Extra Coordinate Research Articles

Holographic duality for 3d spin-3 gravity coupled to scalar field

The 3d spin-3 gravity theory is holographically dual to a 2d W3-extended CFT. In a large-c limit the symmetry algebra of the CFT reduces to SU(1,2)×SU(1,2). On ground of symmetry the dual bulk space–time will be given by an 8d group manifold SU(1,2). Hence we need to introduce five extra coordinates in addition to three ordinary ones. The 3d space–time is a 3d hyper-surface Σ embedded at constant values of the extra variables. Operators in the CFT at the boundary of Σ are expressed in terms of W descendants of the operators at the boundary of Σ0, where the extra variables vanish. In this paper it is shown that AdS/CFT correspondence for a scalar field coupled to 3d spin-3 gravity is realized in this auxiliary 8d space. A bulk-to-boundary propagator of a scalar field is found and a generating functional of boundary two-point functions of scalar W-descendant operators is obtained by using the classical action for the scalar field. Classically, the scalar field must satisfy both Klein–Gordon equation and a third-order differential equation, which are related to the quadratic and cubic Casimir operators of su(1,2). It is found that the coefficient function of the derivatives of the scalar field in the latter equation is the spin-3 gauge field, when restricted to the hypersurface. An action integral in the 8d auxiliary space for the 3d spin-3 gravity coupled to a scalar field is presented. In general, this 8d auxiliary space is a deformation of the manifold SU(1,2). An 8d local frame is introduced and the equations of motion for the 8d connections Aμ, A‾μ are solved. By restricting those solutions onto Σ, flat connections in 3d SL(3,R)×SL(3,R) Chern–Simons theory are obtained and new 3d black hole solutions with and without spin-3 charge are found by this method.

Numerical comparisons of sandwich viscoelastic beam models1

Composite structures with elastic layers and viscoelastic core have been used as a passive damping treatment applied to reduce vibration amplitudes. In the design phase of this type of damping technique, many aspects ranging from computer modeling to laboratory tests should be considered. Due to the frequency dependency of mechanical properties of these materials, time domain based models for viscoelastic materials are not as numerous as frequency domain based methods. Usually, time domain methods introduce extra dissipation coordinates or internal variables as Golla-Hughes-McTavish (GHM) method and also Anelastic Displacement Field (ADF) method. In this paper, these methods are evaluated by comparing theirs results to the ones obtained by means of theoretical analysis. Processing time are also evaluated highlighting advantages and disadvantages of these methodologies. Finally, these two time domain methods are applied to a real structure, pointing out the facilities and difficulties to simulate an actual situation.

Localization of matter fields that interact with gravity in a non-minimal derivative coupling in the Randall-Sundrummodel

In a brane-world model matter fields are assumed to localize on a (3 + 1)-dimensional brane imbedded in a higher dimensional space-time. However, this assumption is not always fulfilled. In fact, vector fields that couple minimally with gravity as well as matter fields that interact with gauge fields, in addition to gravity in a minimal coupling way are not localized on a brane in the five-dimensional Randall-Sundrum (RS) model [1, 2, 3]. In this paper we will report our analysis on localization properties in the RS brane-world model for the case of matter fields that interact with gravity in a non-minimal derivative coupling. By expressing matter fields as a multiplication of fields in a four-dimensional space-time and a function of the extra dimension coordinate we derive localization conditions and field equations in the function of the extra coordinate and solve the equation. The solution is then applied to analyze field localizations. We obtain that the massless scalar and spinor fields are localizable respectively on the RS brane for decreasing and increasing warp factor respectively.

Advanced parametric space-frequency separated representations in structural dynamics: A harmonic–modal hybrid approach

This paper is concerned with the solution to structural dynamics equations. The technique here presented is closely related to Harmonic Analysis, and therefore it is only concerned with the long-term forced response. Proper Generalized Decomposition (PGD) is used to compute space-frequency separated representations by considering the frequency as an extra coordinate. This formulation constitutes an alternative to classical methods such as Modal Analysis and it is especially advantageous when parametrized structural dynamics equations are of interest. In such case, there is no need to solve the parametrized eigenvalue problem and the space-time solution can be recovered with a Fourier inverse transform. The PGD solution is valid for any forcing term that can be written as a combination of the considered frequencies. Finally, the solution is available for any value of the parameter. When the problem involves frequency-dependent parameters the proposed technique provides a specially suitable method that becomes computationally more efficient when it is combined with a modal representation.

Expressing an Observer in Preferred Coordinates by Transforming an Injective Immersion into a Surjective Diffeomorphism

When designing observers for nonlinear systems, the dynamics of the given system and of the designed observer are usually not expressed in the same coordinates or even have states evolving in different spaces. In general, the function, denoted $\tau$ (or its inverse, denoted $\tau^*$) giving one state in terms of the other is not explicitly known and this creates implementation issues. We propose to round this problem by expressing the observer dynamics in the the same coordinates as the given system. But this may impose to add extra coordinates, problem that we call augmentation. This may also impose to modify the domain or the range of the \augmented $\tau$ or $\tau^*$, problem that we call extension. We show that the augmentation problem can be solved partly by a continuous completion of a free family of vectors and that the extension problem can be solved by a function extension making the image of the extended function the whole space. We also show how augmentation and extension can be done without modifying the observer dynamics and therefore with maintaining convergence. Several examples illustrate our results.

Localization of interacting fields in five-dimensional braneworld models

We study the localization properties of fundamental fields which are coupled to one another through the gauge mechanism both in the original Randall–Sundrum (RS) and in the modified Randall–Sundrum (MRS) braneworld models: scalar–vector, vector–vector, and spinor–vector configuration systems. For this purpose, we derive conditions of localization, namely, the finiteness of integrals over the extra coordinate in the action of the system considered. We also derive field equations for each of the systems and then obtain their solutions corresponding to the extra dimension by a separation of variable method for every field involved in each system. We then insert the obtained solutions into the conditions of localization to seek whether or not the solutions are in accordance with the conditions of localization. We obtain that not all of the configuration systems considered are localizable on the brane of the original RS model while, on the contrary, they are localizable on the MRS braneworld model with some restrictions. In terms of field localizability on the brane, this result shows that the MRS model is much better than the original RS model.

Conformal (2 + 4)-braneworld

The 6D brane model is considered, where matter is trapped on the surface of a (2+4)-hyperboloid, as is suggested by the geometrical structure behind the 4D conformal group. The effective dimension of the bulk spacetime for matter fields is five, with the extra space-like and time-like domains. Using the embedding theory, the presence of the familiar factorizable 5D brane metrics in both domains is shown. These metrics with exponential warp factors are able to provide with the additional reduction of the effective spacetime dimensions down to four. It is demonstrated that the extra (1+1)-space is not simply connected and there is a gap in the range of the extra coordinates. This can explain the stability of the model in the domain with the time-like effective fifth dimension and the appearance of the cosmological constant due to the tachyon condensation. It is found that the model exhibits orbifold symmetry and thus is free from the fermion chirality problem.

Higher-order gravity in higher dimensions: geometrical origins of four-dimensional cosmology?

Determining the cosmological field equations is still very much debated and led to a wide discussion around different theoretical proposals. A suitable conceptual scheme could be represented by gravity models that naturally generalize Einstein theory like higher-order gravity theories and higher-dimensional ones. Both of these two different approaches allow one to define, at the effective level, Einstein field equations equipped with source-like energy-momentum tensors of geometrical origin. In this paper, the possibility is discussed to develop a five-dimensional fourth-order gravity model whose lower-dimensional reduction could provide an interpretation of cosmological four-dimensional matter–energy components. We describe the basic concepts of the model, the complete field equations formalism and the 5-D to 4-D reduction procedure. Five-dimensional f(R) field equations turn out to be equivalent, on the four-dimensional hypersurfaces orthogonal to the extra coordinate, to an Einstein-like cosmological model with three matter–energy tensors related with higher derivative and higher-dimensional counter-terms. By considering the gravity model with f(R)=f_0R^n the possibility is investigated to obtain five-dimensional power law solutions. The effective four-dimensional picture and the behaviour of the geometrically induced sources are finally outlined in correspondence to simple cases of such higher-dimensional solutions.

Modeling soft, permeable matter with the proper generalized decomposition (PGD) approach, and verification by means of nanoindentation.

Understanding sliding and load-bearing mechanisms of biphasic soft matter is crucial for designing synthetic replacements of cartilage, contact-lens materials or coatings for medical instruments. Interstitial fluid pressurization is believed to be the intrinsic load-carrying phenomenon governing the frictional properties. In this study, we have characterized permeability and identified the fluid contribution to the support of load during Atomic Force Microscopy (AFM) nanoindentation of soft polymer brushes in aqueous environments, by means of the Proper Generalized Decomposition (PGD) approach. First, rate-dependent AFM nanoindentation was performed on a poly(acrylamide) (PAAm) brush in an aqueous environment, to probe the purely elastic as well as poro-viscoelastic properties. Second, a biphasic model decoupling the fluid and solid load contributions was proposed, using Darcy's equation for liquid flow in porous media. Using realistic time-dependent simulations requires many direct solutions of the 3D partial differential equation, making modeling very time-consuming. To efficiently alleviate the time-consumption of multi-dimensional modeling, we used PGD to solve a Darcy model defined in a 7D domain, considering all the unknowns and material properties as extra coordinates of the problem. The obtained 7D simulation results were compared to the experimental results by using a direct Newton algorithm, since all sensitivities with respect to the model parameters are readily available. Thus, a simulation-based solution for depth- and rate-dependent permeability can be obtained. From the PGD-based model permeability is calculated, and the velocity- and pressure-fields in the material can be obtained in real-time in 3D by adjusting the parameters to the experimental values. The result is a step forward in understanding the fluid flow, permeability and fluid contributions to the load support of biphasic soft matter.

Curved branes with regular support

We study spacetime singularities in a general five-dimensional braneworld with curved branes satisfying four-dimensional maximal symmetry. The bulk is supported by an analog of perfect fluid with the time replaced by the extra coordinate. We show that contrary to the existence of finite distance singularities from the brane location in any solution with flat (Minkowski) branes, in the case of curved branes there are singularity-free solutions for a range of equations of state compatible with the null energy condition.

Meson effective mass in the isospin medium in hard-wall AdS/QCD model

We study a mass splitting of light vector, axial-vector and pseudoscalar mesons in isospin medium in the framework of hard-wall model. We write an effective mass definition for the interacting gauge fields and scalar field introduced in gauge field theory in the bulk of AdS space-time. Relying on holographic duality we obtain a formula for the effective mass of a boundary meson in terms of derivative operator over the extra bulk coordinate. The effective mass found in this way coincides with the one obtained from finding of poles of the two-point correlation function. In order to avoid introducing distinguished infrared boundaries in the quantisation formula for the different mesons from the same isotriplet we introduce extra action terms at this boundary, which reduces distinguished values of this boundary to the same value. Profile function solutions and effective mass expressions were found for the in-medium $\rho$, $a_1$ and $\pi$ mesons.

Membrane duality revisited

Just as string T-duality originates from transforming field equations into Bianchi identities on the string worldsheet, so it has been suggested that M-theory U-dualities originate from transforming field equations into Bianchi identities on the membrane worldvolume. However, this encounters a problem unless the target space has dimension D=p+1. We identify the problem to be the nonintegrability of the U-duality transformation assigned to the pull-back map. Just as a double geometry renders manifest the O(D,D) string T-duality, here we show in the case of the M2-brane in D=3 that a generalized geometry renders manifest the SL(3)×SL(2) U-duality. In the case of M2-brane in D=4, with and without extra target space coordinates, we show that only the GL(4,R)⋉R4 subgroup of the expected SL(5,R) U-duality symmetry is realized.

Embeddings for general relativity

We present a systematic approach to embed n-dimensional vacuum general relativity in an (n+1)-dimensional pseudo-Riemannian spacetime whose source is either a (non)zero cosmological constant or a scalar field minimally coupled to Einstein gravity. Our approach allows us to generalize a number of results discussed in the literature. We construct all the possible (physically distinct) embeddings in Einstein spaces, including the Ricci-flat ones widely discussed in the literature. We examine in detail their generalization, which—in the framework under consideration—are higher-dimensional spacetimes sourced by a scalar field with flat (constant 0) potential. We use the Kretschmann curvature scalar to show that many embedding spaces have a physical singularity at some finite value of the extra coordinate. We develop several classes of embeddings that are free of singularities, have distinct non-vanishing self-interacting potentials and are continuously connected (in various limits) to Einstein embeddings. We point out that the induced metric possesses scaling symmetry and, as a consequence, the effective physical parameters (e.g., mass, angular momentum, cosmological constant) can be interpreted as functions of the extra coordinate.

The metric of extended Einstein equation and Schwarzschild solution in six dimensions

The metric of a warped space-time with two extra dimensions is established by means of the Einstein equations in six dimensions and the compactification of two extra dimensions on a square. It is shown that at every among two fixed points our manifold reduces to the five-dimensional Randall - Sundrum space-time and the hierarchy problem could be solved. The Schwarzschild solutions have been extended in six dimensions. We also proposed an equation for the mass operator generated from extra dimensional coordinates.

Exceptional field theory: SO(5,5)

We construct Exceptional Field Theory for the group $SO(5,5)$ based on the extended (6+16)-dimensional spacetime, which after reduction gives the maximal $D=6$ supergravity. We present both a true action and a duality-invariant pseudo-action formulations. All the fields of the theory depend on the complete extended spacetime. The U-duality group $SO(5,5)$ is made a geometric symmetry of the theory by virtue of introducing the generalised Lie derivative that incorporates a local duality transformation. Tensor hierarchy appears as a natural consequence of the algebra of generalised Lie derivatives that are viewed as gauge transformations. Upon truncating different subsets of the extra coordinates, maximal supergravities in $D=11$ and $D=10$ (type IIB) can be recovered from this theory.

Cosmological solutions from induced matter model applied to 5D $$f(R,T)$$ f ( R , T ) gravity and the shrinking of the extra coordinate

In this work, I present exact cosmological solutions from Wesson's Induced Matter Model application to a general 5D metric in f(R,T) theory of gravity. The non-conservation of the energy-momentum tensor, predicted by f(R,T) theory, allows the derivation of a relation that describes the time evolution of the extra coordinate, revealing its compactification. It is showed that such a compactification could induce the effects of an accelerated expansion in the observable universe.

Geometrical Applications of Split Octonions

It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties from conventional (3 + 1)-theory (e.g., number of dimensions, existence of maximal velocities, Heisenberg uncertainty, and particle generations). This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonions, the noncompact form of the exceptional Lie groupG2. This group generates specific rotations of (3 + 4)-vector parts of split octonions with three extra time-like coordinates and in infinitesimal limit imitates standard Poincare transformations. In this picture translations are represented by noncompact Lorentz-type rotations towards the extra time-like coordinates. It is shown how theG2algebra’s chirality yields an intrinsic left-right asymmetry of a certain 3-vector (spin), as well as a parity violating effect on light emitted by a moving quantum system. Elementary particles are connected with the special elements of the algebra which nullify octonionic intervals. Then the zero-norm conditions lead to free particle Lagrangians, which allow virtual trajectories also and exhibit the appearance of spatial horizons governing by mass parameters.

イベント空間における閲覧者モデルとアメニティ空間設計

In this paper, we develop a model of the behavior of visitors in an exhibition space, and design an amenity space to reduce congestion. Because the flow of visitors depends on the space layout, it is possible to reduce the congestion by changing the layout of exhibits. To deal with the congestion, a macro model of visitors is required. The flow of visitors is modeled by two-dimensional vector field and the individual behavior is represented by the dynamic model including collision avoidance of individuals. To model walking and stopping behaviors, we introduce an extended dimensional space where the extra coordinate represents the viewing time of each exhibition. The parameters in the proposed model are identified from the measured data of human flow line. To reduce the congestion, the layout of exhibits is optimized based on the proposed model by minimizing the collision avoidance vector. The proposed method is verified by simulations and experiments using swarm robots, which consist of autonomous mobile robots and radio controlled robots.

The primordial explosion of a false white hole from a 5D vacuum

We explore the cosmological consequences of some possible big bang produced by a black-hole with mass M in a 5D extended SdS. Under these particular circumstances, the effective 4D metric obtained by the use of a constant foliation on the extra coordinate is comported as a false white hole (FWH), which evaporates for all unstable modes that have wavelengths bigger than the size of the FWH. Outside the white hole the repulsive gravitational field can be considered as weak, so that the dynamics for fluctuations of the inflaton field and the scalar perturbations of the metric can be linearized.

INFLATIONARY DARK ENERGY FROM A CONDENSATE OF SPINORS IN A 5D VACUUM

What is the physical origin of dark energy? Could this energy be originated by other fields than the inflaton? In this paper, we explore the possibility that the expansion of the universe can be driven by a condensate of spinors. These spinors are free of interactions on five-dimensional (5D) relativistic vacuum in an extended de Sitter spacetime. The extra coordinate is considered as noncompact. After making a static foliation on the extra coordinate, we obtain an effective four-dimensional (4D) (inflationary) de Sitter expansion which describes an inflationary universe. In view of our results, we conclude that the condensate of spinors here studied could be an interesting candidate to explain the presence of dark energy in the early universe.