The Bianchi identity and weak gravitational lensing

We investigate the cross-correlation between galaxy clusters and QSOs using Sloan Digital Sky Survey (SDSS) DR4 ~5000 deg2 data. With photometric redshifts of galaxies, we select galaxy clusters based on the local projected densities of LRGs brighter than Mr' = −22. The QSOs are from the main sample of SDSS QSO spectroscopic survey to i' = 19. A significant positive correlation is found between the clusters and QSOs. Under the assumption that the signal is caused by gravitational lensing, we fit the signal with singular isothermal sphere (SIS) model and NFW profile halo model. The velocity dispersion σv = 766 km s−1 is derived for the best-fit of SIS model. Best-fit for the NFW model requires the dark matter halo mass within 1.5 h−1 Mpc to be 4.6×1014 h−1 M⊙. The mass parameter Ωcl of the cluster sample is deduced as 0.077 with the SIS model and 0.083 with the NFW model. Our results of Ωcl are smaller than those given by Croom & Shanks and by Myers et al.

Kilonovae are generally believed to originate from the ejecta of binary neutron stars (NSs) or black hole–NS mergers. Free neutrons might be retained in the outermost layer of the ejecta to produce a precursor via β decay. During the propagation of kilonovae to observers, a small percentage of them might be gravitationally lensed by foreground objects. In this paper, three lens models, i.e., the point-mass model, the singular isothermal sphere (SIS) model, and the Chang–Refsdal model, were taken into consideration to explore the light curves and polarizations of gravitationally lensed kilonovae. We found that, if the time delay between two images exceeds the ejecta-heating timescale for the lens mass ∼1010 M ⊙ in the SIS model, a tiny bump-like signal will be generated in the light curve, and the total luminosity will be magnified in all cases. The polarization of lensed kilonovae is significantly enhanced in most cases. Future detections of lensed kilonovae will impose constraints on the morphology of the ejecta and aid in the determination of the nature of compact object mergers and the search for strong gravitational lenses.

We study the geometrical properties of null congruences generated by an aligned null direction of the Weyl tensor (WAND) in spacetimes of Weyl and Ricci type N (possibly with a non-vanishing cosmological constant) in an arbitrary dimension. We prove that a type N Ricci tensor and a type III or N Weyl tensor have to be aligned. In such spacetimes, the multiple WAND has to be geodetic. For spacetimes with type N aligned Weyl and Ricci tensors, the canonical form of the optical matrix in the twisting and non-twisting cases is derived and the dependence of the Weyl and the Ricci tensors and Ricci rotation coefficients on the affine parameter of the geodetic null congruence generated by the WAND is obtained.

Weak gravitational lensing studies of galaxy clusters often assume a spherical cluster model to simplify the analysis, but some recent studies have suggested this simplifying assumption may result in large biases in estimated cluster masses and concentration values, since clusters are expected to exhibit triaxiality. Several such analyses have, however, quoted expressions for the spatial derivatives of the lensing potential in triaxial models, which are open to misinterpretation. In this paper, we give a clear description of weak lensing by triaxial NFW galaxy clusters and also present an efficient and robust method to model these clusters and obtain parameter estimates. By considering four highly triaxial NFW galaxy clusters, we re-examine the impact of simplifying spherical assumptions and found that while the concentration estimates are largely unbiased except in one of our traixial NFW simulated clusters, for which the concentration is only slightly biased, the masses are significantly biased, by up to 40%, for all the clusters we analysed. Moreover, we find that such assumptions can lead to the erroneous conclusion that some substructure is present in the galaxy clusters or, even worse, that multiple galaxy clusters are present in the field. Our cluster fitting method also allows one to answer the question of whether a given cluster exhibits triaxiality or a simple spherical model is good enough.

We present weak lensing mass estimates of seven shear-selected galaxy cluster candidates from the deep lens survey. The clusters were previously identified as mass peaks in convergence maps of 8.6 deg2 of R-band imaging, and followed up with X-ray and spectroscopic confirmation, spanning a redshift range 0.19-0.68. Most clusters contained multiple X-ray peaks, yielding 17 total mass concentrations. In this paper, we constrain the masses of these X-ray sources with weak lensing, using photometric redshifts from the full set of BVRz' imaging to properly weight background galaxies according to their lensing distance ratios. We fit both NFW and singular isothermal sphere profiles, and find that the results are insensitive to the assumed profile. We also show that the results do not depend significantly on the assumed prior on the position of the mass peak, but that this may become an issue in future larger samples. The inferred velocity dispersions for the extended X-ray sources range from 250 to 800 km s–1, with the exception of one source for which no lensing signal was found. This work further establishes shear selection as a viable technique for finding clusters, but also highlights some unresolved issues such as determination of the mass profile center without biasing the mass estimate, and fully accounting for line-of-sight projections. A follow-up paper will examine the mass-X-ray scaling relations of these clusters.

In this paper we have investigated some properties of the gravitational fields of magnetic type, using timelike congruences and the Bianchi identities. Null eigendirections of the conformal tensor are determined without any assumption regarding the nature of the sources of the gravitational field. Some relations between the kinematical quantities of the timelike congruences and the curvature of space-time have been derived. Our results reduce to those of empty space when the Ricci tensor is taken to be zero. These emptyspace equations lead us to a scalar potential which satisfies the field equation of Newton's theory of gravitation.

A list is given of a canonical set of the Newman–Penrose quantities ΦAB, the tetrad components of the trace-free Ricci tensor, for each Plebański class according to Plebański’s classification of this tensor. This comparative list can easily be extended to cover the classification in tetrad language of any second-order, trace-free, symmetric tensor in a space-time. A fourth-order tensor which is the product of two such tensors was defined by Plebański and used in his classification. This has the same symmetries as the Weyl tensor. The Petrov classification of this tensor, here called the Plebański tensor, is discussed along with the classification of the Ricci tensor. The use of the Plebański tensor in a couple of areas of general relativity is also briefly discussed.

IT is well known that by contraction one can deduce from the Bianchi identities in Riemannian geometry the following result: From this it follows that if: which is the usual condition for an Einstein space, then: This implies: for the conformal and protective curvature tensors in Eisenhart's1 notation. The gravitational significance of (3) and (4) does not appear to have been examined so far, although (4i) has received some attention in a particular context2.

If the space-time has a space-like Killing symmetry, the three-dimensional spin coefficient technique, which was useful for time-like symmetry, can be applied again. This paper contains the spin coefficient forms of the Einstein equation and the Ricci and Weyl tensors.

The locally rotationally symmetric tilted perfect fluid Bianchi type V cosmological model provides examples of future geodesically complete spacetimes that admit a ‘kinematic singularity’ at which the fluid congruence is inextendible but all frame components of the Weyl and Ricci tensors remain bounded. We show that for any positive integer n there are examples of Bianchi type V spacetimes admitting a kinematic singularity such that the covariant derivatives of the Weyl and Ricci tensors up to the nth order also stay bounded. We briefly discuss singularities in classical spacetimes.

The higher dimensional Weyl curvature induces on the brane a new source of gravity. This Weyl fluid of geometrical origin (reducing in the spherically symmetric, static configuration to a dark radiation and dark pressure) modifies space-time geometry around galaxies and has been shown to explain the flatness of galactic rotation curves. Independent observations for discerning between the Weyl fluid and other dark matter models are necessary. Gravitational lensing could provide such a test. Therefore we study null geodesics and weak gravitational lensing in the dark radiation dominated region of galaxies in a class of spherically symmetric brane-world metrics. We find that the lensing profile in the brane-world scenario is distinguishable from dark matter lensing, despite both the brane-world scenario and dark matter models fitting the rotation curve data. In particular, in the asymptotic regions light deflection is 18% enhanced as compared to dark matter halo predictions. For a linear equation of state of the Weyl fluid we further find a critical radius, below which brane-world effects reduce, while above it they amplify light deflection. This is in contrast to any dark matter model, the addition of which always increases the deflection angle.

The determination of the Petrov type of the Weyl tensor for a given space-time metric is important in exact solution theory, in particular in the equivalence problem where it forms an important part in the algorithm for determining whether two space-time metrics are locally isometric. The determination of the Segre type of the Ricci tensor (not discussed here) is also an important part of the algorithm. It should be noted that most Ricci tensor classification algorithms (see Joly and MacCallum 1990) require the determination of the Petrov type of the Plebanski tensor. Thus the determination of the Petrov type of tensor with Weyl tensor symmetries is seen to be particularly fundamental especially in the context of the equivalence problem. .

This lecture completes the discussion of the basic formalism of the proposed generalization of General Relativity based on local automorphism invariance. It contains the proposed Lagrangian for this theory and the concomitant field equations. The fundamental Lagrangian is taken to be in the canonical gauge theory form (quadratic in the total curvature), and the coupling constant therefore is dimensionless. The dynamical variables of the theory are taken to be the spinor, the spin connection, the vierbeins, the Clifford connection, and the drehbeins. Internal consistency of the field equations, as well as congruency with the Bianchi identities, is demonstrated. The relationship of this theory to usual Einsteinian gravity is discussed. Since the proposed Lagrangian is quadratic in the total curvature, it does not contain the usual Einstein-Hilbert term (linear in the Ricci scalar curvature). However, the Clifford connection contains a scalar and a pseudoscalar field. When these fields are extracted from the Clifford connection, the Lagrangian is found to contain a Higgs type potential in these fields, as well as a coupling of these fields to the Ricci scalar. The minimum of the potential occurs for non-zero values of these fields and this induces a term in the Lagrangian linear in the Ricci curvature. In quantum language we would say that the automorphism gauge fields C α obtain non-zero vacuum expectation values: $$\left\langle {C_\alpha } \right\rangle = \sqrt {8\alpha _f } M_p \hat \gamma _\alpha ,$$ (26.1) where \(\hat \gamma _\alpha\) are a set of global (spacetime independent) matrices satisfying the Clifford algebra defining relation, M p is the Planck mass, and \(\alpha _f = \frac{{f^2 }} {{4\pi }}\) where f is the coupling constant for the automorphism gauge fields.KeywordsField EquationBianchi IdentityEinstein GravityRicci ScalarSpin ConnectionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter contains a 56-page-long list of formulae from the calculus on manifolds, and tables concerning different topics: Lie groups, Lie algebras and symmetric spaces, a list of Poincaré polynomials of compact simple Lie groups, an overview of real forms of classical complex simple Lie algebras and their corresponding simple Lie groups, a table of irreducible Riemannian symmetric spaces of type I and III, a table of Riemannian symmetric spaces of classical type with noncompact isotropy group, etc. One can find the formulae for Christoffel symbols, the curvature tensor, Bianchi identities, Ricci tensor, the basic differential operators, the expression for conformal changes of Riemannian metrics, Cartan structure equations for pseudo-Riemannian manifolds, and many more. Several of these formulae are used throughout the book; others are not, but they have been included since such a collection might prove useful as an aide-mémoire, also to lecturers and researchers.KeywordsRelevant TheoryChristoffel SymbolRiemannian Symmetric SpaceMoving CoframeCartan StructureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The metric is an important tensorial object that introduces more structure into a (differential) manifold. The metric coefficients allow for example to determine the length of parameter curves in the manifold and make it possible to relate corresponding co- and contravariant objects defined on the manifold. Thus the inner product of either co- or contravariant quantities and in particular the angle between two vectors or (covectors) may be computed. Adopting the Ricci postulate of vanishing covariant derivative of the metric coefficients results in a decomposition of the fully covariant connection into its Riemann part, that depends exclusively on the metric, and the contortion, that depends exclusively on the torsion. Moreover, in this case the length of vectors and their angle with respect to geodesics are preserved upon parallel transport. Based on the metric coefficients the fully covariant curvature tensor, displaying left and right skew symmetry, is introduced and the corresponding modifications of the Bianchi identities are highlighted. Likewise the metric allows to introduce the mixed-variant Ricci tensor together with corresponding identities. Finally the metric enables to compute the Ricci scalar from the previous curvature tensors. The particular case of a symmetric, metrically connected manifold represents a Riemann geometry with associated Riemann curvature tensor as fundamental for example in the Einstein Theory of General Relativity/Gravitation. A generalization is obtained by allowing for metric manifolds with non-vanishing covariant derivative of its metric coefficients. The presence of the non-metricity is then subsequently reflected by extra terms in the decomposition of the connection and in the explicit representation of the curvature tensor, the Ricci tensor, the Ricci scalar, and the corresponding Bianchi identities for the various curvature quantities.