Scalar Invariants Research Articles

The Invar tensor package

The Invar package is introduced, a fast manipulator of generic scalar polynomial expressions formed from the Riemann tensor of a four-dimensional metric-compatible connection. The package can maximally simplify any polynomial containing tensor products of up to seven Riemann tensors within seconds. It has been implemented both in Mathematica and Maple algebraic systems. Program summary Program title:Invar Tensor Package Catalogue identifier:ADZK_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZK_v1_0.html Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 136 240 No. of bytes in distributed program, including test data, etc.:2 711 923 Distribution format:tar.gz Programming language:Mathematica and Maple Computer:Any computer running Mathematica versions 5.0 to 5.2 or Maple versions 9 and 10 Operating system:Linux, Unix, Windows XP RAM:30 Mb Word size:64 or 32 bits Classification:5 External routines:The Mathematica version requires the xTensor and xPerm packages. These are freely available at http://metric.iem.csic.es/Martin-Garcia/xAct Nature of problem:Manipulation and simplification of tensor expressions. Special attention on simplifying scalar polynomial expressions formed from the Riemann tensor on a four-dimensional metric-compatible manifold. Solution method:Algorithms of computational group theory to simplify expressions with tensors that obey permutation symmetries. Tables of syzygies of the scalar invariants of the Riemann tensor. Restrictions:The present versions do not fully address the problem of reducing differential invariants or monomials of the Riemann tensor with free indices. Running time:Less than a second to fully reduce a monomial of the Riemann tensor of degree 7 in terms of independent invariants.

Strain Dependence of the Critical Temperature $T_{c}$ of Polycrystalline ${\rm Nb}_{3}{\rm Sn}$ Superconductors

The polycrystalline nature of Nb3Sn introduces crystal orientation as a possible variable in the strain dependence of the critical properties. It is shown that a dependence of strain sensitivity on crystal orientation implies a dependence on strain of the distribution of Tc, with implications for the functional dependence of strain sensitivity on the strain tensor. The distribution of Tc is found to have an influence on the characteristics of the resistive transition of Tc . Calculations of the resistive transition are done under alternative assumptions for the crystal orientation dependence of strain sensitivity. Available data on the strain dependence of the transition of Tc are contradictory and inconclusive. If strain sensitivity were found to be independent of grain orientation, that would imply a strain dependence on the entire deviatoric strain tensor, through the scalar invariants, including the tetragonal and shear parts. This would be a surprising conclusion given the emphasis on tetragonal strain in Nb3Sn .

Minimum mass–radius ratio for charged gravitational objects

We rigorously prove that for compact charged general relativistic objects there is a lower bound for the mass–radius ratio. This result follows from the same Buchdahl type inequality for charged objects, which has been extensively used for the proof of the existence of an upper bound for the mass–radius ratio. The effect of the vacuum energy (a cosmological constant) on the minimum mass is also taken into account. Several bounds on the total charge, mass and the vacuum energy for compact charged objects are obtained from the study of the Ricci scalar invariants. The total energy (including the gravitational one) and the stability of the objects with minimum mass–radius ratio is also considered, leading to a representation of the mass and radius of the charged objects with minimum mass–radius ratio in terms of the charge and vacuum energy only.

Some conformal and projective scalar invariants of Riemannian manifolds

It is proved that, on any closed oriented Riemannian n-manifold, the vector spaces of conformal Killing, Killing, and closed conformal Killing r-forms, where 1 ≤ r ≤ n − 1, have finite dimensions t r , k r , and p r , respectively. The numbers t r are conformal scalar invariants of the manifold, and the numbers k r and p r are projective scalar invariants; they are dual in the sense that t r = t n−r and k r = p n−r . Moreover, an explicit expression for a conformal Killing r-form on a conformally flat Riemannian n-manifold is given.

Instability of the negative mass Schwarzschild naked singularity

We study the negative mass Schwarzschild spacetime, which has a naked singularity, and show that it is perturbatively unstable. This is achieved by first introducing a modification of the well-known Regge–Wheeler–Zerilli approach to black hole perturbations to allow for the presence of a ‘kinematic’ singularity that arises for negative masses, and then exhibiting exact exponentially growing solutions to the linearized Einstein's equations. The perturbations are smooth everywhere and behave nicely around the singularity and at infinity. In particular, the first-order variation of the scalar invariants can be made arbitrarily small everywhere as compared to the zeroth order terms. Our approach is also compared to a recent analysis that leads to a different conclusion regarding the stability of the negative mass Schwarzschild spacetime. We also comment on the relevance of our results to the stability of a more general negative mass, nakedly singular spacetimes.

Invariant Measures of Closeness to Linear Dependency of Screw Systems

This paper describes a new, general analytical method of determining the closeness to linear dependence of any system of lines and screws, giving n invariant scalars for an n-system of screws. The method proposed evaluates the maximum available scalar or Klein product between the unit screw under test and any unit combination of screws reciprocal to all of the set except the test screw. Examples are given of the application of this theory to the behaviour of a partially complete system of kinematic restraint such as those used in manufacturing fixtures of a simple structure and of a serial robotic manipulator.

Kinematic and dynamic analysis of open-loop mechanical systems using non-linear recursive formulation

In this paper, a non-linear recursive formulation is developed for kinematic and dynamic analysis of open-loop mechanical systems. The non-linear equations of motion are developed for deformable links that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars and matrices that depend on the spatial co-ordinates as well as the assumed displacement field, and these time invariant quantities represent the dynamic coupling between the rigid-body modes and elastic deformations. A new recursive formulation is presented for solving equations of motion for open-loop chains consisting of interconnected rigid and deformable open-loop mechanical systems. This formulation is expressed by the recursive relationships and the generalized non-linear equations for deformable mechanical systems to obtain a large system of loosely coupled equations of motion. The main processor program consists of three main modules: constraint module, mass module and force module. The constraint module is used to numerically evaluate the relationship between the absolute and joint accelerations. The mass module is used to numerically evaluate the system mass matrix as well as the non-linear Coriolis and centrifugal forces associated with the absolute, joint and elastic co-ordinates. Simultaneously, the force module is used to numerically evaluate the generalized external and elastic forces associated with the absolute, joint and elastic co-ordinates. Computational efficiency is achieved by taking advantage of the structure of the resulting system of loosely coupled equations. The solution techniques used in this investigation yield a much smaller operations count and can more efficiently implement in any computer. The algorithms and solutions presented in this paper are illustrated by using an industrial robotic manipulator system. The numerical results using this formulation are also presented and discussed in this paper. Copyright © 2006 John Wiley & Sons, Ltd.

On spacetimes with constant scalar invariants

We study Lorentzian spacetimes for which all scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant (CSI spacetimes). We obtain a number of general results in arbitrary dimensions. We study and construct warped product CSI spacetimes and higher-dimensional Kundt CSI spacetimes. We show how these spacetimes can be constructed from locally homogeneous spaces and VSI spacetimes. The results suggest a number of conjectures. In particular, it is plausible that for CSI spacetimes that are not locally homogeneous the Weyl type is II, III, N or O, with any boost weight zero components being constant. We then consider the four-dimensional CSI spacetimes in more detail. We show that there are severe constraints on these spacetimes, and we argue that it is plausible that they are either locally homogeneous or that the spacetime necessarily belongs to the Kundt class of CSI spacetimes, all of which are constructed. The four-dimensional results lend support to the conjectures in higher dimensions.

Homogeneity and Curvatures of Geodesic Spheres

We study some scalar curvature invariants on geodesic spheres and use them to characterize several kinds of Riemannian manifolds such as homogenous manifolds and in particular, the two-point homogeneous spaces and the Damek-Ricci spaces.

Invariants of the Riemann tensor: A classical approach

This paper describes the start of an investigation into the application of classical invariant theory to scalar polynomial invariants of the Riemann tensor. In particular, the classical methods of enumerating invariants are discussed with the aim, not achieved in this paper, of verifying Sneddon's result, obtained by explicit calculation of the invariants that the dimension of a Hilbert basis is 38.

Recursive Newton–Euler formulation for flexible dynamic manufacturing analysis of open-loop robotic systems

The main objective of this paper is to develop a recursive formulation for the flexible dynamic manufacturing analysis of open-loop robotic systems. The nonlinear generalized Newton–Euler equations are used for flexible bodies that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars, vectors and matrices that depend on the spatial coordinates as well as the assumed displacement fields. These time invariant quantities represent the dynamic manufacturing couplings between the rigid body motion and elastic deformation. This formulation applies recursive procedures with the generalized Newton–Euler equations for flexible bodies to obtain a large, loosely coupled system equation describing motion in flexible manufacturing systems. The techniques used to solve the system equations can be implemented in any computer system. The algorithms presented in this investigation are illustrated using cylindrical joints for open-loop robotic systems, which can be easily extended to revolute, slider and rigid joints. The recursive Newton–Euler formulation developed in this paper is demonstrated with a robotic system using cylindrical mechanical joints.

Nonlinear Recursive Formulation for Kinematic and Dynamic Analysis of Robotic Manufacturing Systems

The objective of this paper is to develop a nonlinear recursive formulation for the dynamic analysis of robotic manufacturing systems. The nonlinear recursive equations are used for open-loop flexible manipulators that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars, vectors and matrices that depend on the spatial coordinates as well as the assumed displacement fields, and these time invariant quantities represent the dynamic manufacturing couplings between the rigid body motion and elastic deformation. This formulation applies recursive procedures with the nonlinear equations for flexible manipulators to obtain a large, loosely coupled system equation of motion in robotic manufacturing systems. The numerical techniques used to solve for the system equations of motion can be more efficiently implemented in any computer systems. The algorithms presented in this investigation are illustrated by using standard mechanical joints for robotic manufacturing systems that can be easily extended to other special joints. The nonlinear recursive formulation developed in this paper is illustrated by a robotic manufacturing system using standard revolute mechanical joints.

Covariant irreducible parametrization of electromagnetic fields in arbitrary space-time

We present a new unified covariant description of electromagnetic field properties for an arbitrary space-time. We derive a complete set of irreducible components describing a six-dimensional electromagnetic field from the Maxwell and metric tensors using the symmetry group SL(2,C). For the special case of a flat space-time metric the components are shown to correspond to the scalar invariants of the electromagnetic field, the energy-momentum-stress tensor and in addition, three new tensors expressing physical observables of rank two and four, respectively. We make a physical interpretation of one of the new rank two tensor as describing a classical intrinsic spin of the electromagnetic field.

Fermions as sources of accelerated regimes in cosmology

In this work it is investigated if fermionic sources could be responsible for accelerated periods during the evolution of a universe where a matter field would answer for the decelerated period. The self-interaction potential of the fermionic field is considered as a function of the scalar and pseudo-scalar invariants. Irreversible processes of energy transfer between the matter and gravitational fields are also considered. It is shown that the fermionic field could behave like an inflaton field in the early universe and as dark energy for an old universe.

Gravitational field of relativistic gyratons

The metric ansatz is used to describe the gravitational field of a beam-pulse of spinning radiation (gyraton) in an arbitrary number of spacetime dimensions D. First we demonstrate that this metric belongs to the class of metrics for which all scalar invariants constructed from the curvature and its covariant derivatives vanish. Next, it is shown that the vacuum Einstein equations reduce to two linear problems in (D-2)-dimensional Euclidean space. The first is to find the static magnetic potential created by a point-like source. The second requires finding the electric potential created by a point-like source surrounded by given distribution of the electric charge. To obtain a generic gyraton-type solution of the vacuum Einstein equations it is sufficient to allow the coefficients in the corresponding harmonic decompositions of solutions of the linear problems to depend arbitrarily on retarded time and substitute the obtained expressions in the metric ansatz. We discuss properties of the solutions for relativistic gyratons and consider special examples.

AFFINE CURVATURE HOMOGENEOUS 3-DIMENSIONAL LORENTZ MANIFOLDS

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members of the family are geodesically complete, others are not. All have vanishing scalar invariants.

Invariant Strain Analysis of the Critical Temperature<tex>$rm T_rm c$</tex>of<tex>$rm Nb_3rm Sn$</tex>

An analysis has been made that incorporates the strain dependence of the critical temperature T/sub c/ of Nb/sub 3/Sn in a calculation using the formalism of strong superconductivity. The critical temperature in the absence of strain depends on the harmonic phonon spectrum. The value of T/sub c/ can be related to the acoustic phonon modes. It is found that the phonon frequency characteristics can be determined, in a simple model, from a scalar strain energy potential that is a function of the invariants of the strain tensor under the cubic symmetry group and that includes harmonic and anharmonic terms. The strain dependent anharmonic phonon spectrum is found to result in the strain dependence of T/sub c/. A calculation of the normalized strain dependence T/sub c/(/spl epsiv/)/T/sub cm/ is presented for assumed wire and tape composite conductors and shown to agree with measurements from the literature over the full range of strain. The correspondence of the various strain scalar invariants with various aspects of the typical strain profile is discussed. The analysis contains the influence of the hydrostatic strain on T/sub c/ and relates the observed asymmetry in the strain profile between compressive and tensile intrinsic strain to the principle part of the third strain invariant.

Diagrammatic Young projection operators for U(n)

We utilize a diagrammatic notation for invariant tensors to construct the Young projection operators for the irreducible representations of the unitary group U(n), prove their uniqueness, idempotency, and orthogonality, and rederive the formula for their dimensions. We show that all U(n) invariant scalars (3n−j coefficients) can be constructed and evaluated diagrammatically from these U(n) Young projection operators. We prove that the values of all U(n)3n−j coefficients are proportional to the dimension of the maximal representation in the coefficient, with the proportionality factor fully determined by its Sk symmetric group value. We also derive a family of new sum rules for the 3−j and 6−j coefficients, and discuss relations that follow from the negative dimensionality theorem.

Tensor representations and solutions of constitutive equations for viscoelastic fluids

An approach is developed using tensor representations to assess and characterize both the transient behavior and equilibrium states of viscoelastic fluid constitutive equations in viscometric flows. The methodology is based on the replacement of the differential constitutive equation for the deviatoric part of the viscoelastic stress tensor by an equivalent and more tractable set of differential equations for the characteristic scalar invariants. In the case of planar flows, this equivalence leads to an explicit, closed-form analytic solution for the time evolution of the extra-stress tensor that is formally expressed as a second-order fluid relation, with time-dependent coefficients. As a validation of the approach, an analysis of the transient and equilibrium system characteristics of fluid flows described by the corotational Jeffreys model and general Oldroyd-type constitutive equations is presented.

Some remarks on generalised multipole expansions

Models for the mutual potential energy between two molecules proposed in the scientific literature often contain a sum of inverse-power interactions involving pairs of sites belonging to the two particles; in turn, these quantities are functions of a few scalar invariants involved in the problem at hand, and one is often interested in directly obtaining an explicit expression of the potential in terms of the latter; the extensively studied two-centre multipole expansion for the mutual electrostatic energy between two charge distributions is a well-known example of this procedure and of its restrictions. We consider here another, less widely known and possibly complementary, approach, proposed by Sebek some years ago [J. Sebek, Czech. J. Phys. B 38, 1185 (1988)]; the resulting formulae show that this procedure can become computationally favourable for sufficiently high molecular symmetry.