Symmetric energy-momentum tensor of spinor fields

We construct a symmetric energy-momentum tensor for spinor fields with an arbitrary Lagrange function on Riemannian space-time manifolds. We show that this tensor can be considered as the metric energy-momentum tensor.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 2, pp. 306–314, August, 1996.     

Lower bounds for eigenvalues of the Dirac-Witten operator

We get optimal lower bounds for the eigenvalues of the Dirac-Witten operator of compact(with or without boundary) spacelike hypersurfaces of Lorentian manifold satisfying certain conditions,just in terms of the mean curvature and the scalar curvature and the spinor energy-momentum tensor. In the limiting case,the spacelike hypersurface is either maximal and Einstein manifold with positive scalar curvature or Ricci-flat manifold with nonzero constant mean curvature.     

Enhancing energy conserving methods

Recent observations [5] indicate that energy-momentum methods might be better suited for the numerical integration of highly oscillatory Hamiltonian systems than implicit symplectic methods. However, the popular energy-momentum method, suggested in [3], achieves conservation of energy by a global scaling of the force field. This leads to an undesirable coupling of all degrees of freedom that is not present in the original problem formulation. We suggest enhancing this energy-momentum method by splitting the force field and using separate adjustment factors for each force. In case that the potential energy function can be split into a strong and a weak part, we also show how to combine an energy conserving discretization of the strong forces with a symplectic discretization of the weak contributions. We demonstrate the numerical properties of our method by simulating particles that interact through Lennard-Jones potentials and by integrating the Sine-Gordon equation.This work was partly supported by NIH Grant P41RR05969, DOE/NSF Grant DE-FG02-91-ER25099/DMS-9304268, and NSF GCAG/HPCC ASC-9318159.     

Gyroscopic control and stabilization

Summary In this paper, we consider the geometry of gyroscopic systems with symmetry, starting from an intrinsic Lagrangian viewpoint. We note that natural mechanical systems with exogenous forces can be transformed into gyroscopic systems, when the forces are determined by a suitable class of feedback laws. To assess the stability of relative equilibria in the resultant feedback systems, we extend the energy-momentum block-diagonalization theorem of Simo, Lewis, Posbergh, and Marsden to gyroscopic systems with symmetry. We illustrate the main ideas by a key example of two coupled rigid bodies with internal rotors. The energy-momentum method yields computationally tractable stability criteria in this and other examples.     

Galerkin-based energy-momentum time-stepping schemes for anisotropic hyperelastic materials

This paper is motivated by dynamic simulations of fiber-reinforced materials in light-weight structures, and has two goals. First of all, the introduction of energy-momentum schemes for nonlinear anisotropic materials, based on GALERKIN approximations in space and time, assumed strain approximations in time and superimposed algorithmic stress fields (compare [1]). Second, to show a variationally consistent design of energy-momentum schemes using a differential variational principle. We develop a discrete variational principle leading to energy-momentum schemes as discrete EULER-LAGRANGE equations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Translation Invariant Massive Nelson Model: II. The Continuous Spectrum below the Two-Boson Threshold

In this paper we continue the study of the energy-momentum spectrum of a class of translation invariant, linearly coupled, and massive Hamiltonians from non-relativistic quantum field theory. The class contains the Hamiltonians of E. Nelson (J Math Phys 5:1190–1197, 1964) and H. Fröhlich (Adv Phys 3:325–362, 1954). In Møller (Ann Henri Poincaré 6:1091–1135, 2005; Rev Math Phys 18:485–517, 2006) one of us previously investigated the structure of the ground state mass shell and the bottom of the continuous energy-momentum spectrum. Here we study the continuous energy-momentum spectrum itself up to the two-boson threshold, the threshold for energetic support of two-boson scattering states. We prove that non-threshold embedded mass shells have finite multiplicity and can accumulate only at thresholds. We furthermore establish the non-existence of singular continuous energy-momentum spectrum. Our results hold true for all values of the particle-field coupling strength but only below the two-boson threshold. The proof revolves around the construction of a certain relative velocity vector field used to construct a conjugate operator in the sense of Mourre.     

Lower Bounds for the Eigenvalues of the Dirac Operator: Part I. The Hypersurface Dirac Operator

We give optimal lower bounds for the hypersurface Diracoperator in terms of the Yamabe number, the energy-momentum tensor andthe mean curvature. In the limiting case, we prove that the hypersurfaceis an Einstein manifold with constant mean curvature.     

Casimir effect in nonstationary regions with Friedmann metric

A study is made of the two-dimensional problem of finding the energy-momentum tensor of a scalar massless field with symmetric nonstationary boundary conditions in the case of a nonstationary Friedmann metric. A combined method is proposed for regularizing the energy-momentum tensor — point splitting and Abel-Plana summation. A method is proposed for determining the wave functions and energy-momentum tensor when boundary conditions are specified on characteristics. The solution of these problems in two-dimensional spacetime makes it possible to obtain explicit expressions for the energy-momentum tensor.St. Petersburg University of Economics and Finance. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 3, pp. 450–457, December, 1994.     

Scalar field vacuum polarization on homogeneous spaces with an invariant metric

We develop a method for calculating vacuum expectation values of the energy-momentum tensor of a scalar field on homogeneous spaces with an invariant metric. Solving this problem involves the method of generalized harmonic analysis based on the method of coadjoint orbits.     

Vacuum polarization of a scalar field on lie groups and homogeneous spaces

We propose a method for calculating vacuum means of the scalar field energy-momentum tensor on Lie groups and homogeneous spaces. We use the generalized harmonic analysis based on the method of coadjoint representation orbits.     

Positive Energy-Momentum Theorem for AdS-Asymptotically Hyperbolic Manifolds

The aim of this paper is to prove a positive energy-momentum theorem under the (well known in general relativity) dominant energy condition, for AdS-asymptotically hyperbolic manifolds. These manifolds are by definition endowed with a Riemannian metric and a symmetric 2-tensor which respectively tend to the metric and second fundamental form of a standard hyperbolic slice in Anti-de Sitter space-time. There exists a positive mass theorem for asymptotically hyperbolic spin Riemannian manifolds (with zero extrinsic curvature), and we present an extension of this result for the non zero extrinsic curvature case. Communicated by Sergiu Klainerman Submitted: January 15, 2006 Accepted: January 15, 2006     

Radiating metric, retarded time coordinates of Kerr-Newman-de Sitter black holes and related energy-momentum tensor

The charged rotating metric in de Sitter space, derived by Mallett and used by Koberlein, is shown incorrect. Mallett’s metric and his energy-momentum tensor do not satisfy the Einstein-Maxwell field equations with a cosmological term in the nonradiating and radiating Kerr-Newman-de Sitter case. The corresponding correct metric and the radiating energy-momentum tensor are given.     

Persistance d’orbites périodiques relatives dans les systèmes hamiltoniens symétriques

It was known to Poincaré that a non-degenerate periodic orbit in a Hamiltonian system persists to nearby energy-levels. In this Note, we consider the analogous problem for relative periodic orbits in symmetric Hamiltonian systems. We show that non-degenerate relative periodic orbits also persist when shifting to nearby values of the energy-momentum map, under the hypothesis that the group of symmetries acts freely.     

Equation of motion for a spherically-symmetric shell in the relativistic theory of gravity

In the framework of the relativistic theory of gravity, the equation of motion for a spherically-symmetric singular shell is derived and integrated in the first approximation of the Newton potential U = m/r. We use the covariant energy-momentum conservation law for matter in the effective Riemannian space and, independently, the energy-momentum conservation law for the matter + gravity system in Minkowski space. For the problem under consideration, we show the equivalence of our approach to the classical formalism of singular shells.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 344–352, May, 1996.Translated by A. M. Semikhatov.     

Refinement of the gravitational energy definition

We analyze the error arising in the well-known “energy” method for calculating the central density of a white dwarf with maximum possible mass in the framework of the theory of tetrad energy-momentum complexes of the gravitational field. We choose the preferable complex by comparing the central density calculated using each of the three complexes under study with its value obtained by numerically integrating the Einstein equations.     

On the relation between ADM and Bondi energy-momenta Ⅲ-perturbed radiative spatial infinity

In a vacuum spacetime equipped with the Bondi's radiating metric which is asymptotically flat at spatial infinity including gravitational radiation (Condition D), we establish the relation between the ADM total energy-momentum and the Bondi energy-momentum for perturbed radiative spatial infinity. The perturbation is given by defining the "real" time as the sum of the retarded time, the Euclidean distance and certain function f.     

On the relation between ADM and Bondi energymomentaⅢ-perturbed radiative spatial infinity

In a vacuum spacetime equipped with the Bondi's radiating metric which is asymptotically flat at spatial infinity including gravitational radiation (Condition D),we establish the relation between the ADM total energy-momentum and the Bondi energy-momentum for perturbed radiative spatial infinity.The perturbation is given by defining the"real"time as the sum of the retarded time,the Euclidean distance and certain function f.     

A comparison of three methods of constructing Lyapunov functions

Chetayev's effective method [1] for constructing Lyapunov functions in the form of a set of first integrals of the equations of perturbed motion has been widely used since the 1950s in Russia. In the 1980s the energy-Casimir method [2] was developed in the U.S.A. as well as the energy-momentum method [3], employed for Hamiltonian systems. A comparison of these methods for systems with a finite number of degrees of freedom has shown that the energy-Casimir method is a more complicated version of Chetayev's method, while the energy-momentum method is essentially the Routh-Lyapunov method [4,5], stated in modern geometrical language. Some examples are considered.     

On stability problems of the space elevator

A long dumb-bell satellite in the radial configuration represents a simple model of a Space Elevator. The dumb-bell moves with the angular velocity of the planet, but not relative to it. To calculate the stability of such a relative equilibrium we used the reduced energy-momentum method. We considered dumb-bell satellite models with rigid and elastic, but massless, rods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

重建极性连续统理论的基本定律和原理(Ⅹ)——主均衡定律

通过对诸主均衡定律和应用Noether定理得出的守恒定律进行比较,自然地导出微极连续统力学的1个统一的主均衡定律和6个物理上可能的均衡方程．其中,通过扩展众所周知和惯用的能量动量张量的概念,得到相当一般的定名为能量-动量的、能量-角动量的和能量-能量的守恒定律和均衡方程．显然,在这后3种情况下的主均衡定律中,物理场量是难以凭借直觉假定出来的．最后,作为特殊情形,直接推演出若干现有的结果．