Third-order tensor potentials for the Riemann and Weyl tensors

The representations of the Riemann and the Weyl tensors of a four-dimensional Riemannian manifold through covariant derivatives of third-order potentials are examined in detail. The Weyl tensor always admits a completely general representation whereas the Riemann tensor does not. Nevertheless there exists a class of Riemannian manifolds whose Riemann tensors may be calculated in terms of potentials; in this connection, specific examples are exhibited explicitly. The possibility of introducing gauges on the potentials is reexamined in connection with the previous result. New properties of the representations are also discussed.     

Weyl,curvature, ricci,and metric tensor symmetries

Weyl symmetries for some specific spherically symmetric static spacetimes are derived and compared with metric, Ricci, and curvature tensor symmetries.     

On dual properties of the Weyl tensor

We present some new identities satisfied by double and triple products of the Weyl tensorW _v and its dual .Partially supported by C.N.Pq.     

On the Newtonian limit of the Weyl tensor

In this note we wish to complement some recent work in the cosmological literature concerning the Weyl conformal curvature tensor and its parts. In particular, we shall give a clear-cut definition of the Newtonian limits of electric and magnetic parts of the Weyl tensor. We also discuss that only a subset of the relativistic equations is needed to obtain a closed system of equations in the Newtonian limit.     

Eigenvalue estimates for the Dirac operator depending on the Weyl tensor

We prove new lower bounds for the first eigenvalue of the Dirac operator on compact manifolds whose Weyl tensor or curvature tensor, respectively, is divergence-free. In the special case of Einstein manifolds, we obtain estimates depending on the Weyl tensor.     

The magnetic part of the Weyl tensor,and the expansion of discrete universes

We examine the effect that the magnetic part of the Weyl tensor has on the large-scale expansion of space. This is done within the context of a class of cosmological models that contain regularly arranged discrete masses, rather than a continuous perfect fluid. The natural set of geodesic curves that one should use to consider the cosmological expansion of these models requires the existence of a non-zero magnetic part of the Weyl tensor. We include this object in the evolution equations of these models by performing a Taylor series expansion about a hypersurface where it initially vanishes. At the same cosmological time, measured as a fraction of the age of the universe, we find that the influence of the magnetic part of the Weyl tensor increases as the number of masses in the universe is increased. We also find that the influence of the magnetic part of the Weyl tensor increases with time, relative to the leading-order electric part, so that its contribution to the scale of the universe can reach values of \(\sim \)1%, before the Taylor series approximation starts to break down.     

Magnetic fields and the Weyl tensor in the early universe

We have solved the Einstein–Maxwell equations for a class of metrics with constant spatial curvature by considering only a primordial magnetic field as source. We assume a slight modification of the Tolman averaging relations so that the energy–momentum tensor of this field possesses an anisotropic pressure component. This inhomogeneous magnetic universe is isotropic and its time evolution is guided by the usual Friedmann equations. In the case of a flat universe, the space-time metric is free of singularities (except the well-known initial singularity at \(\text {t} = 0\) ). It is shown that the anisotropic pressure of our model has a straightforward relation to the Weyl tensor. We then analyze the effect of this new ingredient on the motion of test particles and on the geodesic deviation of the cosmic fluid.     

Nonlinear phase field theory for fracture and twinning with analysis of simple shear

A phase field theory for coupled twinning and fracture in single crystal domains is developed. Distinct order parameters denote twinned and fractured domains, finite strains are addressed and elastic nonlinearity is included via a neo-Hookean strain energy potential. The governing equations and boundary conditions are derived; an incremental energy minimization approach is advocated for prediction of equilibrium microstructural morphologies under quasi-static loading protocols. Aspects of the theory are analysed in detail for a material element undergoing simple shear deformation. Exact analytical and/or one-dimensional numerical solutions are obtained in dimensionless form for stress states, stability criteria and order parameter profiles at localized fractures or twinning zones. For sufficient applied strain, the relative likelihood of localized twinning vs. localized fracture is found to depend only on the ratio of twin boundary surface energy to fracture surface energy. Predicted criteria for shear stress-driven fracture or twinning are often found to be in closer agreement with test data for several types of real crystals than those based on the concept of theoretical strength.     

The neutrino energy-momentum tensor and the Weyl equations in curved space-time

In general, a first order Lagrangian gives rise to second order Euler-Lagrange equations. However, there are important examples where the associated Euler-Lagrange equations are of first order only, the Weyl neutrino equations being of this type. In this paper we therefore consider first order spinor Lagrangians which give rise to firstorder Euler-Lagrange equations. Specifically, the most general first order spinor field equations of rank one in curved space-time which are derivable from a first order Lagrangian of the same type are explicitly constructed. Subject to a certain restriction, the Weyl neutrino equation is the only possibility. Furthermore, if the spinor field satisfies the Weyl neutrino equation, then the associated energy momentum tensor is the conventional neutrino energymomentum tensor.     

Third-order tensor potentials for the Riemann and Weyl tensors. II: Singular solutions

The analysis of the admissibility of a potential representation for the Riemann tensor is here continued. As in the preceding paper, the starting point is to regard the relationship between the Riemann tensor and its possible potential as a system of partial differential equations determining the unknown potential. The first result, strengthening a previous conclusion, is that there never exist ordinary solutions. Surprisingly enough, in a four-dimensional Riemannian manifold the existence of singular solutions is established without requiring any integrability condition. Possible applications and generalizations are also suggested.     

全球风场构建涡度、散度的新方法

目前在数值预报中通常利用风场借助于差分方法来构建涡度和散度场,这个问题涉及到观测资料求微分的问题,从数学上来说,此问题是不适定的.在有限区域上构建时可以采用一维数值微分来实现,但此方法在端点部分的资料必须是精确的,本文提出了新的方法,该方法借助于周期函数的一维数值微分,并用该方法应用到全球风场构建涡度、散度中去,同时与通常的差分方法进行了比较,利用涡度、散度计算了流函数和势函数,然后用流函数和势函数重构初始风场.结果表明,本文提出的方法算法稳定、可行且计算精度比差分方法高,为应用到全球气象资料的诊断分析及预报中提出了新的思路.     

Geometrical statistics of the vorticity vector and the strain rate tensor in rotating turbulence

We report results on the geometrical statistics of the vorticity vector obtained from experiments in electromagnetically forced rotating turbulence. A range of rotation rates Ω is considered, from non-rotating to rapidly rotating turbulence with a maximum background rotation rate of Ω = 5 rad/s (with Rossby number much smaller than unity). Typically, the Taylor-scale Reynolds number in our experiments is around Re_λ ≈ 100. The measurement volume is located in the centre of the fluid container above the bottom boundary layer, where the turbulent flow can be considered locally statistically isotropic and horizontally homogeneous for the non-rotating case, see L.J.A. van Bokhoven, H.J.H. Clercx, G.J.F. van Heijst, and R.R. Trieling, Experiments on rapidly rotating turbulent flows, Phys. Fluids 21 (2009) 096601. Based on the full set of velocity derivatives, measured in a Lagrangian way by three-dimensional (3D) particle tracking velocimetry, we have been able to quantify statistically the effect of system rotation on several flow properties. For the range of rotation rates considered, the experimental results show how the turbulence evolves from almost isotropic 3D turbulence (Ω ? 0.2 rad/s) to quasi-two-dimensional turbulence (Ω ≈ 5.0 rad/s), and how this is reflected by several statistical quantities. In particular, we have studied the orientation of the vorticity vector with respect to the three eigenvectors of the local strain rate tensor and with respect to the vortex stretching vector. Additionally, we have quantified the role of system rotation on the self-amplification terms of the enstrophy and strain rate equations and the direct contribution of the background rotation on these evolution equations. The main effect of background rotation is the strong reduction of extreme events and related (strong) reduction of the skewness of PDFs of several quantities, for example, the intermediate eigenvalue of the strain rate tensor and the enstrophy self-amplification term.     

Weyl fields,quantum integrability and conformal invariant field theories

We show that some Weyl field theories arise as a quantum “linear” problem associated to some Kac-Moody algebras. We relate this quantum “linear” problem to the conformal invariant field theories studied by Dashen and Frishman and to the WZW field theory.