Original Field Equations Research Articles

Weyl–invariant scalar–tensor gravities from purely metric theories

We describe a method to generate scalar–tensor theories with Weyl symmetry, starting from arbitrary purely metric higher derivative gravity theories. The method consists in the definition of a conformally-invariant metric g^μν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{g}_{\\mu \ u }$$\\end{document}, that is a rank (0,2)-tensor constructed out of the metric tensor and the scalar field. This new object has zero conformal weight and is given by ϕ2/Δgμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^{2/\\Delta }g_{\\mu \ u }$$\\end{document}, where (-Δ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-\\Delta )$$\\end{document} is the conformal dimension of the scalar. As gμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_{\\mu \ u }$$\\end{document} has conformal dimension of 2, the resulting tensor is trivially a conformal invariant. Then, the generated scalar–tensor theory, which we call the Weyl uplift of the original purely metric theory, is obtained by replacing the metric by g^μν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{g}_{\\mu \ u }$$\\end{document} in the action that defines the original theory. This prescription allowed us to define the Weyl uplift of theories with terms of higher order in the Riemannian curvature. Furthermore, the prescription for scalar–tensor theories coming from terms that have explicit covariant derivatives in the Lagrangian is discussed. The same mechanism can also be used for the derivation of the equations of motion of the scalar–tensor theory from the original field equations in the Einstein frame. Applying this method of Weyl uplift allowed us to reproduce the known result for the conformal scalar coupling to Lovelock gravity and to derive that of Einsteinian cubic gravity. Finally, we show that the cancellation of the volume divergences in the theory given by the conformal scalar coupling to Einstein–Anti-de Sitter gravity is achieved by the Weyl uplift of the original theory augmented by counterterms, which is relevant in the framework of conformalrenormalization.

New Frontiers of Human Wisdom: Information, Communication and Consciousness Empowered Decision-Making under the Broader Realm of Reality

History of human creative and innovative pursuits is, at times, highlighted with instances of some most intellectually accomplished minds’ biggest regrets about their own intuitive wisdom and scientific discoveries ultimately proving to be unwarranted during some future course of modern observations and scientific developments. Most notable of all is Albert Einstein’s self-admitted, suppressing his own intuitive conviction, the so-called ‘Greatest blunder ever made in life’ in 1917 about his failure to accept the validation of his own equations of the ‘cosmological constant’ (usually denoted by the Greek capital letter lambda: Λ) in his original field equations of general relativity, which ultimately turned out to be in fact quite consistent with current observations in view of the fundamental role that this cosmological constant now appears to be playing in the modern cosmology and closely associated to the concepts of dark energy and quintessence. This instance reinforces the very deep-rooted, fundamental, and the pioneering role played by intuitive consciousness as a guiding instinct or force in a majority of pure creative enterprises and pursuits that humans have been engaged with since time immemorial. In this context, the present chapter including sub-chapters aims at presenting a dossier of concepts and applications of Information, Communication, and Consciousness Science going beyond conventional human and materialistic endeavors of present-day society. It entails and invokes a systemic holistic perspective towards resolving many issues and challenges that the global society is currently up fronted with. A systems perspective of cognitive consciousness holds that cognition can be explained by means of a continuous dynamic system in which all the elements are interrelated. The domain or system of cognitive consciousness can be hypothesized to be one such system where numbers of endogenous and exogenous variables have dynamic interplay impacting the physiology, minds, and meditations of individuals and Consciousness practitioners. Hence, understanding the non-linearity of this form of evolutionary Cognitive-Physical-Psychological dynamic system should be paramount. In the same manner, Quantum consciousness can be understood as a Neuronal-Spiritual phenomenon that is inseparably associated and in communion with the dynamic Genetic-physical existence of a human being. Probing further deep also reveals a unique phenomenon of Super causality as a sum total of all creational causalities and retro-causalities. It can be said that Complex dynamical systems are neither completely rigid nor fully random; instead, they display a unique balance of ‘integration’, cohesion, and robustness at the global level and at the same time, differentiation and multiple realizability at the component levels. Hence, in the domain of consciousness too, we see causation, in which upstream determinants at the higher-order consciousness (Spiritual) levels influence and regulate events at lower levels of human cognitive consciousness. It was only as a result of cutting edge physics, particularly Quantum Physics, the very fundamentals of all existence (perceived) so far that has confirmed again and again that Consciousness is in fact the very ground of Being. There is nothing else apart from Consciousness, from which arises mind as does matter. Physicists, of late, have also come to realize that the universe is interconnected in much subtler ways than had once thought. In quantum physics, the observer and the observed can no longer be separated and the whole is more fundamental than the part. According to this view, consciousness imposes downward causation. In other words, our free will is real as intention.

Third order extensions of 3d Chern–Simons interacting to gravity: Hamiltonian formalism and stability

We consider inclusion of interactions between 3d Einstein gravity and the third order extensions of Chern–Simons. Once the gravity is minimally included into the third order vector field equations, the theory is shown to admit a two-parameter series of symmetric tensors with on-shell vanishing covariant divergence. The canonical energy–momentum is included into the series. For a certain range of the model parameters, the series include the tensors that meet the weak energy condition, while the canonical energy is unbounded in all the instances. Because of the on-shell vanishing covariant divergence, any of these tensors can be considered as an appropriate candidate for the right hand side of Einstein's equations. If the source differs from the canonical energy momentum, the coupling is non-Lagrangian while the interaction remains consistent with any of the tensors. We reformulate these not necessarily Lagrangian third order equations in the first order formalism which is covariant in the sense of 1+2 decomposition. After that, we find the Poisson bracket such that the first order equations are Hamiltonian in all the instances, be the original third order equations Lagrangian or not. The brackets differ from canonical ones in the matter sector, while the gravity admits the usual PB's in terms of ADM variables. The Hamiltonian constraints generate lapse, shift and gauge transformations of the vector field with respect to these Poisson brackets. The Hamiltonian constraint, being the lapse generator, is interpreted as strongly conserved energy. The matter contribution to the Hamiltonian constraint corresponds to 00-component of the tensor included as a source in the right hand side of Einstein equations. Once the 00-component of the tensor is bounded, the theory meets the usual sufficient condition of classical stability, while the original field equations are of the third order.

Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes

We present weighted covariant derivatives and wave operators for perturbations of certain algebraically special Einstein spacetimes in arbitrary dimensions, under which the Teukolsky and related equations become weighted wave equations. We show that the higher dimensional generalization of the principal null directions are weighted conformal Killing vectors with respect to the modified covariant derivative. We also introduce a modified Laplace–de Rham-like operator acting on tensor-valued differential forms, and show that the wave-like equations are, at the linear level, appropriate projections off shell of this operator acting on the curvature tensor; the projection tensors being made out of weighted conformal Killing–Yano tensors. We give off shell operator identities that map the Einstein and Maxwell equations into weighted scalar equations, and using adjoint operators we construct solutions of the original field equations in a compact form from solutions of the wave-like equations. We study the extreme and zero boost weight cases; extreme boost corresponding to perturbations of Kundt spacetimes (which includes near horizon geometries of extreme black holes), and zero boost to static black holes in arbitrary dimensions. In 4D our results apply to Einstein spacetimes of Petrov type D and make use of weighted Killing spinors.

On the weak‐field approximation of nonlinear vacuum electrodynamics of the Plebański class

Plebański's class of nonlinear vacuum electrodynamics is considered, which is for several reasons of interest at the present time. In particular, the question is answered under which circumstances Maxwell's original field equations are recovered approximately and which ‘post‐Maxwellian’ effects could arise. To this end, a weak field approximation method is developed, allowing to calculate ‘post‐Maxwellian’ corrections up to Nth order. In some respect, this is analogue of determining ‘post‐Newtonian’ corrections from relativistic mechanics by a low velocity approximation. As a result, we got a series of linear field equations that can be solved order by order. In this context, the solutions of the lower orders occur as source terms inside the higher order field equations and represent a ‘post‐Maxwellian’ self‐interaction of the electromagnetic field, which increases order by order. It becomes apparent that one has to distinguish between problems with and without external source terms because without sources also high frequency solutions can be approximately described by Maxwell's original equations. The higher order approximations, which describe ‘post‐Maxwellian’ effects, can give rise to experimental tests of Plebańksi's class. Finally, two boundary value problems are discussed to have examples at hand.

Symmetry operators and decoupled equations for linear fields on black hole spacetimes

In the class of vacuum Petrov type D spacetimes with cosmological constant, which includes the Kerr-(A)dS black hole as a particular case, we find a set of four-dimensional operators that, when composed off shell with the Dirac, Maxwell and linearized gravity equations, give a system of equations for spin weighted scalars associated with the linear fields, that decouple on shell. Using these operator relations we give compact reconstruction formulae for solutions of the original spinor and tensor field equations in terms of solutions of the decoupled scalar equations. We also analyze the role of Killing spinors and Killing–Yano tensors in the spin weight zero equations and, in the case of spherical symmetry, we compare our four-dimensional formulation with the standard 2 + 2 decomposition and particularize to the Schwarzschild-(A)dS black hole. Our results uncover a pattern that generalizes a number of previous results on Teukolsky-like equations and Debye potentials for higher spin fields.

An analytical study on the instability phenomena during the phase transitions in a thin strip under uniaxial tension

In the experiments on stress-induced phase transitions in SMA strips, several interesting instability phenomena have been observed, including a necking-type instability (associated with the stress drop), a shear-type instability (associated with the inclination of the transformation front) and an orientation instability (associated with the switch of the inclination angle). In order to shed more lights on these phenomena, in this paper we conduct an analytical study. We consider the problem in a three-dimensional setting, which implies that one needs to study the difficult problem of solution bifurcations of high-dimensional nonlinear partial differential equations. By using the smallness of the maximum strain, the thickness and width of the strip, we use a methodology, which combines series expansions and asymptotic expansions, to derive the asymptotic normal form equations, which can yield the leading-order behavior of the original three-dimensional field equations. An important feature of the second normal form equation is that it contains a turning point for the localization (necking) solution of the first equation. It is the presence of such a turning point which causes the inclination of the phase transformation front. The WKB method is used to construct the asymptotic solutions, which can capture the shear instability and the orientation instability successfully. Our analytical results reveal that the inclination of the transformation front is a phenomenon of localization-induced buckling (or phase-transition-induced buckling as the localization is caused by the phase transition). Due to the similarities between the development of the Luders band in a mild steel and the stress-induced transformations in a SMA, the present results give a strong analytical evidence that the former is also caused by macroscopic effects instead of microscopic effects. Our analytical results also reveal more explicitly the important roles played by the geometrical parameters.

Can self-organisation emerge through dynamic neural fields computation?

In this paper, dynamic neural fields (DNFs) are used to develop key features of a cortically-inspired computational module. Under the perspective of designing computational systems that can exhibit the flexibility and genericity of the cortical substrate, using neural field as the competition layer for self-organising modules has to be considered. However, despite the fact that they serve as a biologically-inspired model, applying DNFs to drive self-organisation is not straightforward. In order to address that issue, an original method for evaluating neural field equations is proposed, based on statistical measurements of the field behaviour in some scenarios. Limitations of classical neural field equations are then quantified, and an original field equation is proposed to overcome these difficulties. The performance of the proposed field model is discussed in comparison with some previously considered models, leading to the promotion of the proposed model as a suitable mean for processing competition in cortex-like computation for cognitive systems.

Bifurcation to a corner-like formation in a slender nonlinearly elastic cylinder: asymptotic solution and mechanism

In this paper, we study the corner-like formation in a slender nonlinearly elastic cylinder due to compression. Mathematically, this is a very challenging problem: one needs to study the bifurcation of the nonlinear field equations (complicated nonlinear partial differential equations) and show that there is a bifurcation leading to a solution with a corner-like profile. We also aim to obtain the asymptotic expression for this post-bifurcation solution. As far as we know, there is no available analytical method to obtain the post-bifurcation solution of nonlinear partial differential equation(s). Here we use a novel approach to tackle the present problem. Through a method of coupled series–asymptotic expansions, we manage to derive the normal form equation of the original nonlinear field equations, which can be written as a singular ordinary differential equation (ODE) system (the vector field has a singularity at one point). With welding end conditions, the problem is reduced to study the boundary-value problem of a singular ODE system. It seems that such a boundary-value problem of a singular ODE system was not formulated or studied before in the context of the present problem. With the help of phase planes, we manage to solve such a boundary-value problem. It turns out that there is a vertical singular line in phase planes, which causes the bifurcation to a corner-like solution. The expression for this post-bifurcation solution is also obtained. From the analytical results obtained, we reveal that the coupling effect of the material nonlinearity and geometrical size is the physical mechanism that causes the formation of a corner-like profile.

Low energy effective gravitational equations on a Gauss-Bonnet brane

We present effective gravitational equations at low energies in a ${Z}_{2}$-symmetric braneworld with the Gauss-Bonnet term. Our derivation is based on the geometrical projection approach, and we solve iteratively the bulk geometry using the gradient expansion scheme. Although the original field equations are quite complicated due to the presence of the Gauss-Bonnet term, our final result clearly has the form of the Einstein equations plus correction terms, which is simple enough to handle. As an application, we consider homogeneous and isotropic cosmology on the brane. We also comment on the holographic interpretation of bulk gravity in the Gauss-Bonnet braneworld.

Nodally exact Ritz discretizations of 1D diffusion–absorption and Helmholtz equations by variational FIC and modified equation methods

This article presents the first application of the Finite Calculus (FIC) in a Ritz-FEM variational framework. FIC provides a steplength parametrization of mesh dimensions, which is used to modify the shape functions. This approach is applied to the FEM discretization of the steady-state, one-dimensional, diffusion–absorption and Helmholtz equations. Parametrized linear shape functions are directly inserted into a FIC functional. The resulting Ritz-FIC equations are symmetric and carry a element-level free parameter coming from the function modification process. Both constant- and variable-coefficient cases are studied. It is shown that the parameter can be used to produce nodally exact solutions for the constant coefficient case. The optimal value is found by matching the finite-order modified differential equation (FOMoDE) of the Ritz-FIC equations with the original field equation. The inclusion of the Ritz-FIC models in the context of templates is examined. This inclusion shows that there is an infinite number of nodally exact models for the constant coefficient case. The ingredients of these methods (FIC, Ritz, MoDE and templates) can be extended to multiple dimensions

Integrability of Generalized (Matrix) Ernst Equations in String Theory

The integrability structures of the matrix generalizations of the Ernst equation for Hermitian or complex symmetric $d\times d$-matrix Ernst potentials are elucidated. These equations arise in the string theory as the equations of motion for a truncated bosonic parts of the low-energy effective action respectively for a dilaton and $d\times d$ - matrix of moduli fields or for a string gravity model with a scalar (dilaton) field, U(1) gauge vector field and an antisymmetric 3-form field, all depending on two space-time coordinates only. We construct the corresponding spectral problems based on the overdetermined $2d\times 2d$-linear systems with a spectral parameter and the universal (i.e. solution independent) structures of the canonical Jordan forms of their matrix coefficients. The additionally imposed conditions of existence for each of these systems of two matrix integrals with appropriate symmetries provide a specific (coset) structures of the related matrix variables. An equivalence of these spectral problems to the original field equations is proved and some approach for construction of multiparametric families of their solutions is envisaged.

Invariant solutions of a supersymmetric fluid model

In this paper, we present a method for constructing invariant solutions of the supersymmetric Chaplygin gas in (1 + 1) dimensions. This approach is based on the use of a generalized Legendre transformation, through which we transform the original field equations into a new set of equations involving the velocity and sound speed of the fluid as independent variables. We describe the Lie symmetry properties of the equations in both coordinate systems, and make use of a systematic subgroup classification to determine certain classes of group-invariant solutions of the transformed field equations. Where it is possible, the Legendre transformation is applied in reverse in order to obtain equivalent solutions of the standard field equations. A number of analytic solutions of the supersymmetric Chaplygin gas in one spatial dimension are found. In addition, certain basic elements of a possible extension of our method to the supersymmetric Chaplygin gas in two spatial dimensions have been formulated. In particular, the (2 + 1)-dimensional analogue of the transformed equation has been determined, and some of its Lie point symmetries have been identified. For a certain specific case, it has been demonstrated how an invariant solution can be inverted and then extended to a solution of the planar supersymmetric model.

Study of a system for the isothermal separation of components in a binary alloy with change of phase

An isothermal model describing the separation of the components of a binary metallic alloy is considered. A process of phase transition is also assumed to occur in the solder; hence, the state of the material is described by two order parameters, i.e. the concentration c of the first component and the phase field ϕ. A physical derivation is provided starting from energy balance considerations. The resulting system of PDEs consists of a rather regular second-order parabolic equation for ϕ coupled with a fourth-order relation of Cahn-Hilliard type for c with constraint and solution-dependent mobility. Global existence of solutions is proved and several regularity properties are discussed under more restrictive assumptions on the physical parameters. Continuous dependence on data is shown in a special case. An asymptotic analysis of the model is also performed, yielding at the limit step a coupling of the original phase field equation with a Stefan-like system for c.

Relativistic Schrödinger Theory and the Hartree–Fock Approach

Within the framework of Relativistic Schrodinger Theory (RST), the scalar two-particle systems with electromagnetic interactions are treated on the basis of a non-Abelian gauge group U(2) which is broken down to the Abelian subgroup U(1)×U(1). In order that the RST dynamics be consistent with the (non-Abelian) Maxwell equations, there arises a compatibility condition which yields cross relationships for the links between the field strengths and currents of both particles such that self-interactions are eliminated. In the non-relativistic limit, the RST dynamics becomes identical to the well-known Hartree–Fock equations (for spinless particles). Consequently the original RST field equations may be considered as the relativistic generalization of the Hartree–Fock equations, and the “exchange interactions” of the conventional theory (induced by the anti-symmetrization postulate) do reappear here as ordinary gauge interactions due to a broken symmetry.

DiscussionDesign method for bolted connections loaded perpendicular-to-grain: Discussion

This paper and recent papers by the same authors (Quenneville and Mohammad 2000; Mohammad and Quenneville 2001) present a tremendous amount of valuable laboratory data on the subject of bolted connection behaviour in timber. The observations of behaviour and failure mechanisms are particularly relevant. The authors present new design equations, based on failure mechanisms, to predict connection strength — an improvement on the current code requirements found in O86.1 (CSA 1994). These new equations were shown to produce improved results in the paper’s figures; however, the equations lack a detailed analytical basis. All bolted connections, regardless of the material, will develop a three-dimensional stress field — the magnitudes of the stresses depend on the connection geometry and material properties. The ductility in the steel connector also plays a role. In the wood member, the stress distribution is complicated by material orthotropy and nonlinear behaviour of wood in compression. The connection behaviour is further complicated by brittle failure of wood in tension and in shear. The authors provide between three and six equations to predict the connection strength for different connection con figurations loaded perpendicular-to-grain, when strength is governed by bearing. They provide only one additional equation to predict connection strength when splitting governs. The authors later conclude that, based on their tests, tension perpendicular-to-grain splitting was found to govern connection strength in most cases. From their conclusion, one would expect that their proposed equation for the prediction of brittle failures would have been developed using a rigorous analysis of the stresses in a bolted connection, rather than curve-fitting to an expression originally devel oped to predict the strength of timber rivets. Foschi and Longworth (1975) used a combination of material nonlinearity and the weakest link theory to predict the strength of timber rivets undergoing brittle failure. This work forms the basis of the current O86.1 rivet design procedure. Their one-dimensional finite element model predicted stresses around each rivet over a series of load steps. At each step, the stresses were analyzed to determine if the tension or shear stresses exceeded the allowable stresses according to the Weibull weakest link theory (Weibull 1939). The model also pinpointed the governing mode of failure (i.e., tension perpendicular-to-grain or shear). This model was specifically developed for timber rivets — the boundary conditions for the governing displacement field equations were chosen specifically for the rivet connection analyzed at that time. Quenneville and Mohammad modified Foschi’s (1973) governing equation but gave no direct indication that the original field equations can be applied to the design of multiple-bolt connections. There are a number of reasons to believe that recalibrating

Kinematic reduction of reaction-diffusion fronts with multiplicative noise: derivation of stochastic sharp-interface equations.

We study the dynamics of generic reaction-diffusion fronts, including pulses and chemical waves, in the presence of multiplicative noise. We discuss the connection between the reaction-diffusion Langevin-like field equations and the kinematic (eikonal) description in terms of a stochastic moving-boundary or sharp-interface approximation. We find that the effective noise is additive and we relate its strength to the noise parameters in the original field equations, to first order in noise strength, but including a partial resummation to all orders which captures the singular dependence on the microscopic cutoff associated with the spatial correlation of the noise. This dependence is essential for a quantitative and qualitative understanding of fluctuating fronts, affecting both scaling properties and nonuniversal quantities. Our results predict phenomena such as the shift of the transition point between the pushed and pulled regimes of front propagation, in terms of the noise parameters, and the corresponding transition to a non-Kardar-Parisi-Zhang universality class. We assess the quantitative validity of the results in several examples including equilibrium fluctuations and kinetic roughening. We also predict and observe a noise-induced pushed-pulled transition. The analytical predictions are successfully tested against rigorous results and show excellent agreement with numerical simulations of reaction-diffusion field equations with multiplicative noise.

Optimal design method for microwave device using time domain method and design sensitivity analysis. I. FETD case

In this paper, a new optimal shape design method for microwave passive devices for better performances in a broadband is proposed, which is based on vector finite element time domain (FETD) method and design sensitivity analysis (DSA). The design sensitivity is evaluated using an adjoint variable method. While the original field equation is an initial-value problem, the adjoint equation is an terminal-value problem. To transform the terminal-value problem into a initial-value problem, a backward time scheme is used. To verity its usefulness, a two-dimensional E-plane waveguide T-junction is analyzed.

Transverse patterns in nondegenerate intracavity second-harmonic generation

Pattern formation is theoretically investigated in the process of second-harmonic generation in an externally driven plane-plane optical cavity which uses phase matching of type II. It is shown that in this system a symmetry-breaking polarization instability at the fundamental frequency can lead to the spontaneous appearance of self-organized states, which can be periodic or quasiperiodic in space. A weakly nonlinear analysis in the vicinity of the bifurcation point where these patterns emerge is developed by deriving both a complex order parameter equation of the Swift-Hohenberg type close to resonance, and a set of amplitude equations for the neutral modes far from resonance. The amplitude equations reveal the existence of multistability and pattern coexistence, and show that, besides the most common periodic patterns such as rolls, squares, and hexagons, quasipatterns with an arbitrary orientational order can be selected by the nonlinearity in this system. Numerical simulations of the original field equations are presented and the main predictions of the weakly nonlinear analysis are compared with the numerical results.

On a Stationary Cosmology in the Sense of Einstein's Theory of Gravitation

A world is to be considered stationary in the sense of general relativity if the coefficients of its metric are independent of time in a coordinate system in which the masses are at rest on average. The remark on the system of coordinates is important because time itself is no invariant notion but is taken only in the sense of proper time. Our definition is unique, in the form given above. On the other hand it is also possible to have points where no matter is present. At such points we may place a test body of infinitesimally small mass and analyse whether it remains at rest in our coordinate system. A necessary and sufficient condition for this is that the time lines of our coordinate system are geodesics. Therefore the static solution given by de Sitter is not an example of a stationary world. The Schwarzschild line element which, from a cosmological point of view, is a world with a single central body can also not be considered a stationary solution. Indeed, there are no stationary solutions which are also spherically symmetric for the original field equations. The only such solution for the cosmological equations is Einstein's cylinder world. It is, to my knowledge, the only stationary world known so far. In that case the average matter density and the total mass of the world has to have a well defined value given by the cosmological constant which doubtless would be purely coincidental and is thus not a satisfactory assumption. In the following we shall discuss a new solution which is in accord with the original field equations without the need of an a priori relation between mass and cosmological constant. However, we shall find that its mass cannot be less than the mass of the cylinder world.