Congruence Of Curves Research Articles

Congruence of Curves in Weyl-Otsuki Spaces

In this paper, we study congruence of curves in Weyl-Otsuki spaces using the Ricci's coefficients of that congruence in orthogonal case. We first prove that the Ricci’s coefficients determine the regular general connection of an Otsuki space. Then we give the condition for these coefficients in Weyl-Otsuki spaces to be skew-symmetric in the first two indices as in Riemannian spaces. We obtain the necessary and sufficient conditions for the curves of a congruence to be geodesic, normal and irrotational, respectively. Finally, we prove that if a congruence satisfy the equation, and any two of the conditions that to be geodesic, normal and irrotational, then it also satisfies the other third one.

Kinematics in metric-affine geometry

In a given geometry, the kinematics of a congruence of curves is described by a set of three quantities called expansion, rotation, and shear. The equations governing the evolution of these quantities are referred to as kinematic equations. In this paper, the kinematics of congruence of curves in a metric-affine geometry are analysed. Without assuming an underlying theory of gravity, we derive a generalised form of the evolution equations for expansion, namely, Raychaudhuri equation (timelike congruences) and Sachs optical equation (null congruences). The evolution equations for rotation and shear of both timelike and null congruences are also derived. Generalising the deviation equation, we find that torsion and non-metricity contribute to a relative acceleration between neighbouring curves. We briefly discuss the interpretation of the expansion scalars and derive an equation governing angular diameter distances. The effects of torsion and non-metricity on the distances are found to be dependent on which curves are chosen as photon trajectories. We also show that the rotation of a hypersurface orthogonal congruence (timelike or null) is a purely non-Riemannian feature.

Asymptotic structure with a positive cosmological constant

This is the second of two papers (Fernández-Álvarez F and Senovilla J M M 2021 Class. Quantum Grav. 39 165011) that study the asymptotic structure of space–times with a non-negative cosmological constant Λ. This paper deals with the case Λ > 0. Our approach is founded on the ‘tidal energies’ built with the Weyl curvature and, specifically, we use the asymptotic super-Poynting vector computed from the rescaled Bel–Robinson tensor at infinity to provide a covariant, gauge-invariant, criterion for the existence, or absence, of gravitational radiation at infinity. The fundamental idea we put forward is that the physical asymptotic properties are encoded in , where the first element of the triplet is a three-dimensional manifold, the second is a representative of a conformal class of Riemannian metrics on , and the third element is a traceless symmetric tensor field on . The full set of physically relevant properties of the space–time cannot be characterised at infinity without taking D ab into consideration, and our radiation criterion takes this fully into account. We similarly propose a no-incoming radiation criterion based also on the triplet and on radiant supermomenta deduced from the rescaled Bel–Robinson tensor too. We search for news tensors encoding the two degrees of freedom of gravitational radiation and argue that any news-like object must be associated to, and depends on, two-dimensional cross-sections of . We identify one component of news for every such cross-section and present a general strategy to find the second component, which depends on the particular physical situation. We put in connection the radiation condition and the news-like tensors with the directional structure of the gravitational field at infinity and the criterion of no-incoming radiation. We also introduce the concept of equipped by endowing the conformal boundary with a selected congruence of curves which may be determined by the algebraic structure of the asymptotic Weyl tensor. We also define a group of asymptotic symmetries preserving the new structures. We consider the limit Λ → 0, and apply all our results to selected exact solutions of Einstein field equations in order to illustrate their validity.

Hamilton–Jacobi approach for linearly acceleration-dependent Lagrangians

We develop a constructive procedure for arriving at the Hamilton–Jacobi framework for the so-called affine in acceleration theories by analysing the canonical constraint structure. We find two scenarios in dependence of the order of the emerging equations of motion. By properly defining generalized brackets, the non-involutive constraints that originally arose, in both scenarios, may be removed so that the resulting involutive Hamiltonian constraints ensure integrability of the theories and, at the same time, lead to the right dynamics in the reduced phase space. In particular, when we have second-order derivatives equations of motion we are able to detect the gauge invariant sector of the theory by using a suitable approach based on the projection of the Hamiltonians onto the tangential and normal directions of the congruence of curves in the configuration space. Regarding this, we also explore the generators of canonical and gauge transformations of these theories. Further, we briefly outline how to determine the Hamilton principal function S for some particular setups. We apply our findings to some representative theories: a Chern–Simons-like theory in (2+1)-dim, an harmonic oscillator in 2D and, the geodetic brane cosmology emerging in the context of extra dimensions.

Singularity theorems and the inclusion of torsion in affine theories of gravity

We extend the scope of the Raychaudhuri-Komar singularity theorem of General Relativity to affine theories of gravity with and without torsion. We first generalize the existing focusing theorems using time-like and null congruences of curves which are hypersurface orthogonal, showing how the presence of torsion affects the formation of focal points in Lorentzian manifolds. Considering the energy conservation on a given affine gravity theory, we prove new singularity theorems for accelerated curves in the cases of Lorentzian manifolds containing perfect fluids or scalar field matter sources.

On the Singularities of Families of Curve Congruences on Lorentzian Surfaces

In Nabarro (2011), we define and study the families of ??-conjugate curve congruences ????i${\mathcal {L}\mathcal {C}}^{i}_{\alpha }$ and the families of reflected ??-conjugate curve congruences ????i${\mathcal {L}\mathcal {R}}^{i}_{\alpha }$, i = 1, 2, associated to a self-adjoint operator ?? on a smooth and oriented surface M endowed with a Lorentzian metric. These families parametrize parts of the pencils of forms that link the equation of the ??-asymptotic (resp. ??-characteristic) curves and that of the ??-principal curves. There is a crucial difference with the Riemannian case due to the existence of lightlike curves. In this paper, we study the generic local singularities in the members of these families and describe the way they bifurcate within the families.

On the ( $$1+3$$ 1 + 3 ) threading of spacetime with respect to an arbitrary timelike vector field

We develop a new approach on the (1+3) threading of spacetime $(M, g)$ with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial tensor fields and on the Riemannian spatial connection $\nabla^{\star}$, which behave as $3D$ geometric objects. We obtain new formulas for local components of the Ricci tensor field of $(M, g)$ with respect to the threading frame field, in terms of the Ricci tensor field of $\nabla^{\star}$ and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri's equation enables us to prove Lemma 6.2, which completes a well known lemma used in the proof of Penrose-Hawking singularity theorems.Finally, we apply the new $(1+3)$ formalism to the study of the dynamics of a Kerr-Newman black hole.

Invariants of Riemannian curves in dimensions 2 and 3

In this article, a complete and geometrical description of minimal sets of differential invariants in 2- and 3-dimensional Riemannian manifolds is given in terms of the dimension of the isometry group. These invariants provide a solution to the (local) problem of congruence of curves so that a final generalization of Frenet's theory in low dimensional manifolds is reached.

A Class of Conformal Curves in the Reissner–Nordström Spacetime

A class of curves with special conformal properties (conformal curves) is studied on the Reissner-Nordstr\"om spacetime. It is shown that initial data for the conformal curves can be prescribed so that the resulting congruence of curves extends smoothly to future and past null infinity. The formation of conjugate points on these congruences is examined. The results of this analysis are expected to be of relevance for the discussion of the Reissner-Nordstr\"om spacetime as a solution to the conformal field equations and for the global numerical evaluation of static black hole spacetimes.

Families of curve congruences on Lorentzian surfaces and pencils of quadratic forms

We define and study families of conjugate and reflected curve congruences associated to a self-adjoint operator on a smooth and oriented surface M endowed with a Lorentzian metric g. These families trace parts of the pencil joining the equations of the -asymptotic and the -principal curves, and the pencil joining the -characteristic and the -principal curves, respectively. The binary differential equations (BDEs) of these curves can be viewed as points in the projective plane. Using the polar lines of various BDEs with respect to the conic of degenerate quadratic forms, we obtain geometric results on the above pencils and their relation with the metric g, on the type of solutions of a given BDE, of its -conjugate equation and on BDEs with orthogonal roots.

SPACETIME AS A DEFORMABLE SOLID

In this letter we discuss the possibility of treating the spacetime by itself as a kind of deformable body for which we can define a fundamental lattice, just like atoms in crystal lattices. We show three signs pointing in that direction. We simulate the spacetime manifold by a very specific congruence of curves and use the Landau–Raychadhuri equation to study the behavior of such a congruence. The lattice appears because we are forced to associate to each curve of the congruence a sort of fundamental "particle". The world-lines of these particles should be identified with the congruence fulfilling the spacetime manifold. The conclusion is that when describing the deformations of the spacetime the Einstein equations emerge and the spacetime metric should be treated as a secondary (not fundamental) object of the theory.

Families of surfaces and conjugate curve congruences

Abstract Given a smooth and oriented surface M in the Euclidean space ℝ3, the conjugate curve congruence Cα is a family of pairs of foliations on M that links the lines of curvature and the asymptotic curves of M. This family is first introduced in [Fletcher, Geometrical problems in computer vision, Liverpool University, 1996] and is studied in [Bruce, Fletcher, Tari, Contemp. Math. 354: 1–18, 2004, Bruce, Tari, Trans. Amer. Math. Soc. 357: 267–285, 2005]. When the surface M = M 0 is deformed in a 1-parameter family of surfaces Mt , we obtain a 2-parameter family of conjugate curve congruence Cα,t . We study in this paper the generic local singularities in C α 0,0 and the way they bifurcate in the family Cα,t , with (α, t) close to (α 0, 0).

Dupin indicatrices and families of curve congruences

We study a number of natural families of binary differential equations (BDE's) on a smooth surface M in R-3. One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDE's, another between the characteristic and principal BDE's. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets ( given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE. More generally, we consider a natural class of BDE's on such a surface M, and show how the pencil of BDE's joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDE's are intimately related.

Congruences of Dislocations in Continuously Dislocated Crystals

The time-dependent congruences of Volterra-type dislocations are investigated based on the generalized formulas of Frenet in a Riemannian space. The analysis is applied to the description of congruences of edge and mixed dislocations consistent with a continuous distribution of dislocations for which its material space is an equidistant Riemannian space. In particular, the principal congruences of dislocations are considered. The kinematics of congruences of mixed dislocations endowed with univocally defined local slip planes is discussed. It is shown that the geometry of such congruences of dislocations admits a class of nonlinear evolution equations describing the curvature and torsion of a congruence of curves in a Riemannian space. Additional conditions imposed on the derived system of equations in order to describe the evolution of curvature and torsion of congruences of edge dislocations are proposed. In the static case, an expression is given for shear stresses required to bend prismatic edge dislocations of torsion zero located on the totally geodesic crystal surfaces. It follows that the congruence of these dislocations is endowed with a finite self-energy function.

Affine projection tensor geometry: Decomposing the curvature tensor when the connection is arbitrary and the projection is tilted

This paper constructs the geometrically natural objects which are associated with any projection tensor field on a manifold with any affine connection. The approaches to projection tensor fields which have been used in general relativity and related theories assume normal projection tensors of codimension one and connections which are metric compatible and torsion-free. These assumptions fail for projections onto lightlike curves or surfaces and other situations where degenerate metrics occur as well as projections onto two-surfaces and projections onto space–time in the higher dimensional manifolds of unified field theories. This paper removes these restrictive assumptions. One key idea is to define two different ‘‘extrinsic curvature tensors’’ which become equal for normal projections. In addition, a new family of geometrical tensors is introduced: the cross-projected curvature tensors. In terms of these objects, projection decompositions of covariant derivatives, the full Riemann curvature tensor and the Bianchi identities are obtained and applied to perfect fluids, timelike curve congruences, string congruences, and the familiar 3+1 analysis of the spacelike initial value problem of general relativity.

A generalization of the Mariot theorem on congruences of null geodesics

It is shown that the property of a congruence of curves to consist of null geodesics can be defined in terms of a distribution of a co-dimension one, without reference to the conformal structure of the underlying differen­tiable manifold: if k is the vector field tangent to the congruence and k is a 1-form characterizing the distribution, then the congruence is said to be null if k ˩ k = 0 and geodesic if, and only if, k ∧£ k = 0. The geodesic property of the congruence, on an n -dimensional manifold, means that if F is an ( n —2)-form such that k ˩ F = 0 and k ∧ F = 0, then k ∧ d F = 0. A twisting geodesic null congruence on S 1 ᵡ S 2 l + 1 , associated with the Hopf fibration S 2 l + 1 → CP l , is constructed as an illustration.

A GENERALIZATION FOR LAGUERRE FUNCTION OF A HYPERSURFACE IN A RIEMANNIAN MANIFOLD

A generalization relative to a congruence of curves for the Laguerre function of a hyper­ surface in a Riemannian space has obtained by Nirmala (1965) . In this article we generalize the Laguerre function relative to any vector field without using a congruence of curves.

Numerical construction of space-time

Traditional analytic methods have not been powerful enough to solve for strongfield time-dependent solutions of the nonlinear Einstein field equations. For this reason, numerical solution of the equations seemed to be a natural approach. Building on the earlier work inspired by DeWitt & Misner, my colleagues and I have developed a practical algorithm for constructing space-times. The point of view is that space-time is decomposed into spacelike time slices and a congruence of curves threading these slices. The time coordinate is that scalar function which is constant on each time slice, while spatial coordinates are constant along the members of the congruence. The systematic 3+1 Hamiltonian approach was exhaustively studied by Arnowitt et al . (1962); the application to solution of the field equations was recently stressed by Smarr & York (1978 a, b ).

On Δ‐principal directions of a congruence of curves in a FINSLER hypersurface

AbstractIn the existing literature of FINSLER spaces, it has been stressed [ELIOPOULOS 1959, RUND 1956] that the process of Δ‐differentiation leads to the use of DUPIN's indicatrix in finding out the principal directions at a point of a hypersurface which are indeterminate. The process of Δ‐differentiation [4, 7] requires the use of the osculating DUPIN's indicatrix corresponding to a line‐element of FINSLER hypersurface which leads to the linear eigen value problem and thereby helps in determining the principal directions of a congruence of curves. This fact increases the scope of the theory of FINSLER spaces to a considerable extent. In this paper, therefore, an attempt has been made to find the Δ‐principal directions, generalized EULER's theorem, minimal congruences, Δ‐geodesic principal directions and Δ‐absolute curvature of the congruence with respect of a curve of Fn −1.

Fluid flows obtained by geometrical characterizations

A problem in fluid mechanics is considered solved in the usual sense, if we can describe fairly explicitly certain functions such as pressure, density, velocity, etc. over a suitable domain. From such a complete description of a “fluid flow” we can pick out certain purely geometrical attributes such as the family of constant pressure surfaces, or the family of curves determined by the streamlines. In actual fact, of course, we don’t often come up with complete explicit solutions of fluid mechanics problems, and often start our consideration of a problem by focusing our attention on the geometry-for instance, we might begin by assuming the flow is “two dimensional.” Such an assumption is usually motivated by a given physical situation. If we are more concerned with obtaining a general understanding of the nature of solutions in fluid mechanics rather than solving a specific problem, then any one of an endless number of geometrical assumptions might be imposed. Thus we might look for flows whose streamlines form normal congruences of curves, or whose principal normal congruences form normal congruences [I], etc. Rather than pick specific geometrical properties we might ask more general questions of the type “to what extent does the specification of the geometry of a flow determine the flow ?” Or “to what extent do the equations governing the flow restrict the possible shapes of the streamlines, constant pressure surfaces, or other intrinsic geometrical configurations ?” The study of the differential geometry of fluids is old. The spirit is represented (and most results presented) in Refs. [l-9] and references therein. The present contribution is in that spirit. We shall be concerned with steady nondissipative compressible fluid flow