Letter: On the Newtonian Limit in Gravity Models with Inverse Powers of R

I reconsider the problem of the Newtonian limit in nonlinear gravity models in the light of recently proposed models with inverse powers of R. Expansion around a maximally symmetric local background with curvature scalar R ₀ > 0 gives the correct Newtonian limit on length scales R ₀ ^–1/2 if the gravitational Lagrangian satisfies f(R ₀)f(R₀) 1, and I propose two models with f(R ₀) = 0.     

Limiting behavior of asymptotically flat gravitational fields

Couch and Torrence suggest that the vacuum Einstein equations admit a larger class of asymptotically flat solutions than those exhibiting the peeling property. Starting with the assumption that , (d/dr) and (/x ^A ) , wherex ^A (A = 2, 3) are angular coordinates, they show that , where ₁ 2 and ₁<₀; , where ₂ 1 and ₁< ₁; and ₄ and ₃ peel as they would under the stronger peeling conditions. The Winicour-Tamburino energy-momentun and angular momentum integrals for these solutions, in general, diverge. In fact, since Couch and Torrence determine only the radial dependence of the solution, it is not clear that the solutions are well defined. We find that the stronger assumption , (d/dr) , and (/x ^A ) does result in well-defined solutions for which both the energy-momentum and angular momentum intergrals are not only finite but result in the same expressions as are obtained for peeling space-times. This assumption appears to be the minimal assumption that is necessary for investigating outgoing radiation at null infinity.In part based on a dissertation by Stephanie Novak and submitted to Syracuse University in partial fulfillment of the requirement for the Ph.D. degree.     

On equivalent-neighbour,random site models of disordered systems

Models of random systems whose Hamiltonian reads , where and _i ,=1,...,n are independent, identically distributed random variables are discussed.J _ij are assumed to be symmetric, with respect toJ ₀, random variables and also symmetric functions of components of . A question of dependence of a phase diagram on a probability distribution of is addressed. A class of distributions and interactionsJ _ij , which give rise to phase diagrams called typical is selected. Then a problem of obtaining typical phase diagrams, containing a certain region with an infinite number of pure phases, is studied.     

Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann–Hilbert problem approach is used to derive the leading-order asymptotics as of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation ( NLSE), , with finite-density initial data

A Generalisation of Brinkmann's Theorem

Brinkmann has shown that conformally related Einstein spaces are either Ricci flat pp-waves or conformally flat spaces with one as de Sitter space and the other being flat. We herewith show that if g_ikand are distinct metrics for which , then both represent (generalised) pp-waves; and w_,iis a homothetic conformal Killing vector of g_ik. It is further observed that in the situation we can have many (and also non-conformally flat) conformally related solutions, unlike in Einstein spaces. In particular for perfect fluid coupling, we show that the conformally related solutions have to be Friedman-Robertson-Walker metrics with equation of state as , or pp-waves depending on whether w_,iis timelike, spacelike or null.     

Einstein equations and Fierz-Pauli equations with self-interaction in quantum gravity

The Einstein equations can be written as Fierz-Pauli equations with self-interaction, together with the covariant Hilbert-gauge condition, where W denotes the covariant wave operator and G _ik the Einstein tensor of the metric g _ik collecting all nonlinear terms of Einstein's equations. As is known, there do not, however, exist plane-wave solutions _ik(z)with g ^ik Z,_i Z,_k=0of these equations such that what is essential to the introduction of gravitons is not satisfied in general relativity. This nonexistence corresponds with the uncertainty relation,p(g*)²(x)³h hG/ c ³ concerning the total nonlinear gravitational field g ^*ik =g ^k + ^k .     

Observables and Statistical Maps

This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the -effect algebra of effects (fuzzy events) and the set of probability measures on a measurable space . An observable is defined, where is the value space of X. It is noted that there exists a one-to-one correspondence between states on and elements of and between observables and -morphisms from to . Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived.     

BV Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of(W) Strings

Given a simple, simply laced, complex Lie algebra corresponding to the Lie group G, let be thesubalgebra generated by the positive roots. In this Letter we construct aBV algebra whose underlying graded commutative algebra is given by the cohomology, with respect to , of the algebra of regular functions on G with values in . We conjecture that describes the algebra of allphysical (i.e., BRST invariant) operators of the noncritical string. The conjecture is verified in the two explicitly known cases, ₂ (the Virasoro string) and ₃ (the string).     

On General Plane Fronted Waves. Geodesics

A general class of Lorentzian metrics, , , with any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connectedness and multiplicity causal character of connecting geodesics. These results are independent of the possibility of a full integration of geodesic equations. Variational and geometrical techniques are applied systematically. In particular, we prove that the asymptotic behavior of H(x,u) with x at infinity determines many properties of geodesics. Essentially, a subquadratic growth of H ensures geodesic completeness and connectedness, while the critical situation appears when H(x,u) behaves in some direction as , as in the classical model of exact gravitational waves.     

Ergodic states in a non commutative ergodic theory

Using the Godement mean of positive-type functions over a groupG, we study -abelian systems { , } of aC*-algebra and a homomorphic mapping of a groupG into the homomorphism group of . Consideration of the Godement mean off(g)U _g withf a positive-type function overG andU a unitary representation ofG first yields a generalized mean-ergodic theorem. We then define the Godement mean off(g) ( _g (A)) withA and a covariant representation of the system { , } for which theG-invariant Hilbert space vectors are cyclic and study its properties, notably in relation with ergodic and weakly mixing states over . Finally we investigate the discrete spectrum of covariant representations of { , } (i.e. the direct sum of the finite-dimensional subrepresentations of the associated representations ofG).On leave of absence from Istituto di Fisica G. Marconi Piazzale delle Scienze 5 — Roma.     

Quantum-Logics-Valued Measure Convergence Theorem

In this paper, the following quantum-logic valued measure convergence theorem is proved: Let (L ₁, 0, 1) be a Boolean algebra, (L ₂, , , 0, 1) be a quantum logic and { _n : n N} be a sequence of s-bounded (L ₂, , , 0, 1)-valued measures which are defined on (L ₁, 0, 1). If for each a (L ₁, 0, 1), { _n (a)} _{n N} is an order topology Cauchy sequence, when {v(a)} convergent to 0, { _n (a)} is order topology convergent to 0 for each n N, where v is a nonnegative finite additive measure which is defined on (L ₁, 0, 1), then when {v(a)} convergent to 0, { _n (a)} are order topology convergent to 0 uniformly with respect to n N.     

Effective Potential for \mathcal{P}\mathcal{T}-Symmetric Quantum Field Theories

Recently, a class of -invariant scalar quantum field theories described by the non-Hermitian Lagrangian = () ² +g ² (i) was studied. It was found that there are two regions of . For <0 the -invariance of the Lagrangian is spontaneously broken, and as a consequence, all but the lowest-lying energy levels are complex. For 0 the -invariance of the Lagrangian is unbroken, and the entire energy spectrum is real and positive. The subtle transition at =0 is not well understood. In this paper we initiate an investigation of this transition by carrying out a detailed numerical study of the effective potential V _eff (_c) in zero-dimensional spacetime. Although this numerical work reveals some differences between the <0 and the >0 regimes, we cannot yet see convincing evidence of the transition at =0 in the structure of the effective potential for -symmetric quantum field theories.     

Local properties of equilibrium states and the particle spectrum in quantum field theory

It is shown that in a quantum field theory describing free, scalar particles with masses m _i , i there exist locally normal equilibrium states with finite energy density for all temperature >0 if and only if . This result lends support to the conjecture that the nuclearity criterion proposed by Buchholz and Wichmann is sensitive to the thermodynamical properties of field-theoretic models.     

Preferred Invariant Symplectic Connections on Compact Coadjoint Orbits

Let be the Haag--Kastler net generated by the (2) chiral current algebra at level 1. We classify the SL(2, )-covariant subsystems by showing that they are all fixed points nets ^H for some subgroup H of the gauge automorphisms group SO(3) of . Then, using the fact that the net ₁ generated by the (1) chiral current can be regarded as a subsystem of , we classify the subsystems of ₁. In this case, there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem .     

A\overline {Poincar\acute{e}} gauge theory of gravity

We develop a gauge theory of gravity on the basis of the principal fiber bundle over the four-dimensional space-timeM with the covering groupP¯ ₀ of the proper orthochronous Poincaré group. The field components are constructed with the connection coefficients , and with a Higgs-type field. A Lorentz metricg is introduced with , which are then identified with the components of duals of the Vierbein fields. Associated with there is a spinor structure onM. For Lagrangian densityL, which is a function of , ,, matter field , and oftheir first derivatives, we give the conditions imposed by the requirement of the gauge invariance. The Lagrangian densityL is restricted to be of the formL =L ^tot (, T ^klm ,R ^klmn , _k , ), in whichT ^klm ,R ^klmn are the field strengths of , , respectively. Identities and conservation laws following from the gauge invariance are given. Particularly noteworthy is the fact that the energy momentum conservation law follows from theinternal translational invariance. The field equation of is automatically satisfied, if those of and of are both satisfied. The possible existence of matter fields with intrinsic energy momentum is pointed out. When is a field with vanishing intrinsic energy momentum, the present theory practically agrees with the conventional Poincaré gauge theory of gravity, except for the seemingly trivial terms in the expression of the spin-angular momentum density. A condition leading to a Riemann-Cartan space-time is given. The field holds a key position in the formulation.     

The equation of motion and gravitational radiation of an ideal fluid sphere moving in a gravitational background

On the basis of an approximation method developed in a previous paper the motion of an ideal fluid sphere in a weak gravitational background is investigated. The sphere is assumed to be small in the sense that its radius is small compared with the change of the background . Furthermore the deformations of the sphere when accelerated by the background are assumed to be small compared with the extension of the sphere in the absence of acceleration. In the lowest mixed order (mixed of the background and the retarded potentials of the sphere in lowest order) the equation of motion is yielded by integrating Einstein's conservation law of energy and momentum over the world-tube of the sphere. One obtains an equation of motion for the center of the sphere that is identical with the geodesic line linearized in . In the case of a static background of a localized matter distribution it is shown that Einstein's energy-momentum complex formed with the retarded potentials from the accelerated motion of the sphere in lowest order (lowest mixed order) leads to an outgoing radiation of gravitational energy. All radiation terms can be expressed in terms of the background and the world-line of the center of the sphere.     

Time dependent gravitational potential in the universe and some consequences

In this paper, assuming a linear change of the gravitational potential V in the universe, i.e. , some consequences are obtained. 1. The Hubble red shift is explained by the potential difference between the considered galaxy long time ago and the observer at this epoch. 2. The anomalous acceleration a _P from the spacecraft Pioneer 10 and 11 [1] is explained. 3. The deformations of the trajectories of planets are studied. It is shown that the planetary orbits are not axially symmetric and the angle from the perihelion to the aphelion is , while the angle from the aphelion to the perihelion is , where is the orbital period. There is no perihelion precession caused by the time dependent gravitational potential V. The quotient of two consecutive orbital periods ₁ and ₂ is equal to This formula is tested for the pulsars B1885+09 and B1534+12, and the results are good.This revised version was published online in April 2005. The publishing date was inserted.     

Projectively Equivariant Symbol Calculus

The spaces of linear differential operators acting on -densities on and the space of functions on which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect (ⁿ) of vector fields of ⁿ. However, these modules are isomorphic as sl(n + 1,)-modules where is the Lie algebra of infinitesimal projective transformations. In addition, such an -equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the -equivariant symbol map to study the of kth-order linear differential operators acting on -densities, for an arbitrary manifold M and classify the quotient-modules .     

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.