In the critical dimension n=4, this result yields a closed formula for the global conformal invariant ∫MQ4dvol (for closed M) and various decompositions of Q4, which are analogs of the Alexakis/Deser-Schwimmer type decompositions of global conformal invariants. These results involve a series of obvious local conformal invariants of the embedding M4↪X5 (defined in terms of the Weyl tensor and the trace-free second fundamental form) and a non-trivial local conformal invariant C. In turn, we identify C as a linear combination of two local conformal invariants J1 and J2. We also observe that these are special cases of local conformal invariants for hypersurfaces in backgrounds of general dimension. Moreover, in the critical dimension n=4, a linear combination of J1 and J2 can be expressed in terms of obvious local conformal invariants of the embedding M↪X. This finally reduces the non-trivial part of the structure of Q4 to the non-trivial invariant J1. For totally umbilic M, the invariants Ji vanish, and the formula for P4 substantially simplifies. For closed M4↪R5, we relate the integrals of Ji to functionals of Guven and Graham-Reichert. Moreover, we establish a Deser-Schwimmer type decomposition of the Graham-Reichert functional of a hypersurface M4↪X5 in general backgrounds. In this context, we find one further local conformal invariant J3. Finally, we derive an explicit formula for the singular Yamabe energy of a closed M. The resulting explicit formulas show that it is proportional to the total extrinsic fourth-order Q-curvature. This observation confirms a special case of a general fact and serves as an additional cross-check of our main result. We carefully discuss the relations of our results to the recent literature, in particular to the work of Blitz, Gover and Waldron.

Conformal renormalization of scalar-tensor theories

We study a conformally coupled scalar-tensor theory with a quartic potential possessing local conformal symmetry up to a boundary term. We show that requiring the restoration of the full local conformal symmetry fixes the counterterms that render the on-shell action finite. The building block of the resulting action is a conformally covariant tensor which is constructed out of the metric and the scalar field and it has the same conformal weight as the Weyl tensor. This allows us to obtain the counterterms for the scalar-tensor sector in a closed form. The finiteness of the conformally complete version of the action is suggestive on the validity of the conformal renormalization prescription. We extend this theory by adding the conformal gravity action and also the Einstein-AdS action written in MacDowell-Mansouri form. Even though the latter breaks the conformal symmetry, we find that the action is still renormalized provided a suitable falloff of the scalar field when considering asymptotically locally anti-de Sitter solutions. Black hole solutions in these theories are studied, for which the Hawking temperature and the partition function to first order in the saddle-point approximation are calculated, providing a concrete example of this renormalization scheme.

Geometry, conformal Killing-Yano tensors and conserved “currents”

In this paper we discuss the construction of conserved tensors (currents) involving conformal Killing-Yano tensors (CKYTs), generalising the corresponding constructions for Killing-Yano tensors (KYTs). As a useful preparation for this, but also of intrinsic interest, we derive identities relating CKYTs and geometric quantities. The behaviour of CKYTs under conformal transformations is also given, correcting formulae in the literature. We then use the identities derived to construct covariantly conserved “currents”. We find several new CKYT currents and also include a known one by Penrose which shows that “trivial” currents are also useful. We further find that rank-n currents based on rank-n CKYTs k must have a simple form in terms of dk. By construction, these currents are covariant under a general conformal rescaling of the metric. How currents lead to conserved charges is then illustrated using the Kerr-Newman and the C-metric in four dimensions. Separately, we study a rank-1 current, construct its charge and discuss its relation to the recently constructed Cotton current for the Kerr-Newman black hole.