Effective Field Equations Research Articles

Effective wave motion in periodic discontinua near spectral singularities at finite frequencies and wavenumbers

We consider the effective wave motion, at spectral singularities such as corners of the Brillouin zone and Dirac points, in periodic continua intercepted by compliant interfaces that pertain to e.g. masonry and fractured materials. We assume the Bloch-wave form of the scalar wave equation (describing anti-plane shear waves) as a point of departure, and we seek an asymptotic expansion about a reference point in the wavenumber-frequency space using wavenumber separation as the perturbation parameter. Using the concept of broken Sobolev spaces to cater for the presence of kinematic discontinuities, we next define the “mean” wave motion via inner product between the Bloch wave and an eigenfunction (at specified wavenumber and frequency) for the unit cell of periodicity. With such projection-expansion approach, we obtain an effective field equation, for an arbitrary dispersion branch, near apexes of “wavenumber quadrants” featured by the first Brillouin zone. For completeness, we investigate asymptotic configurations featuring both (a) isolated, (b) repeated, and (c) nearby eigenvalues. In the case of repeated eigenvalues, we find that the “mean” wave motion is governed by a system of wave equations and Dirac equations, whose size is given by the eigenvalue multiplicity, and whose structure is determined by the participating eigenfunctions, the affiliated cell functions, and the direction of wavenumber perturbation. One of these structures is shown to describe the so-called Dirac points – apexes of locally conical dispersion surfaces – that are relevant to the generation of topologically protected waves. In situations featuring clusters of tightly spaced eigenvalues, the effective model is found to entail a Dirac-like system of equations that generates “blunted” conical dispersion surfaces. We illustrate the analysis by numerical simulations for two periodic configurations in R2 that showcase the asymptotic developments in terms of (i) wave dispersion, (ii) forced wave motion, and (iii) frequency- and wavenumber-dependent phonon behavior.

A New Black Hole Solution in Conformal Dilaton Gravity on a Warped Spacetime

An exact time-dependent solution of a black hole is found in a conformally invariant gravity model on a warped Randall-Sundrum spacetime, by writing the metric . Here, represents the “un-physical” spacetime and ω the dilaton field, which will be treated on equal footing as any renormalizable scalar field. In the case of a five-dimensional warped spacetime, we thereafter write . The dilaton field can be used to describe the different notion the in-going and outside observers have of the Hawking radiation by using different conformal gauge freedom. The disagreement about the interior of the black hole is explained by the antipodal map of points on the horizon. The free parameters of the solution can be chosen in such a way that is singular-free and topologically regular, even for ω → 0 . It is remarkable that the 5D and 4D effective field equations for the metric components and dilaton fields can be written in general dimension n = 4,5. From the exact energy-momentum tensor in Eddington-Finkelstein coordinates, we are able to determine the gravitational wave contribution in the process of evaporation of the black hole. It is conjectured that, in context of quantization procedures in the vicinity of the horizon, unitarity problems only occur in the bulk at large extra-dimension scale. The subtraction point in an effective theory will be in the UV only in the bulk, because the use of a large extra dimension results in a fundamental Planck scale comparable with the electroweak scale.

Parity violation in Poincaré gauge gravity

We analyze the parity violation issue in the Poincaré gauge theory of gravity for the two classes of models which are built as natural extensions of the Einstein–Cartan theory. The conservation laws of the matter currents are revisited and we clarify the derivation of the effective Einstein field equation and the structure of the effective energy–momentum current for arbitrary matter sources.

Single graviton loop contribution to the self-mass of a massless, conformally coupled scalar on a de Sitter background

We use a simplified formalism to re-compute the single graviton loop contribution to the self-mass of a massless, conformally coupled scalar on de Sitter background which was originally made by Boran, Kahya and Park [1-3]. Our result resolves the problem with the flat space correspondence limit that was pointed out by Fr\"ob [4]. We discuss how this computation will be used in a long-term project to purge the linearized effective field equation of gauge dependence.

Brane universe and holography in the spacetime of a charged AdS dilaton black hole

In the background of a charged AdS dilaton black hole, we investigate the movement of a self-graviting 3-brane and relevant holographic effects as the brane move close to the AdS boundary. The induced metric on brane corresponds to an exact FLRW geometry, while the evolution of brane is determined by Israel junction condition and the effective Einstein field equation on brane together. When the brane approaches the AdS boundary, AdS/CFT correspondence implies that a radiation dominated FLRW-universe ($P=\frac{1}{3}\rho$) should be given. According to the holographic renormalization procedure, we involve an appropriate surface counterterm into the gravitational action for achieving $P=\frac{1}{3}\rho$ on brane. This surface counterterm also plays a important role in caculating the mass of charged AdS dilaton black hole. Finally, we obtain the thermodynamic quantities and give an extend Cardy-Verlinde formula on brane.

Optimizing microstructure of a poroelastic layer with cylindrical pores for absorption properties

The present work addresses the problem of multi-scale characterization and optimization of absorption properties of poroelastic materials. Considering the microstructural parameters, we compute the effective properties and field equations within the framework of Biot’s theory. Asymptotic homogenization method is used for computing the effective properties which are then applied to solve an acoustic problem. The solution of the latter problem enables the choice of the objective function for the optimization procedure of the optimal design of an unit cell. Preliminary numerical studies show that different skeleton rigidity (soft, stiff or perfectly rigid) may lead to distinct acoustic behavior and optimal parameters for the poroelastic panel.

Scale-dependent planar anti-de Sitter black hole

In this work, we investigate four-dimensional planar black-hole solutions in anti-de Sitter spacetimes in light of the so-called scale-dependent scenario. To obtain this new family of solutions, the classical couplings of the theory, i.e., the gravitational coupling G0 and the cosmological constant $ \Lambda_{0}$, are not taken to be fixed values anymore. Thus, those classical parameters evolve to functions which change along the “height” coordinate, z . The effective Einstein field equations are solved, and the results are analyzed and compared with the classical counterpart. Finally, some thermodynamic properties of the presented scale-dependent black hole are investigated.

Stable Cosmological Kaluza\u2013Klein Spacetimes

We consider the Einstein flow on a product manifold with one factor being a compact quotient of 3-dimensional hyperbolic space without boundary and the other factor being a flat torus of fixed arbitrary dimension. We consider initial data symmetric with respect to the toroidal directions. We obtain effective Einsteinian field equations coupled to a wave map type and a Maxwell type equation by the Kaluza–Klein reduction. The Milne universe solves those field equations when the additional parts arising from the toroidal dimensions are chosen constant. We prove future stability of the Milne universe within this class of spacetimes, which establishes stability of a large class of cosmological Kaluza–Klein vacua. A crucial part of the proof is the implementation of a new gauge for Maxwell-type equations in the cosmological context, which we refer to as slice-adapted gauge.

Scale-dependent rotating BTZ black hole

This work presents a generalization of the rotating black hole in two plus one dimensions, in the light of scale-dependent gravitational couplings. In particular, the gravitational coupling kappa _0 and the cosmological term varLambda _0 are not forced to be constants anymore. Instead, kappa and varLambda are allowed to change along the radial scale r. The effective Einstein field equations of this problem are solved by assuming static rotational symmetry and by maintaining the usual structure of the line element. For this generalized solution, the asymptotic behavior, the horizon structure, and the thermodynamic properties are analyzed.

Scale-dependent (2+1)-dimensional electrically charged black holes in Einstein-power-Maxwell theory

In this work we extend and generalize our previous work on the scale dependence at the level of the effective action of black holes in the presence of non-linear electrodynamics. In particular, we consider the Einstein-power-Maxwell theory without a cosmological constant in (2+1) dimensions, assuming a scale dependence of both the gravitational and the electromagnetic coupling and we investigate in detail how the scale-dependent scenario affects the horizon and thermodynamic properties of the classical black holes for any value of the power parameter. In addition, we solve the corresponding effective field equations imposing the “null energy condition” in order to obtain analytical solutions. The implications of quantum corrections are also briefly discussed.

Einstein’s gravity from a polynomial affine model

We show that the effective field equations for a recently formulated polynomial affine model of gravity, in the sector of a torsion-free connection, accept general Einstein manifolds—with or without cosmological constant—as solutions. Moreover, the effective field equations are partially those obtained from a gravitational Yang–Mills theory known as Stephenson–Kilmister–Yang theory. Additionally, we find a generalization of a minimally coupled massless scalar field in General Relativity within a ‘minimally’ coupled scalar field in this affine model. Finally, we present a brief (perturbative) analysis of the propagators of the gravitational theory, and count the degrees of freedom. For completeness, we prove that a Birkhoff-like theorem is valid for the analyzed sector.

Deducing cosmological observables from the S matrix

We study one loop quantum gravitational corrections to the long range force induced by the exchange of a massless scalar between two massive scalars. The various diagrams contributing to the flat space S-matrix are evaluated in a general covariant gauge and we show that dependence on the gauge parameters cancels at a point considerably {\it before} forming the full S-matrix, which is unobservable in cosmology. It is possible to interpret our computation as a solution to the effective field equations --- which could be done even in cosmology --- but taking account of quantum gravitational corrections from the source and from the observer.

Field dynamics on the trapping horizon in Vaidya spacetime

In this article, we shed some light on the field theoretic aspect of the spherically symmetric trapping horizon in Vaidya spacetime. The effective field equations are that of a Chern-Simons theory coupled to bulk sources through two different couplings, one is purely geometric and the other is matter dependent. This is an effective generalization of the equilibrium horizon scenario where the Chern-Simons theory is coupled to the bulk geometry through a constant matter-independent coupling. Further, we note that the field equations pulled-back to a cross-section of the horizon is inadequate to manifest the nature of the horizon. Hence, contrary to the usual practice, the evolution equation needs to be considered at least while passing on to the quantum theory.

Modified magnetomechancial model in the constant and low intensity magnetic field based on J–A theory**Project supported by the Major Program of Sichuan Province Science and Technology Plan, China (Grant No. 2015SZ0010) and the Scientific Research Foundation of Sichuan Province, China (Grant No. 2014GZ0121).

The existing magnetomechancial models cannot explain the different experimental phenomena when the ferromagnetic specimen is respectively subjected to tension and compression stress in the constant and low intensity magnetic field, especially in the compression case. To promote the development of magnetomechancial theory, the energy conservation equation, effective magnetic field equation, and anhysteretic magnetization equation of the original Jiles–Atherton (J–A) theory are elucidated and modified, an equation of the local equilibrium status is employed and the differential expression of the modified magnetomechancial model based on the modified J–A theory is established finally. The effect of stress and plastic deformation on the magnetic parameters is analyzed. An excellent agreement is achieved between the theoretic predictions by the present modified model and the previous experimental results. Comparing with the calculation results given by the existing models and experimental results, it is seen indeed that the modified magnetomechanical model can describe the different magnetization features during tension-release and compression-release processes much better, and is the only one which can accurately reflect the experimental observation that the magnetic induction intensity reverses to negative value with the increase of the compressive stress and applied field.

Modeling plastic deformation effect on the hysteresis loops of ferromagnetic materials based on modified Jiles-Atherton model

Plastic deformation is one of the most important features that affect the hysteresis magnetic properties of steels, because it changes the dislocation density and affects domain-wall movement and pinning. In order to model the effect of plastic deformation on the magnetic properties, the prevailing Jiles-Atherton (J-A) theory is extensively used. However, the J-A models in a series of papers published by Jiles et al. are not completely consistent. As a result, there exists no uniform formula of magneto-plastic model established by different researchers, based on different J-A models, and various versions given by different mathematic expressions of magneto-plastic model often create difficulty in discriminating the accuracies and effectivenesses of the analyzed results. Therefore, it is necessary to establish an accurate and reasonable magneto-plastic model. In this paper, on the basis of magnetization mechanism of ferrimagnet and plastic deformation model, the effects of plastic deformation on the magnetic characteristic parameters adopted in magneto-plastic model, such as dislocation density, pinning coefficient and scaling constant, are analyzed and the relationship between them is first established. Then, by contrasting the fitting formula of the anhysteretic magnetization curve, the energy conservation equation and the effective magnetic field equation established by different researchers, several queries are proposed, and the irrationality and inaccuracy of the existing magneto-plastic model are elucidated, such as the mixing of anhysteresis magnetization and magnetization, the unreasonably regarding the irreversible magnetization energy as actual total magnetization energy. Thus, the energy conservation equation, the effective magnetic field equation and the anhysteretic magnetization equation are modified, and the differential expression of the magneto-plastic model is re-derived finally. Comparing the results calculated by the existing magneto-plastic models with the experimental results, it is seen indeed that a more sharp change of magnetization appears at small plastic deformation, then, the values of magnetization decrease more slowly with the increase of plastic deformation than those from the models respectively proposed by Li Jian-Wei, Leng Jian-Cheng and Wang Zheng-Dao; the saturation magnetization and residual magnetization decrease with the increase of plastic deformation, the coercive force is increased oppositely and the trend to reach the saturation magnetization becomes gentler, which is more exactly consonant with experiment observation than that calculated by the Sablik's model; additionally, the hysteresis loops of the plastically deformed carbon-steel samples calculated by the modified magneto-plastic model are also in better agreement with the experimental results than those from the existing models. Consequently, the modification is effective, and the modified magneto-plastic model is more accurate to simulate the plastic deformation effect on the magnetic property of ferromagnetic material.

A macroscopically effective Lorentz gauge theory of gravity

Following the ideas of effective field theories, we derive classically effective field equations of the recently developed Lorentz gauge theory of gravity. It is shown that Newton’s gravitational constant emerges as an effective coupling parameter if an extremely small length is integrated out of the underlying theory. The linear version of the effective theory is shown to be fully consistent with the Newtonian gravity. We also derive a numerical solution for the interior of a star and show that in the nonlinear regions, the behavior of the effective theory deviates from the predictions of general relativity.

Spherically symmetric brane in a bulk of f(R) and Gauss–Bonnet gravity

Effective gravitational field equations on a four-dimensional brane embedded in a five-dimensional bulk have been considered. Using the Einstein–Hilbert action along with the Gauss–Bonnet correction term, we have derived static spherically symmetric vacuum solution to the effective field equations, first order in the Gauss–Bonnet coupling parameter. The solution so obtained, has one part corresponding to general relativity with an additional correction term, proportional to the Gauss–Bonnet coupling parameter. The correction term modifies the spacetime structure, in particular, the location of the event horizon. Proceeding further, we have derived effective field equations for gravity with Gauss–Bonnet correction term and a static spherically symmetric solution has been obtained. In this case the Gauss–Bonnet term modifies both the event and cosmological horizon of the spacetime. There exists another way of obtaining the brane metric—expanding the bulk gravitational field equations in the ratio of bulk to brane curvature scale and assuming a separable bulk metric ansatz. It turns out that static, spherically symmetric solutions obtained from this perturbative method can be matched exactly, with the solutions derived earlier. This will hold for Einstein–Hilbert plus Gauss–Bonnet as well as for f(R) with the Gauss–Bonnet correction. Implications of these results are discussed.

Solutions on a brane in a bulk spacetime with Kalb–Ramond field

Effective gravitational field equations on a brane have been derived, when the bulk spacetime is endowed with the second rank antisymmetric Kalb–Ramond field. Since both the graviton and the Kalb–Ramond field are closed string excitations, they can propagate in the bulk. After deriving the effective gravitational field equations on the brane, we solve them for a static spherically symmetric solution. It turns out that the solution so obtained represents a black hole or naked singularity depending on the parameter space of the model. The stability of this model is also discussed. Cosmological solutions to the gravitational field equations have been obtained, where the Kalb–Ramond field is found to behave as normal pressure free matter. For certain specific choices of the parameters in the cosmological solution, the solution exhibits a transition in the behaviour of the scale factor and hence a transition in the expansion history of the universe. The possibility of accelerated expansion of the universe in this scenario is also discussed.

Quantum scalar corrections to the gravitational potentials on de Sitter background

We employ the graviton self-energy induced by a massless, minimally coupled (MMC) scalar on de Sitter background to compute the quantum corrections to the gravitational potentials of a static point particle with a mass $M$. The Schwinger-Keldysh formalism is used to derive real and causal effective field equations. When evaluated at the one-loop order, the gravitational potentials exhibit a secular decrease in the observed gravitational coupling $G$. This can also be interpreted as a (time dependent) anti-screening of the mass $M$.

Effective gravitational field equations on m-brane embedded in n-dimensional bulk of Einstein and $$f(\mathcal {R})$$ f ( R ) gravity

We have derived effective gravitational field equations on a lower dimensional hypersurface (known as a brane), placed in a higher dimensional bulk spacetime for both Einstein and $f(\mathcal{R})$ gravity theories. We have started our analysis on $n$-dimensional bulk from which the effective field equations on a $(n-1)$-dimensional brane has been obtained by imposing $Z_{2}$ symmetry. Subsequently, we have arrived at the effective equations in $(n-2)$-dimensions starting from the effective equations for $(n-1)$ dimensional brane. This analysis has been carried forward and is used to obtain the effective field equations in $(n-m)$-dimensional brane, embedded in a $n$-dimensional bulk. Having obtained the effective field equations in Einstein gravity, we have subsequently generalized the effective field equation in $(n-m)$-dimensional brane which is embedded the $n$-dimensional bulk spacetime endowed with $f(\mathcal{R})$ gravity. We have also presented applications of our results in the context of Einstein and $f(\mathcal{R})$ gravity. In both the cases we have discussed vacuum static spherically symmetric solutions as well as solutions in cosmological context. Implications are also discussed.