Singularity-free Solutions Research Articles

Global singularity-free solution of the Iordanskii force problem

We present a derivation of the transverse force acting on a hydrodynamic vortex in the presence of an incoming sound wave from a global solution of the scattering problem, using the method of matched asymptotic expansions. The solution presented includes a detailed treatment of the interaction of the incident wave with the vortex core, and is free from the singularities in the momentum exchange between vortex and sound wave which have led to contradictory results for the value of the transverse force in the literature.

ON THE SUPERSTRING HAMILTONIAN IN THE FRIEDMANN SPACE–TIME

The ten-dimensional effective Lagrangian [Formula: see text] for the gravitational sector of the heterotic superstring theory is known up to quartic higher-derivative order [Formula: see text]. In cosmology, the reduced, four-dimensional line element assumes the Friedmann form ds2 = dt2 - a(t)2dx2, where t is comoving time and a(t) ≡ a0eα(t) is the radius function of the three-space dx2, whose curvature is k = 0, ± 1. The four-Lagrangian can then be expressed as the power-series [Formula: see text], where ˙ ≡ d/dt, from which the field equation can be derived by the method of Ostrogradsky. Here, we determine the coefficients Λ0, An, Bn, Cn, and Kn, which are all non-vanishing in general. We recover the previously obtained, high-curvature, anti-de Sitter vacuum state [Formula: see text] with effective cosmological constant Λ = {18/[175ζ(3) - 1/2]}1/3A r κ-2, whose existence makes it possible to envisage a singularity-free and horizon-free cosmological solution, stable to linear perturbations. It is interesting that all the coefficients of quartic origin arise from the near-cancellation of sums of opposite sign but magnitude f ≈ (28.6–369) times larger than the answer. They thus exhibit a slight asymmetry with regard to positive and negative energies, the anti-de Sitter vacuum being characterized by positive Nordström energy, and therefore only accessible at high curvatures. This vacuum state is a Bose–Einstein condensate of non-interacting gravitons at zero temperature, which, referred to comoving time, can only be formulated after the Wick rotation t → ±iτ, resulting in an imaginary horizon.

Singularities at the moving contact line. Mathematical, physical and computational aspects

The near-field dynamics in the moving contact-line problem described in the framework of several alternative theories is examined in the general case of finite capillary numbers against the most basic criteria to be satisfied by a mathematical model. It is shown that both types of so-called ‘slip models’ invariably fail the criteria: they lead to solutions either with a pressure singularity at the contact line or with the flow kinematics qualitatively different from that observed in experiments, or both. The same criteria applied to an early developed theory of dynamic wetting as an interface formation process are shown to be satisfied: the model leads to a singularity-free solution with the correct ‘rolling’ kinematics. The flow exhibits no spurious low-velocity region near the contact line which is unavoidable in all ‘slip models’. The pressure at the contact line remains finite, and the dynamic contact angle, being part of the solution, depends on the flow field and not only on the contact-line speed. This last feature accounts for the effect of ‘hydrodynamic assist of dynamic wetting’ reported in recent experiments. The implications of the results for numerical algorithms incorporating different models for the moving contact line are discussed.

Capillary breakup of liquid threads: a singularity-free solution

The process of capillary breakup of a thread of Newtonian liquid is considered theoretically in the simplest case where the thread is surrounded by an inviscid, dynamically passive gas. The goal is to remove the singularities inherent in the known solutions to the problem obtained in the framework of the standard model and explain some puzzling qualitative features of the process observed in experiments. The analysis is based on the idea that, since the known solutions indicate that the rate at which fresh free-surface area is created tends to infinity as breakup is approached, one has that the surface tension, whose relaxation to equilibrium is always associated with a small but finite relaxation time, is bound to deviate from its equilibrium value in the process of breakup. This gives rise to a surface-tension gradient which starts to pull the liquid thread apart (the flow-induced Marangoni effect), whilst the role of the capillary pressure-driven squeezing of the liquid out of the neck diminishes as the surface tension in the minimal cross-section decreases. An earlier developed theory incorporating the interface formation process is applied without any ad hoc alterations and analysed in the framework of the slender-jet approximation. The resulting solution is singularity-free and allows one to describe some previously unexplained features of experiment by Kowalewski (1996, Fluid Dyn. Res., 17, 121).

THE SPACE OF COSMOLOGICAL SPACE–TIMES

I first focus on how to best describe the space of cosmological space–times and what its essential properties are and some comments on the dynamical behavior revealed by studies that will be described in more detail by John Wainwright. I will then relate this both to observations and to anthropic issues (i.e. the possible existence of observers). This space includes some viable singularity free solutions which will be briefly described, thus posing the issue of the tension between very special initial conditions and the existence of initial singularities. I will conclude with remarks on the issue of realized infinities in this context and the concept of multiverses.

A singularity-free solution of Friedmann universe in the discrete spacetime

An idealized Friedmann cosmological model in the discrete spacetime is consist of collection of dust particles obtained in paper［1］ and proved to be singu larity-free.

Numerical Investigation of a Class of Nonlinear Schrödinger Equations

This paper numerically investigates the space-localized spherically symmetric, stationary, and singularity-free solutions of the Nonlinear Schrödinger equation when the nonlinearity is a step function. Previously no-node solutions have been obtained analytically. Here, it is shown that localized stationary solutions with one node and two nodes also exist.

Blown-up p-branes and the cosmological constant

We consider a blown-up 3-brane, with the resulting geometry R(3,1) × S(N − 1), in an infinite-volume bulk with N > 2 extradimensions. The action on the brane includes both an Einstein term anda cosmological constant. Similar set-ups have been proposed both toreproduce the 4D gravity on the brane, and to solve the cosmologicalconstant problem. Here we obtain a singularity-free solution toEinstein's equations everywhere in the bulk and on the brane, whichallows us to address these question explicitly. One finds, however,that the proper volume of S(N − 1) and the cosmological constant onthe brane have to be fine-tuned relative to each other, thus thecosmological constant problem is not solved. Moreover the scalarpropagator on the brane behaves four-dimensionally over aphenomenologically acceptable range only if the warp factor on thebrane is huge, which aggravates the weak scale–Planck scalehierarchy problem.

Regular cosmological bouncing solutions in low energy effective action from string theories

The possibility of obtaining singularity free cosmological solutions in four dimensional effective actions motivated by string theory is investigated. In these effective actions, in addition to the Einstein-Hilbert term, the dilatonic and the axionic fields are also considered as well as terms coming from the Ramond-Ramond sector. A radiation fluid is coupled to the field equations, which appears as a consequence of the Maxwellian terms in the Ramond-Ramond sector. Singularity free bouncing solutions in which the dilaton is finite and strictly positive are obtained for models with flat or negative curvature spatial sections when the dilatonic coupling constant is such that $\omega < - 3/2$, which may appear in the so called $F$ theory in 12 dimensions. These bouncing phases are smoothly connected to the radiation dominated expansion phase of the standard cosmological model, and the asymptotic pasts correspond to very large flat spacetimes.

No Go Theorem for Self Tuning Solutions With Gauss-Bonnet Terms

We consider self tuning solutions for a brane embedded in an anti de Sitter spacetime. We include the higher derivative Gauss-Bonnet terms in the action and study singularity free solutions with finite effective Newton's constant. Using the methods of Csaki et al, we prove that such solutions, when exist, always require a fine tuning among the brane parameters. We then present a new method of analysis in which the qualitative features of the solutions can be seen easily without obtaining the solutions explicitly. Also, the origin of the fine tuning is transparent in this method.

Singularity-Free Cosmological Solutions with Non-Rotating Perfect Fluids

The paper establishes the result that solutions of the type described in the title of the article are in essence only those that have been already presented in the literature provided the acceleration vector is hypersurface orthogonal. The procedure adopted in the paper is somewhat novel - while the usual practice is to display an exact solution and then to examine whether it is singularity free, the present paper discovers the conditions which a singularity free solution of the desired type must satisfy. There is no attempt to obtain exact solutions. Simply, the conditions that were ad-hoc introduced in the deduction of singularity free solutions are here shown to follow from the requirement of non-singularity.

A general solution for the position, velocity, and acceleration of hyperredundant planar manipulators

AbstractA new approach for the solution of the position, velocity, and acceleration of hyperredundant planar manipulators following any twice‐differentiable desired path is presented. The method is singularity free, and provides a robust solution even in the event of mechanical failure of some of the robot actuators. The approach is based on defining virtual layers, and dividing them into virtual/real three‐link or four‐link subrobots. It starts by solving the inverse kinematic problem for the subrobot located in the lowest virtual layer, which is then used to solve the inverse kinematic equations for the subrobots located in the upper virtual layers. An algorithm is developed that provides a singularity‐free solution up to the full extension through a configuration index. The configuration index can be interpreted as the average of the determinants of the Jacobians of the subrobots. The equations for the velocities and accelerations of the manipulator are solved by extending the same approach, and it is shown that the value of the configuration index is critical in maintaining joint velocity continuity. The inverse dynamic problem of the robot is also solved to obtain the torques required for the robot actuators to accomplish their tasks. Computer simulations of several hyperredundant manipulators using the proposed method are presented as numerical examples. © 2002 John Wiley &amp; Sons, Inc.

Spinor field in a Bianchi type-I universe: Regular solutions

Self-consistent solutions to the nonlinear spinor field equations in General Relativity has been studied for the case of Bianchi type-I (B-I) space-time. It has been shown that, for some special type of nonliearity the model provides regular solution, but this singularity-free solutions are attained at the cost of broken dominant energy condition in Hawking-Penrose theorem. It has also been shown that the introduction of $\Lambda$-term in the Lagrangian generates oscillations of the B-I model, which is not the case in absence of $\Lambda$ term. Moreover, for the linear spinor field, the $\Lambda$ term provides oscillatory solutions, those are regular everywhere, without violating dominant energy condition. Key words: Nonlinear spinor field (NLSF), Bianch type -I model (B-I), $\Lambda$ term PACS 98.80.C Cosmology

Classical analogous of quantum cosmological perfect fluid models

Quantization in the minisuperspace of a gravity system coupled to a perfect fluid, leads to a solvable model which implies singularity free solutions through the construction of a superposition of the wavefunctions. We show that such models are equivalent to a classical system where, besides the perfect fluid, a repulsive fluid with an equation of state p Q = ρ Q is present. This leads to speculate on the true nature of this quantization procedure. A perturbative analysis of the classical system reveals the condition for the stability of the classical system in terms of the existence of an anti-gravity phase.

Geons and (4+1) Gravity

Gravitational-electromagnetic entities “geons” are singularity-free solutions of the Einstein-Maxwell equations. These structures in cylindrical symmetry are considered here through the noncompactified Kaluza-Klein theory which describes geometrically the gravitation field and its sources.

Einstein-Langevin and Einstein-Fokker-Planck equations for Oppenheimer-Snyder gravitational collapse in a spacetime with conformal vacuum fluctuations

A consistent extension of the Oppenheimer-Snyder gravitational collapse formalism is presented which incorporates stochastic, conformal, vacuum fluctuations of the metric tensor. This results in a tractable approach to studying the possible effects of vacuum fluctuations on collapse and singularity formation. The motivation here, is that it is known that coupling stochastic noise to a classical field theory can lead to workable methodologies that accommodate or reproduce many aspects of quantum theory, turbulence or structure formation. The effect of statistically averaging over the metric fluctuations gives the appearance of a deterministic Riemannian structure, with an induced non-vanishing cosmological constant arising from the nonlinearity. The Oppenheimer-Snyder collapse of a perfect fluid or dust star in the fluctuating or `turbulent' spacetime, is reformulated in terms of nonlinear Einstein-Langevin field equations, with an additional noise source in the energy-momentum tensor. The smooth deterministic worldlines of collapsing matter within the classical Oppenheimer-Snyder model, now become nonlinear Brownian motions due to the backreaction induced by vacuum fluctuations. As the star collapses, the matter worldlines become increasingly randomized since the backreaction coupling to the vacuum fluctuations is nonlinear; the input assumptions of the Hawking-Penrose singularity theorems should then be violated. Solving the nonlinear Einstein-Langevin field equation for collapse - via the Ito interpretation - gives a singularity-free solution, which is equivalent to the original Oppenheimer solution but with higher-order stochastic corrections; the original singular solution is recovered in the limit of zero vacuum fluctuations. The `geometro-hydrodynamics' of noisy gravitational collapse, were also translated into an equivalent mathematical formulation in terms of nonlinear Einstein-Fokker-Planck (EFP) continuity equations with respect to comoving coordinates: these describe the collapse as a conserved flow of probability. A solution was found in the dilute limit of weak fluctuations where the EFP equation is linearized. There is zero probability that the star collapses to a singular state in the presence of background vacuum fluctuations, but the singularity returns with unit probability when the fluctuations are reduced to zero. Finally, an EFP equation was considered with respect to standard exterior coordinates. Using the thermal Brownian motion paradigm, an exact stationary or equilibrium solution was found in the infinite standard time relaxation limit. The solution gives the conditions required for the final collapsed object (a black hole) to be in thermal equilibrium with the background vacuum fluctuations. From this solution, one recovers the Hawking temperature without using field theory. The stationary solution then seems to correspond to a black hole in thermal equilibrium with a fluctuating conformal scalar field; or the Hawking-Hartle state.

Singularity free dilaton-driven cosmologies and pre-little-bangs

There are no reasons why the singularity in the growth of the dilaton coupling should not be regularised, in a string cosmological context, by the presence of classical inhomogeneities. We discuss a class of inhomogeneous dilaton-driven models whose curvature invariants are all bounded and regular in time and space. We prove that the non-space-like geodesics of these models are all complete in the sense that none of them reaches infinity for a finite value of the affine parameter. We conclude that our examples represent truly singularity-free solutions of the low energy beta functions. We discuss some symmetries of the obtained solutions and we clarify their physical interpretation. We also give examples of solutions with spherical symmetry. In our scenario each physical quantity is everywhere defined in time and space, the big-bang singularity is replaced by a maximal curvature phase where the dilaton kinetic energy reaches its maximum. The maximal curvature is always smaller than one (in string units) and the coupling constant is also smaller than one and it grows between two regimes of constant dilaton, implying, together with the symmetries of the solutions, that higher genus and higher curvature corrections are negligible. We argue that our examples describe, in a string cosmological context, the occurrence of ``little bangs''(i.e. high curvature phases which never develop physical singularities). They also suggest the possibility of an unexplored ``pre-little-bang'' phase.

Quantum fluctuations in radiation dominated anisotropic cosmology

Using the metric conformal transformation and simple path integral, Feynman propagator method, for computing its quantum fluctuations, we analyse the radiation dominated anisotropic Bianchi Type I cosmology. We proceed to show that the quantum conformal fluctuations diverge at the classical spacetime singularity, suggesting that a singularity free solution can exist in anisotropic cosmology in the quantum regime.

The Acoustic Analogy and the Prediction of the Noise of Rotating Blades

The acoustic analogy was introduced into acoustics by Lighthill in 1952 to understand and predict the noise generated by the jet of an aircraft turbojet engine. The idea behind the acoustic analogy is simple but powerful. The entire noise generation process is mathematically reduced to the study of wave propagation in a quiescent medium with the effect of flow replaced by quadrupole sources. In jet noise theory, Lighthill was able to obtain significant and useful qualitative results from the acoustic analogy. The acoustic analogy has influenced the theoretical and experimental research on jet noise since the early 1950s. This paper, however, focuses on another area in which the acoustic analogy has had a significant impact, namely, the prediction of the noise of rotating machinery. The governing equation for this problem was derived by Ffowcs Williams and Hawkings in 1969. This equation is a wave equation for perturbation density with three source terms, which have become known as thickness, loading, and the quadrupole source terms, respectively. The Ffowcs Williams–Hawkings (FW–H) equation has been used for the successful prediction of the noise of helicopter rotors, propellers, and fans. Several reasons account for the success and popularity of the acoustic analogy. First, the problems of acoustics and aerodynamics are separated. Second, because the FW–H equation is linear, powerful analytical methods from linear operator theory can be used to obtain closed-form solutions. Third, advances in digital computers and computational fluid dynamics algorithms have resulted in high-resolution near-field aerodynamic calculations that are suitable for noise prediction. We present some of the mathematical results for noise prediction based on the FW–H equation, including examples for helicopter rotors. In particular, we discuss the prediction of blade-vortex interaction noise and high-speed impulsive noise of helicopter rotors. For high-speed propellers, we briefly discuss the derivation of a singularity-free solution of the FW–H equation for a supersonic panel on a blade.

Development of virtual link method for the solution of hyper-redundant spatial robots

This article presents a new method for the kinematics of hyper-redundant spatial robots, called the virtual link method. The method is based on dividing the robot into sub-systems consisting of three or four links. Virtual links are established between the first and the last nodes of each sub-system. A hypothetical robot with reduced of degrees of freedom is developed using the virtual links. The method then solves for the kinematics of the virtual link system followed by the sub-systems. The virtual link method also provides a singularity-free solution by predicting their occurrence and taking appropriate action. An approaching singularity is reflected by a drop in the determinant of the Jacobian of the system below a threshold value. In this situation, the method either chooses a different virtual link configuration or switches to a displacement distribution scheme. © 1996 John Wiley & Sons, Inc.