Energy and momentum in general relativity

In general relativity, conservation of energy and momentum is expressed by an equation of the form /x= 0, where –gT represents the total energy, momentum, and stress. This equation arises from the divergence formula dV _v = (/x ^v )d ⁴ d. Here we show that this formula fails to account properly for the system of basis vectors e(x). We obtain the (invariant) divergence formula e dV _v = e (/x ^v + )d ⁴ d. Conservation of energy and momentum is therefore expressed by the covariant equation (/x ^v ) + = 0. We go on to calculate the variation of the action under uniform displacements in space-time. This calculation yields the covariant equation of conservation, as well as the fully symmetric energy tensor . Finally, we discuss the transfer of energy and momentum, within the context of Einstein's theory of gravitation.     

Absolute parallelism: Spherical symmetry and singularities

A one-parameter class of compatible field equations of n-frames h ^a is considered, characterized by the fact that after the substitution h ^a = HPH ^a (H = detH ^a , p is the constant depending on the dimensionality n and on the equation parameter ), the equations can be written in a trilinear form which contains the matrix H_a and its derivatives, but not H ^a . The equations remain regular for degenerated matrices H_a, if r = rank H_a ^< 2, the points with r < n being singular (if n > 1 + 1/). It is shown that there exist spherically symmetric solutions with singularities appearing for r < n; as the singularity develops, there appears a domain bounded by singular points in which, for certain values of and n, the field h ^a becomes complex.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 47–52, September, 1991.The author thanks I. L. Bukhbinder for his interest to this paper and useful discussions during the seminar.     

The twisted SU(3) group. Irreducible *-representations of the C *-algebra C(SμU(3))

Irreducible ^*-representations of C(SU(3)) are constructed for ]0, 1[. It is proved that C(SU(3)) is a type-I C ^*-algebra.     

Fock representations of the affine Lie algebraA 1 (1)

The aim of this note is to show that the affine Lie algebraA ₁ ⁽¹⁾ has a natural family _, ,v of Fock representations on the spaceC[x _i,y _j;i andj ], parametrized by (,v) C ². By corresponding the highest weight _, of _, to each (,), the parameter spaceC ₂ forms a double cover of the weight spaceC₀C –₁ with singularities at linear forms of level –2; this number is (–1)-times the dual Coxeter number. Our results contain explicit realizations of irreducible non-integrable highest wieghtA ₁ ⁽¹⁾ -modules for generic (,v).     

Feasibility of an experiment to determine the branching ratio for the emission of a heavy neutrino after muon capture in3He

The triton energy of the muon capture reaction ³He t+v, where ³ He is the ground state of muonic³He, has been measured in order to investigate a possible heavy v admixture into the flavour with high sensitivity. ³ He has been formed via the pd fusion reaction by stopping ^– in an ionization chamber (IC) filled with an H/D gas mixture of 3% D concentration at a pressure of 161 bar. In a first short experiment 650 triton events were observed yielding an upper limit for the -heavy v mixing strength of 2.3×10^–3 atE _0v=60 MeV.     

The paramagnetic state (μ+e−) in silicon and tellurium at high temperature

The paramagnetic state (⁺e^–) in Si and Te was observed in a longitudinal magnetic field. The mean lifetimes of these states were obtained: _Si = 1.45(3) s, _Te = 12.5(8) s at 290 K, _Te = 12(2) s at 250 K.     

On Convergence to Equilibrium Distribution, II. The Wave Equation in Odd Dimensions, with Mixing

The paper considers the wave equation, with constant or variable coefficients in ⁿ , with odd n3. We study the asymptotics of the distribution _t of the random solution at time t as t . It is assumed that the initial measure ₀ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that ₀ satisfies a Rosenblatt- or Ibragimov–Linnik-type space mixing condition. The main result is the convergence of _t to a Gaussian measure as t , which gives a Central Limit Theorem (CLT) for the wave equation. The proof for the case of constant coefficients is based on an analysis of long-time asymptotics of the solution in the Fourier representation and Bernstein's room-corridor argument. The case of variable coefficients is treated by using a version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.     

Exact gravitational field of the infinitely long rotating hollow cylinder

The vacuum line element inside an infinitely long rotating hollow cylinder is the usual flat space line element. It is fitted in a most general way to the general cylindrical vacuum field outside at the singular hypersurfaceR ₀=const, representing the infinitely thin hollow cylinder. With the use of the jump conditions atR ₀=const the surface densities , of which the energy-momentum-stress tensor of the shell consists, are calculated. The physical properties of the cylinder, as derived from the eigenvalues and -vectors of , and the generated gravitational field are discussed in full detail.     

Algebraisch spezielle Einstein-Räume mit einer Bewegungsgruppe

In algebraically special Einstein spaces (R_v=0) with a hypersurfaceorthogonal spacelike Killing vector field _v, the trajectories of the multiple eigen null directions k lie — except one case — in the subspacesV ₃ orthogonal to _v (k=0) and are hypersurface-orthogonal. The solutions with vanishing expansion (k_,;=0, Kundt's class) can be determined explicitly.     

Collisionless kinetic equations for massive particles with spins 1/2 and 0 in gravitational and electromagnetic fields

We define a covariant and gauge-invariant generalization of the Wigner functions of particles with spins 1/2 and 0. The collisionless kinetic equations are obtained for these particles in external gravitational and electromagnetic fields in the quasiclassical approximation; also obtained are the momentum representations of the energy-momentum tensor, current, and spin tensor, taking into account the effects of the spin's interaction with the gravitational field an electromagnetic field. The following notation is used: e and m are the charge and mass of the particles; is Planck's constant; (x) are the covariant-fixed Dirac matrices; ^,=(1/4)[, ]: a(^b)=(1/2) (a b +ab ); [A, B]=A·B – B·A; A,B=A·B+B·A; g(x)=det(g (x));R = –...; the speed of light c=1.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 47–53, September, 1990.The author wishes the thank Yu. G. Ignat'ev and members of the seminar in General Relativistic Statistics and Cosmology of the Kazan' Pedagogical Institute for useful discussions.     

BRST quantization of a scalar particle in a curved background

The BRST formalism is employed to quantize a scalar particle and interactions with an external scalar field (x ) and vector gauge fieldA (x ) in the background of an arbitrary gravitational field. The second-quantized actions are obtained.     

Calculation of Ratios of μ and π Masses to the e Mass for a Simplified Model with a Square-Well Potential

In the present paper, the feasibility of constructing a model of elementary and particles based on the Poincaré model of the electron [1] is examined with allowance for schemes of particle decay. The muon and pion in stopped states are considered as spherical resonators for virtual neutrino quanta excited inside an elastic lepton shell; the number of these quanta is determined by the scheme of the corresponding particle decay. In the model, the muon differs from the electron by two additional quanta of the neutrino field. The e, , and masses are calculated with the help of a single parameter. The ratio of and e masses is about (6c/e ²)^2/3, and the ratio of ° and e masses is (3/2)^2/3. The calculated e, , and ° masses are in the 0.547:105.71:134.963 ratios, which is in good agreement with the available experimental data. The calculated electromagnetic radius of the charged pion (±) = 0.5f is close to that calculated from the experimental data. The neutrino mass _e is found to be m( _e ) 0.02 eV.     

Quantization of the Maxwell field in linear conformal invariant gauges

It is shown that the generalized Lorentz gauges provide all linear conformal invariant gauges, i.e. gauges such that A =0.     

On a Static Solution of the Einstein Equation with Incoming and Outgoing Radiation

The Einstein equation with T = k k + where k, l are null is considered with spherical symmetry and staticity. The solution has a naked singularity and is not asymptotically flat. However, it may be interpreted as an envelope for any static spherical body making it more massive. Such an interpretation and some of its implications are detailed.     

Trilinear generally covariant equations

Field equations for n-frames h ^a that are possible in the theory of absolute parallelism are considered. It is shown that in three cases the equations can, after the substitution h ^a =H^pH ^a (H=det H ^a , p is an n-dependent constant), be written in a trilinear form that contains only the matrix H _a and its derivatives and not H ^a . It is shown that the equations are still regular for degenerate but finite matrices H _a if rank H _a 2.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 22–27, February, 1991.     

Massive spin-two particle in a gravitational field

The spin-two particle is described by a symmetric tensorh subject to the subsidiary conditionsh = h =0. Their covariant generalization and the wave equation have been obtained directly from the Eulerian variational equations by algebraic methods only. In addition to the tensor fieldh a symmetric third-rank tensor = as well as a vector fieldA have been added, neither of which enter in the final result. The Lagrangian function is taken as a linear sum of all combinations which can be constructed from these functions, as well as terms involving the curvature tensor and its two possible contractions. Variation with respect toh , andA independently gives the Euler equations. Combining the various trace equations and choice of arbitrary constants yields the subsidiary conditions, while the Euler equations themselves give the connection between the auxiliary functions and the tensorh as well as the generalization of the wave equationD D h + 2R h -R h -R h +g R h +Rh =m ² h Finally, variation with respect tog yields the energy-momentum tensor.     

Laser damage in silicon photodiodes

Thermal damage of silicon photodiodes exposed to intense optical radiation is investigated. Damage thresholds of Si photodiodes irradiated by 1.06m laser pulses are reported for values of irradiation time,, ranging from 10^–8 to 1s. Threshold laser irradiation produces visible microscopic damage and a permanent degradation in photoresponse. The loss of responsivity is associated with degradation of the detector diode characteristics due to laser-induced heating. The time and wavelength dependence agree with the predictions of a thermal model which treats a semi-infinite material irradiated by a Gaussian laser beam. The energy density thresholds are independent of for short irradiation times and asymptotically approach a limiting behaviour which increases as for long times. They are given by the empirical relationE ₀=65[1+217/tan^–1(258)^1/2] J cm^–2 for 1.06m radiation. The thresholds at short irradiation times of detectors damaged by 1.06m radiation are about 25 times larger than those of detectors exposed to 0.6943m radiation. The greater susceptibility at 0.6943m is attributed to a larger optical absorption coefficient.     

Self-consistent calculation of statics and dynamics of the roughening transition

Statics and dynamics of the modified kinetic discrete Gaussian model are treated selfconsistently using a Gaussian probability assumption. A non-trivial roughening temperatureT _R is found in exactly two dimensions only. The free energyF, the correlation length and the interface roughness h ² are found to behave—lnFlnh ²(T _R –T)^–1 for temperaturesT approachingT _R from below. The linear relaxation rate of the order parameter is found to be proportional to ^–2. As a model for crystal growth, the growth rate depends linearly upon the chemical potential difference aboveT _R , shows a metastable regime belowT _R with a spinodal limit of metastability _c , beyond which oscillatory growth starts. The critical behavior of _c is found to be ln _c –(T _R –T)^–1+O(ln (T _R –T)).     

Heat transfer in unsteady MHD oscillatory flow

Heat transfer in unsteady MHD channel flow (of an incompressible viscous and electrically conducting fluid) under oscillatory pressure gradient when the channel surfaces are conducting and moving with time-dependent velocities has been analysed. The velocity, magnetic field and temperature distributions have been obtained and their numerical results are shown graphically.Symbols u velocity - H _o applied magnetic field - H _x induced magnetic field - T temperature - T ₁ ^* ,T ₂ ^* temperatures of the upper and lower planes - density - p pressure - kinematic viscosity - magnetic diffusivity - electrical conductivity of the fluid - A ^* characteristic velocity - L characteristic length - _e magnetic permeability of the fluid - C _p specific heat - coefficient of viscosity - k thermal conductivity - ₁, ₂ permeabilities of the planes - ₁, ₂ conductivities of the planes - ₁, ₂ conductance ratios of the planes - Pr Prandtl number (=C _p /k) - E Eckert number - M Hartmann number (= _e H _o L/) - R _e Reynolds number (=LA ^*/) - R _m magnetic Reynolds number (= _e LA ^*) - S Pr.E(=S)     

Stochastic coupling and thermodynamic inequalities

Let ₁ and ₂ be thermodynamic Gibbs measures on ^m and ⁿ , respectively. Diffusions are constructed having ₁, and ₂ as invariant measures. These diffusions are then coupled; inequalities between expectations of certain random variables on the two spaces result.Partially supported by NSF-MCS 74-07313-A03