Lanczos Potential for the van Stockung Space-Time

The Lanczos Potential is a theoretical useful tool to find the conformal Weyl curvature tensor C _abcd of a given relativistic metric. In this paper we find the Lanczos potential L _abc for the van Stockung vacuum gravitational field. Also, we show how the wave equation can be combined with spinor methods in order to find this important three covariant index tensor.     

Expanding spherically symmetric models without shear

The integrability properties of the field equationL _xx =F(x)L ² of a spherically symmetric shear-free fluid are investigated. A first integral, subject to an integrability condition onF(x), is found, giving a new class of solutions which contains the solutions of Stephani and Srivastava as special cases. The integrability condition onF(x) is reduced to a quadrature which is expressible in terms of elliptic integrals in general. There are three classes of solution and in general the solution ofL _xx =F(x)L ² can only be written in parametric form. The case for whichF=F(x) can be explicitly given corresponds to the solution of Stephani. A Lie analysis ofL _xx =F(x)L ² is also performed. If a constant vanishes, then the solutions of Kustaanheimo and Qvist and of this paper are regained. For 0 we reduce the problem to a simpler, autonomous equation. The applicability of the Painlevé analysis is also briefly considered.     

Proofs of existence of local potentials for trace-free symmetric 2-forms using dimensionally dependent identities

We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second-order differential equation, e.g., a wave equation in Lorentz signature. Furthermore, we exploit higher-dimensional tensor identities to obtain the analogous results for (m, m)-forms in 2m dimensions.     

Local existence of symmetric spinor potentials for symmetric (3,1)-spinors in Einstein space–times

We investigate the possibility of existence of a symmetric potential H_ABA′B′=H_{(AB)(A′B′)} for a symmetric (3,1)-spinor L_ABCA′, e.g., a Lanczos potential of the Weyl spinor, as defined by the equation L_ABCA′=_(A^B′H_BC)A′B′. We prove that in all Einstein space–times such a symmetric potential H_ABA′B′ exists. Potentials of this type have been found earlier in investigations of some very special spinors in restricted classes of space–times. A tensor version of this result is also given. We apply similar ideas and results by Illge to Maxwell’s equations in a curved space–time.     

关于纯引力共形反常

本文采用几何和拓扑的方法,讨论弯曲时空中的纯引力共形反常,并得到了纯引力共形反常的新的表达式αε^abcdΩ_abΛΩ_cd+βε^abcd H_abΛH_cd,其中α,β是任意常数,Ω_ab与H_ab分别是Riemann曲率2-形式与Thomas引入的共形不变曲率2-形式。这里,第一项正比于Euler类,第二项除了包含通常熟知的Wegl张量平方项以外,还含有其它共形不变的不变量。 关键词：     

In {2 2} vacuums,the null-datum on null rays is exactlyr −5

It is shown, by a simple argument, that in any {2 2} (i.e. type D) vacuum space-time, the complex component of the Weyl curvature (spinor abcd) along any light ray is precisely proportional tor ^–5, wherer is the standard Schwarzschild radial coordinate, or its natural complex generalization for the arbitrary {2 2} case.     

Modules Over the Noncommutative Torus and Elliptic Curves

Using the Weil–Brezin–Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely generated projective modules over the algebra A _θ of the noncommutative torus. We show that such A _θ -modules have a natural interpretation as Moyal deformations of vector bundles over an elliptic curve E _τ , under the condition that the deformation parameter θ and the modular parameter τ satisfy a non-trivial relation.     

WANDs of the black ring

Necessary conditions for various algebraic types of the Weyl tensor in higher dimensions are determined. These conditions are then used to find Weyl aligned null directions for the black ring solution. It is shown that the black ring solution is algebraically special, of type I_i, while locally on the horizon the type is II. One exceptional subclass – the Myers-Perry solution – is of type D.     

Spinor Factorizations of the Bel-Robinson Tensor

Recently Bonilla and Senovilla studied factorizations of the symmetric and tracefree rank four Bel-Robinson tensor T_abcd into two symmetric tracefree rank two tensors. While the Bel-Robinson tensor has the dimension of energy density squared, each of these factors has the dimension of energy density. When the two factors can be chosen to be equal they are called the square root of T_abcd. The approach used was purely tensorial. In this paper we use spinors and show that the factors can be found in a very simple way using the principal null directions of the Weyl tensor. We obtain a factorization of the Weyl spinor into two symmetric rank two spinors, which when multiplied by their complex conjugates give the tracefree and symmetric factors of T_abcd. The factorization is immediately seen to be non-unique in most cases and the number of essentially non-equivalent factorizations becomes clear. It also becomes obvious that the square root only can exist in spacetimes of Petrov types N, D and O, in which cases one can equally well speak about the square root of the Weyl spinor. Explicit formulas for the factors of the Weyl spinor are given for all Petrov types.     

Locally Weyl Invariant Massless Bosonic and Fermionic Spin-1/2 Action in the (Y4, g) Space-Time

We search for a real bosonic and fermionic action in four dimensions which both remain invariant under local Weyl transformations in the presence of contortion and non-metricity tensor. In the presence of the non-metricity tensor the investigation is extended to (W _n, g) space-time while when the torsion is encountered we are restricted to the (U ₄, g) space-time. Our results hold in general for the (Y ₄, g) space-time and we also calculate extra contributions to the conformal gravity.     

Physical states and BRST operators for higher-spin <Emphasis Type="Italic">W</Emphasis> strings

In this paper, we mainly investigate the W _2,s ^M ⊗ W _2,s ^L system, in which the matter and the Liouville subsystems generate the W _2,s ^M and W _2,s ^L algebras, respectively. We first give a brief discussion of the physical states for the corresponding W strings. The lower states are given by freezing the spin-2 and spin-s currents. Then, introducing two pairs of ghost-like fields, we give the realizations of the W _1,2,s algebras. Based on these linear realizations, the BRST operators for the W _2,s algebras are obtained. Finally, we construct new BRST charges of the Liouville system for the W _2,s ^L strings at the specific values of the central charges c: for the W _2,3^L algebra, c=−24 for the W _2,4^L algebra and for the W _2,6^L algebra, at which the corresponding W _2,s ^L algebras are singular.     

Letter: The Non-Existence of a Lanczos Potential for the Weyl Curvature Tensor in Dimensions n ≥ 7

In this paper it is shown that a Lanczos potential for the Weyl curvature tensor does not exist for all spaces of dimension n 7.     

The KT-BRST Complex of a Degenerate Lagrangian System

Quantization of a Lagrangian field system essentially depends on its degeneracy and implies its BRST extension defined by sets of non-trivial Noether and higher-stage Noether identities. However, one meets a problem how to select trivial and non-trivial higher-stage Noether identities. We show that, under certain conditions, one can associate to a degenerate Lagrangian L the KT-BRST complex of fields, antifields and ghosts whose boundary and coboundary operators provide all non-trivial Noether identities and gauge symmetries of L. In this case, L can be extended to a proper solution of the master equation.      

The affine geometry of the Lanczos H-tensor formalism

We identify the fiber-bundle-with-connection structure that underlies the Lanczos H-tensor formulation of Riemannian geometrical structure. We consider linear connections to be type (1,2) affine tensor fields, and we sketch the structure of the appropriate fiber bundle that is needed to describe the differential geometry of such affine tensors, namely the affine frame bundleA ₁ ² M with structure groupA ₁ ² (4) =GL(4) T ₁ ^{2 4} over spacetimeM. Generalized affine connections on this bundle are in 1-1 correspondence with pairs(, K) onM, where thegl(4)-component denotes a linear connection and the T ₁ ^{2 4}-componentK is a type (1,3) tensor field onM. We show that the Lanczos H-tensor arises from a gauge fixing condition on this geometrical structure. The resulting translation gauge, theLanczos gauge, is invariant under the transformations found earlier by Lanczos. The other Lanczos variablesQ _mandq are constructed in terms of the translational component of the generalized affine connection in the Lanczos gauge. To complete the geometric reformulation we reconstruct the Lanczos Lagrangian completely in terms of affine invariant quantities. The essential field equations derived from ourA ₁ ² (4)-invariant Lagrangian are the Bianchi and Bach-Lanczos identities for four-dimensional Riemannian geometry.     

W-geometry of the Toda systems associated with non-exceptional simple Lie algebras

The present paper describes theW-geometry of the Abelian finite non-periodic (conformal) Toda systems associated with theB, C andD series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a generalization of the classical Plücker embedding of theA-case to the flag manifolds associated with the fundamental representations ofB _n ,C _n andD _n , and a direct proof that the corresponding Kähler potentials satisfy the system of two-dimensional finite non-periodic (conformal) Toda equations. It is shown that theW-geometry of the type mentioned above coincide with the differential geometry of special holomorphic (W) surfaces in target spaces which are submanifolds (quadrics) ofCP ^N with appropriate choices ofN. In addition, theseW-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Plücker embedding. These conditions are automatically fulfilled when Toda equations hold.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud.     

Lanczos Spintensor and GHP Formalism

This work shows how the equations which relate the Lanczos potential K _ijr to the conformal tensor C _abpq, can be structured in a simple form when written in terms of the formalism proposed by Geroch-Held-Penrose. As working examples, we present the immediate construction of K _ijk for any spacetime with Petrov type O, N or III. Our findings are in good agreement with already published results, which indicates a relationship between the spin coefficients and the Lanczos spintensor onto the canonical tetrad.     

A regular potential which is nowhere in L 1

A nonnegative potential V: ℝ^v→ℝ is constructed for which V ∉ L _q (G) for any nonempty open G⊂→^v, q>0, and for which nevertheless W _inf2 ^sup1 ∩ Q(V) is dense in W _inf2 ^sup1 , i.e., is a form core for −1/2Δ in L ₂.     

Numerical simulation of diffusive conductivity in Rashba split two-dimensional gas

We numerically model the conductivity of a two-dimensional electron gas (2DEG) in the presence of the Rashba spin–orbit (SO) interaction in the diffusive transport regime. We performed simulation using samples which width W and length L are up to 200 and 30 000, respectively, on a tight-binding square lattice. When the system is in the diffusive regime, the quadratic increase of the conductivity with SO interaction strength λ_SO derived previously by Born approximation is reproduced except for very weak SO interaction. In order to obtain satisfactory agreement between numerical and analytical results, the sample width and length should be much larger than the mean free path ℓ but the length should be shorter than the localization length ξ, e.g. 4ℓW and 10ℓLξ. The anomaly at weak SO interaction is also observed in the conductance fluctuation and the localization length, and is attributed to the finite size crossover from symplectic to orthogonal class with decreasing SO interaction. The typical values of the SO interaction characterizing the crossover obtained for ℓ48 are λ_SO1.0/W and 0.2/W when we impose open and periodic boundary conditions, respectively.     

Double Higgs production at the linear colliders and the probing of the Higgs self-coupling

We study double Higgs production in the e⁺e^? and γγ modes of the linear collider. It is also shown how one can probe the scalar potential in these reactions. We discuss the effective longitudinal W approximation in γγ processes and the W _LW_L luminosities in the two modes of a high-energy linear collider. A generalised non-linear gauge-fixing condition, which is particularly useful for tree-level calculations of electroweak processes for the laser induced collider, is presented. Its connection with the background-field approach to gauge fixing is given.     

Finite-size corrections to the free energy of Coulomb systems with a periodic boundary condition

Classical Coulomb systems ind dimensions (d?2) with a periodic boundary condition, periodW, in the directionx ^(d)are considered. With the other directions of the confining volume of lengthL, it is shown that if the system is in a conducting phase, then the “strip” free energykTf _W ,f _W = ?lim _L→∞ L ^?(d?1) log Z, has the large-W expansion $$f_W \sim Wf_\infty + \frac{{(d/2 - 1)\Gamma (d/2 - 1)}}{{\pi ^{d/2} W^{d - 1} }}\zeta (d) + O\left( {\frac{1}{{W^{d + 1} }}} \right)$$ wherekTf _∞ is the bulk free energy per unit volume, ζ(x) denotes the Riemann zeta function, andΓ(x) denotes the gamma function. With 1/W identified askT, this result is precisely the low-temperature behavior of the free energy of a (d?1)-dimensional Debye solid. This fact is explained in terms of an equivalence between the Coulomb gas and quantum fields. Also, the expansion is verified for some exactly solved models of Coulomb systems in two dimensions.