Nonlinear Gravity Research Articles

Cosmology ofF(R)nonlinear massive gravity

The theory of nonlinear massive gravity can be extended into the F(R) form as developed in Phys.Rev.D90, 064051 (2014). Being free of the Boulware-Deser ghost, such a construction has the additional advantage of exhibiting no linear instabilities around a cosmological background. We investigate various cosmological evolutions of a universe governed by this generalized massive gravitational theory. Specifically, under the Starobinsky ansantz, this model provides a unified description of the cosmological history, from early-time inflation to late-time self-acceleration. Moreover, under viable F(R) forms, the scenario leads to a very interesting dark-energy phenomenology, including the realization of the quintom scenario without any pathology. Finally, we provide a detailed analysis of the cosmological perturbations at linear order, as well as the Hamiltonian constraint analysis, in order to examine the physical degrees of freedom.

Cosmological evolutions ofF(R)nonlinear massive gravity

Recently a new extended nonlinear massive gravity model has been proposed which includes the $F(R)$ modifications to dRGT model.We follow the $F(R)$ nonlinear massive gravity and study its implications on cosmological evolutions. We derive the critical points of the cosmic system and study the corresponding kinetics by performing the phase-plane analysis.

Numerical modeling of 3D fully nonlinear potential periodic waves

A simple and exact numerical scheme for long-term simulations of 3D potential fully nonlinear periodic gravity waves is suggested. The scheme is based on the surface-following nonorthogonal curvilinear coordinate system. Velocity potential is represented as a sum of analytical and nonlinear components. The Poisson equation for the nonlinear component of velocity potential is solved iteratively. Fourier transform method, the second-order accuracy approximation of vertical derivatives on a stretched vertical grid and the fourth-order Runge–Kutta time stepping are used. The scheme is validated by simulation of steep Stokes waves. A one-processor version of the model for PC allows us to simulate evolution of a wave field with thousands degrees of freedom for hundreds of wave periods. The scheme is designed for investigation of nonlinear 2D surface waves, generation of extreme waves, and direct calculations of nonlinear interactions.

Variational space–time (dis)continuous Galerkin method for nonlinear free surface water waves

A new variational finite element method is developed for nonlinear free surface gravity water waves using the potential flow approximation. This method also handles waves generated by a wave maker. Its formulation stems from Miles' variational principle for water waves together with a finite element discretization that is continuous in space and discontinuous in time. One novel feature of this variational finite element approach is that the free surface evolution is variationally dependent on the mesh deformation vis-à-vis the mesh deformation being geometrically dependent on free surface evolution. Another key feature is the use of a variational (dis)continuous Galerkin finite element discretization in time. Moreover, in the absence of a wave maker, it is shown to be equivalent to the second order symplectic Störmer–Verlet time stepping scheme for the free-surface degrees of freedom. These key features add to the stability of the numerical method. Finally, the resulting numerical scheme is verified against nonlinear analytical solutions with long time simulations and validated against experimental measurements of driven wave solutions in a wave basin of the Maritime Research Institute Netherlands.

Three Least-Squares Minimization Approaches to Interpret Gravity Data Due to Dipping Faults

We have developed three different least-squares minimization approaches to determine, successively, the depth, dip angle, and amplitude coefficient related to the thickness and density contrast of a buried dipping fault from first moving average residual gravity anomalies. By defining the zero-anomaly distance and the anomaly value at the origin of the moving average residual profile, the problem of depth determination is transformed into a constrained nonlinear gravity inversion. After estimating the depth of the fault, the dip angle is estimated by solving a nonlinear inverse problem. Finally, after estimating the depth and dip angle, the amplitude coefficient is determined using a linear equation. This method can be applied to residuals as well as to measured gravity data because it uses the moving average residual gravity anomalies to estimate the model parameters of the faulted structure. The proposed method was tested on noise-corrupted synthetic and real gravity data. In the case of the synthetic data, good results are obtained when errors are given in the zero-anomaly distance and the anomaly value at the origin, and even when the origin is determined approximately. In the case of practical data (Bouguer anomaly over Gazal fault, south Aswan, Egypt), the fault parameters obtained are in good agreement with the actual ones and with those given in the published literature.

Dust devil generation

The equations describing axi-symmetric nonlinear internal gravity waves in an unstable atmosphere are derived. A hydrodynamic model of a dust devil generation mechanism in such an atmosphere is investigated. It is shown that in an unstably stratified atmosphere the convective plumes with poloidal motion can grow exponentially. Furthermore, it is demonstrated that these convective plumes in an atmosphere with weak large scale toroidal motion are unstable with respect to three-dimensional dust devil generation.

THREE-DIMENSIONAL SIMULATION THE RUNUP OF NONLINEAR SURFACE GRAVITY WAVES ON SHALLOW COAST

The paper is devoted to three-dimensional numerical modeling of a runup of nonlinear surface gravity waves based on Navier-Stokes equations. The three-dimensional problem is stated, boundary and initial conditions are described. Using split method for physical processes, there has been constructed the discrete model with consideration of cell filling coefficient. The peculiarities of the modeled three-dimensional tank structure are described. There are pre- sented three-dimensional models of stepwise runup of the surface gravity wave upon a sloping shore.

Three-dimensional simulation of the runup of nonlinear surface gravity waves

The runup of nonlinear surface gravity waves is numerically simulated in two and three dimensions on the basis of the Navier-Stokes equations. The three-dimensional problem is formulated, and the boundary and initial conditions are described. The splitting method over physical processes is used to construct a discrete model taking into account the cell occupation coefficient. The runup of nonlinear surface gravity waves is simulated in two dimensions for slopes of various geometries, and the numerical results are analyzed. The structural features of the simulated three-dimensional basin are described. Three-dimensional models for the staged runup of nonlinear surface gravity waves breaking on coastal slopes in shallow water areas are considered.

Covariant approach to the no-ghost theorem in massive gravity

We discuss the no-ghost theorem in the massive gravity in a covariant manner. Using the BRST formalism and St\"{u}ckelberg fields, we first clarify how the Boulware-Deser ghost decouples in the massive gravity theory with Fierz-Pauli mass term. Here we find that the crucial point in the proof is that there is no higher (time) derivative for the St\"{u}ckelberg `scalar' field. We then analyze the nonlinear massive gravity proposed by de Rham, Gabadadze and Tolley, and show that there is no ghost for general admissible backgrounds. In this process, we find a very nontrivial decoupling limit for general backgrounds. We end the paper by demonstrating the general results explicitly in a nontrivial example where there apparently appear higher time derivatives for St\"{u}ckelberg scalar field, but show that this does not introduce the ghost into the theory.

Cosmological investigations of (extended) nonlinear massive gravity schemes with nonminimal coupling

In this paper we investigate the case of nonminimal coupling in the (extended) nonlinear massive gravity theories. We first consider massive gravity in the Brans-Dicke background such that the graviton mass is replaced by ${A}^{2}(\ensuremath{\sigma})m$ where $\ensuremath{\sigma}$ is the Brans-Dicke field and $A(\ensuremath{\sigma})$ is the conformal coupling and show that there is no viable thermal history of the Universe in this case. We then invoke a cubic Galileon term as the nonlinear completion of the $\ensuremath{\sigma}$ Lagrangian and show that there is a stable de Sitter solution in this case. However, the de Sitter is blocked by the matter phase, which is also a simultaneous attractor of the dynamics. The de Sitter phase can, however, be realized by invoking unnatural fine-tunings. We next investigate the cosmology of quasidilaton nonlinear massive gravity with nonminimal coupling. As a generic feature of the nonminimal coupling, we show that the model exhibits a transient phantom phase that is otherwise impossible. While performing the observational data analysis on the models, we find that a small value of the coupling constant is allowed for quasidilaton nonlinear massive gravity. For both of the cases under consideration, it is observed that we have an effective pressure of matter that comes from the constraint equation. For mass-varying nonlinear massive gravity in the Brans-Dicke background, the effective pressure of matter is nonzero, which affects the evolution of the Hubble parameter, thereby spoiling the consistency of the model with data. As for the quasidilaton nonlinear massive gravity, the effective pressure of matter can be kept around zero by controlling the coupling constant, and the model is shown to be fit well with observations.

Dynamics of dispersive long waves in fluids

We obtain many different variable separation solutions for (2+1)-dimensional variable coefficient dispersive long wave equation by means of five different methods, including the multilinear variable separation approach, the projective Ricatti equation method, the extended projective Ricatti equation method, the extended tanh-function method and the improved tanh-function method. However, by careful analysis, we find that variable separation solution obtained by the multilinear variable separation approach includes all variable separation solutions obtained by other four direct methods. Thus variable separation solution for (2+1)-dimensional variable coefficient dispersive long wave equation exists a uniform form. Based on this uniform variable separation solution, we discuss the completely elastic interaction between foldons, the non-completely elastic interaction between bell-like semi-foldon, peaked semi-foldon and foldon, and the completely non-elastic interaction between bell-like semi-foldon and peaked semi-foldon. These results are helpful to analyze more precisely nonlinear and dispersive long gravity waves traveling in two horizontal directions, such as the bubbles on (or under) a fluid surface and folded waves in various ocean waves.

Modeling, observation, and control of hysteresis torsion in elastic robot joints

Hysteresis torsion in elastic robot joints occurs as a coupled nonlinearity due to internal friction, backlash, and nonlinear stiffness, which are coactive inside of mechanical transmission assemblies. The nonlinear joint torsion leads to hysteresis lost motion and can provoke control errors in relation to the joint output at both trajectories tracking and positioning. In this paper, a novel modeling approach for describing the nonlinear input–output behavior of elastic robot joints is proposed together with the observation and control method, which aim to compensate for the relative joint torsion without load sensing. The proposed modeling approach includes the recently developed 2SEP dynamic friction model and Bouc–Wen-like hysteresis model, which is originated from structural mechanics, both arranged according to the assumed torque transmitting structure. The proposed method is evaluated with experiments using the laboratory setup which emulates a single rotary joint under impact of nonlinear elasticities, friction, and gravity.

Rolls of the internal gravity waves in the Earth's atmosphere

Abstract. The effect of the wind shear on the roll structures of nonlinear internal gravity waves (IGWs) in the Earth's atmosphere with the finite vertical temperature gradients is investigated. A closed system of equations is derived for the nonlinear dynamics of the IGWs in the presence of temperature gradients and sheared wind. The solution in the form of rolls has been obtained. The new condition for the existence of such structures was found by taking into account the roll spatial scale, the horizontal speed and wind shear parameters. We have shown that the roll structures can exist in a dynamically unstable atmosphere.

An experimental study of the free evolution of rotating, nonlinear internal gravity waves in a two-layer stratified fluid

AbstractThe temporal evolution of nonlinear large-scale internal gravity waves, in a two-layer flow affected by background rotation, is studied via laboratory experiments conducted in a cylindrical tank, mounted on a rotating turntable. The internal wave field is excited by the relaxation of an initial forced tilt of the density interface ($\eta _{i}$), which generates internal waves, such as Kelvin and Poincaré waves, in response to rotation effects. The behaviour of $\eta _{i}$, in the shore region, is analysed in terms of the background rotation and the nonlinear steepening of the basin-scale waves. The results show that the degeneration of the fundamental Kelvin wave into a solitary-type wave packet is caused by nonlinear steepening and it is influenced by the background rotation. In addition, the physical scales of the leading solitary-type wave are closer to Korteweg–de Vries theory as the rotation increases. Moreover, the nonlinear interaction between the Kelvin wave and the Poincaré wave can transfer energy to higher or lower frequencies than the frequency of the fundamental Kelvin wave, as a function of the background rotation. In particular, a specific normal mode in the off-shore region could be energized by this interaction. Finally, the bulk decay rate of the fundamental Kelvin wave, $\tau _{dk}$, was investigated. The results exhibit that $\tau _{dk}$ is concordant with the Ekman damping time scale when there is no evidence of steepening in the basin-scale waves. However, as nonlinear processes increase, $\tau _{dk}$ shows a strong decrease. In this context, the nonlinear processes play an important role in the decay of the fundamental Kelvin wave, via the energy radiation to other modes. The results reported demonstrate that the background rotation and nonlinear processes are essential aspects in understanding the degeneration and the decay of large-scale internal gravity waves on enclosed basins.

Stability of the Schwarzschild-de Sitter black hole in the dRGT massive gravity theory

The Schwarzschild-de Sitter solution in the Einstein theory with a positive cosmological constant $\Lambda=m^2/\alpha$ becomes an exact solution to the dRGT non-linear massive gravity theory with the mass parameter $m$ when the theory parameters $\alpha$ and $\beta$ satisfy the relation $\beta=\alpha^2$. We study the perturbative behaviour of this black hole solution in the non-linear dRGT theory with $\beta=\alpha^2$. We find that the linear perturbation equations become identical to those for the vacuum Einstein theory when they are expressed in terms of the gauge-invariant variables. This implies that this black hole is stable in the dRGT theory as far as the spacetime structure is concerned in contrast to the case of the bi-Schwarzschild solution in the bi-metric theory. However, we have also found a pathological feature that the general solution to the perturbation equations contain a single arbitrary function of spacetime coordinates. This implies a degeneracy of dynamics in the St\"uckelberg field sector at the linear perturbation level in this background. Physical significance of this degenercy depends on how the St\"uckelberg fields couple observable fields.

Complex singularity of a stokes wave

Two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth can be described by a conformal map of the fluid domain into the complex lower half-plane. Stokes wave is the fully nonlinear gravity wave propagating with the constant velocity. The increase of the scaled wave height $H/\lambda$ from the linear limit $H/\lambda=0$ to the critical value $H_{max}/\lambda$ marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave. Here $H$ is the wave height and $\lambda$ is the wavelength. We simulated fully nonlinear Euler equations, reformulated in terms of conformal variables, to find Stokes waves for different wave heights. Analyzing spectra of these solutions we found in conformal variables, at each Stokes wave height, the distance $v_c$ from the lowest singularity in the upper half-plane to the real line which corresponds to the fluid free surface. We also identified that this singularity is the square-root branch point. The limiting Stokes wave emerges as the singularity reaches the fluid surface. From the analysis of data for $v_c\to 0$ we suggest a new power law scaling $v_c\propto (H_{max}-H)^{3/2}$ as well as new estimate $H_{max}/\lambda \simeq 0.1410633$.

The exp(−Φ(η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations

Periodic and soliton solutions are presented for the (1+1)-dimensional classical Boussinesq equation which governs the evolution of nonlinear dispersive long gravity wave traveling in two horizontal directions on shallow water of uniform depth. The equation is handled via the exp(−Φ(η))-expansion method. It is worth declaring that the method is more effective and useful for solving the nonlinear evolution equations. In particular, mathematical analysis and numerical graph are provided for those solitons, periodic, singular kink and bell type solitary wave solutions to visualize the dynamics of the equation.

NOTE ABOUT HAMILTONIAN FORMALISM FOR GENERAL NONLINEAR MASSIVE GRAVITY ACTION IN STÜCKELBERG FORMALISM

In this paper, we try to prove the absence of the ghosts in case of the general nonlinear massive gravity action in Stückelberg formalism. We argue that in order to find the explicit form of the Hamiltonian it is natural to start with the general nonlinear massive gravity action [S. F. Hassan and R. A. Rosen, Phys. Rev. Lett. 108, 041101 (2012), arXiv:1106.3344 [hep-th]]. We perform the complete Hamiltonian analysis of the Stückelberg form of the minimal the nonlinear gravity action in this formulation and show that the constraint structure is so rich that it is possible to eliminate nonphysical modes. Then, we extend this analysis to the case of the general nonlinear massive gravity action. We find the corresponding Hamiltonian and collection of the primary constraints. Unfortunately we are not able to finish the complete analysis of the stability of all constraints due to the complex form of one primary constraint so that we are not able to determine the conditions under which given constraint is preserved during the time evolution of the system.

Numerical bifurcation and Hopf continuation of Lorenz–Stenflo system

A graphical overview of the bifurcation structures and their behaviour of the Lorenz–Stenflo system (Stenflo 1996 Phys. Scr. 53 83) for describing nonlinear low-periodic short-wavelength acoustic gravity waves are investigated. We utilize the MATCONT2.5.1 package to analyse the detailed bifurcation scenarios as the physical and control parameters are varied. The variety of patterns revealed here has not been studied before. Using forward and backward bifurcations, the Hopf bifurcation and their continuation curves of the system are presented in the rotation and Rayleigh number plane, Prandtl number and rotation plane, and also in the rotation and geometric parameter plane. These bifurcation phenomena and their behaviour may be useful for better prediction of existence and stability of large-scale quasi-stationary structures amid chaotic motion in the atmosphere.

Hartle–Hawking no-boundary proposal in dRGT massive gravity: making inflation exponentially more probable

It is known that the no-boundary proposal in the traditional Einstein gravity does not prefer inflation, that is, the probability of realizing a large number of e-folds is exponentially suppressed. This situation may be changed drastically in a class of nonlinear massive gravity theories recently proposed by de Rham, Gabadadze and Tolley, called dRGT massive gravity. We show that the contribution from the massive gravity sector can enhance the probability of a large number of e-folds substantially for a sufficiently large mass parameter mg comparable to the Hubble parameter during inflation, say mg ≳ 1012 GeV. We illustrate possible models to trigger such a large mass parameter in the early universe while it is negligibly small in the present universe. This opens a new window to explore the inflationary scenario in the context of quantum cosmology.