Curvature Theory Research Articles

Infrared scaling for a graviton condensate

The coupling between gravity and matter provides an intriguing length scale in the infrared for theories of gravity within Einstein-Hilbert action and beyond. In particular, we will show that such an infrared length scale is determined by the number of gravitons Ng≫1 associated to a given mass in the non-relativistic limit. After tracing out the matter degrees of freedom, the graviton vacuum is found to be in a displaced vacuum with an occupation number of gravitons Ng≫1. In the infrared, the length scale appears to be L=Ngℓp, where L is the new infrared length scale, and ℓp is the Planck length. In a specific example, we have found that the infrared length scale is greater than the Schwarzschild radius for a slowly moving in-falling thin shell of matter. We will argue that the appearance of such an infrared length scale in higher curvature theories of gravity, such as in quadratic and cubic curvature theories of gravity, is also expected. Furthermore, we will show that gravity is fundamentally different from the electromagnetic interaction where the number of photons, Np, is the fine structure constant after tracing out an electron wave function.

Novel Methodology for Inflection Circle-Based Synthesis of Straight Line Crank Rocker Mechanism

Abstract Emerging fields like Compact Compliant Mechanisms have created newer/novel situations for application of straight line mechanisms. Many of these situations in Automation and Robotics are multidisciplinary in nature. Application Engineers from these domains are many times uninitiated in involved procedures of synthesis of mechanisms and related concepts of Path Curvature Theory. This paper proposes a predominantly graphical approach using properties of Inflection Circle (IC) to synthesize a crank rocker mechanism for tracing a coupler curve which includes the targeted straight line path. The generated approximate straight line path has acceptable deviation in length, orientation, and extent of approximate nature well within the permissible ranges. Generation of multiple choices for the link geometry is unique to this method. To ease the selection, a trained artificial neural network (ANN) is developed to indicate relative length of various options generated. Using studied unique properties of Inflection Circles a methodology for anticipating the orientation of the straight path vis-à-vis the targeted path is also included. Two straight line paths are targeted for two different crank rockers. Compared with the existing practice of selecting the mechanism with some compromise due to inherent granularity of the data in Atlases, proposed methodology helps in indicating the possibility of completing the dimensional synthesis. The developed solution is well within the design specifications and is without a compromise.

Chiral Anomaly in Non-Relativistic Systems: Berry Curvature and Chiral Kinetic Theory

Chiral anomaly and the novel quantum phenomena it induces have been widely studied for Dirac and Weyl fermions. In most typical cases, the Lorentz covariance is assumed and thus the linear dispersion relations are maintained. However, in realistic materials, such as Dirac and Weyl semimetals, the nonlinear dispersion relations appear naturally. We develop a kinetic framework to study the chiral anomaly for Weyl fermions with nonlinear dispersions using the methods of Wigner function and semi-classical equations of motion. In this framework, the chiral anomaly is sourced by Berry monopoles in momentum space and could be enhanced or suppressed due to the windings around the Berry monopoles. Our results can help understand the chiral anomaly-induced transport phenomena in non-relativistic systems.

Spherically symmetric black holes in the limiting curvature theory of gravity

In this paper we describe a model of a four-dimensional spherically symmetric black hole in a limiting curvature theory of gravity. In this theory the Einstein-Hilbert action is modified by adding to the action terms providing inequality constraints on chosen curvature invariants. We demonstrated that in such a model, instead of formation of the curvature singularity, the spacetime remains regular in the black hole interior. For black holes with gravitational radius much larger than the radius $\ensuremath{\ell}$ of the critical curvature the obtained solutions describe a space exponentially expanding in one direction and periodically oscillating in the other two (spherical) directions.

New symmetry in higher curvature spacetimes

New symmetries have been found in Einstein-Maxwell spacetimes. New symmetries have also been found in imperfect fluid curved spacetimes. We will prove in this paper that we can extend these symmetries to spacetimes with higher curvature terms. Higher curvature theories are in many cases associated to dark energy for instance. We provide further justification for these higher curvature formulations through the existence of a new symmetry.

Feature selection method based on Menger curvature and LDA theory for a P300 brain–computer interface

Objective. Brain–computer interface (BCI) systems decode electroencephalogram (EEG) signals to establish a channel for direct interaction between the human brain and the external world without the need for muscle or nerve control. The P300 speller, one of the most widely used BCI applications, presents a selection of characters to the user and performs character recognition by identifying P300 event-related potentials from the EEG. Such P300-based BCI systems can reach good levels of accuracy but are difficult to use in day-to-day life due to redundancy and noisy signal. A room for improvement should be considered. We propose a novel hybrid feature selection method for the P300-based BCI system to address the problem of feature redundancy, which combines the Menger curvature and linear discriminant analysis. Approach. First, selected strategies are applied separately to a given dataset to estimate the gain for application to each feature. Then, each generated value set is ranked in descending order and judged by a predefined criterion to be suitable in classification models. The intersection of the two approaches is then evaluated to identify an optimal feature subset. Main results. The proposed method is evaluated using three public datasets, i.e. BCI Competition III dataset II, brain/neural computer interaction Horizon dataset, and Lausanne Federal Institute of Technology dataset. Experimental results indicate that compared with other typical feature selection and classification methods, our proposed method has better or comparable performance. Significance. Additionally, our proposed method can achieve the best classification accuracy after all epochs in three datasets. In summary, our proposed method provides a new way to enhance the performance of the P300-based BCI speller.

Bouncing cosmology in the limiting curvature theory of gravity

In this paper we discuss models satisfying the limiting curvature condition. For this purpose we modify the Einstein-Hilbert action by adding a term which restricts the growth of curvature. We analyze cosmological solutions in such models. Namely, we consider a closed contracting homogeneous isotropic universe filled with thermal radiation. We demonstrate that for properly chosen curvature constraints such a universe has a bounce. As a result its evolution is nonsingular and contains a "de Sitter-type" supercritical stage connecting contracting and expanding phases. Possible generalizations of these results are briefly discussed.

Wormhole solutions with NUT charge in higher curvature theories

We present wormholes with a Newman–Unti–Tamburino (NUT) charge that arise in certain higher curvature theories, where a scalar field is coupled to a higher curvature invariant. For the invariants we employ (i) a Gauss–Bonnet term and (ii) a Chern–Simons term, which then act as source terms for the scalar field. We map out the domain of existence of wormhole solutions by varying the coupling parameter and the scalar charge for a set of fixed values of the NUT charge. The domain of existence for a given NUT charge is then delimited by the set of scalarized nutty black holes, a set of wormhole solutions with a degenerate throat and a set of singular solutions.

Universal renormalization procedure for higher curvature gravities in D \u2264 5

We implement a universal method for renormalizing AdS gravity actions applicable to arbitrary higher curvature theories in up to five dimensions. The renormalization procedure considers the extrinsic counterterm for Einstein-AdS gravity given by the Kounterterms scheme, but with a theory-dependent coupling constant that is fixed by the requirement of renormalization for the vacuum solution. This method is shown to work for a generic higher curvature gravity with arbitrary couplings except for a zero measure subset, which includes well-known examples where the asymptotic behavior is modified and the AdS vacua are degenerate, such as Chern-Simons gravity in 5D, Conformal Gravity in 4D and New Massive Gravity in 3D. In order to show the universality of the scheme, we perform a decomposition of the equations of motion into their normal and tangential components with respect to the Poincare coordinate and study the Fefferman-Graham expansion of the metric. We verify the cancellation of divergences of the on-shell action and the well-posedness of the variational principle.

Offset Trajectory Planning of Robot End Effector and its Jerk with Curvature Theory

Some situations that change the parameters of the kinematic structure may cause the robot end effector to deviate [Merlet [2005] Parallel Robots, Vol. 128 (Springer Science & Business Media, Germany)] from the desired trajectory. This effect is called the robustness of the robot by Merlet. One of the ways to correct the robustness is by updating the robot trajectory. The jerk vector of the robot end effector is the third-order positional variation of the TCP and defined as thus the time derivative of the acceleration vector. If there is a high curvature on the transition curve trajectory of robot, then there is a tangential jerk along the trajectory. In this study, the geometrically offset trajectory of the robot end effector from the current trajectory was obtained by using the curvature theory. The angular velocity and angular acceleration of the offset trajectory were calculated. An example of the main trajectory of robot end effector and its offset is given. Also, the jerk of the robot end effector of the offset trajectory was calculated according to the curvature of the trajectory surface in case of a jerk problem caused by a high curvature in the transition curve along the offset trajectory curve.

Two-dimensional black holes in the limiting curvature theory of gravity

In this paper we discuss modified gravity models in which growth of the curvature is dynamically restricted. To illustrate interesting features of such models we consider a modification of two-dimensional dilaton gravity theory which satisfies the limiting curvature condition. We show that such a model describes two-dimensional black holes which contain the de Sitter-like core instead of the singularity of the original non-modified theory. In the second part of the paper we study Vaidya type solutions of the model of the limiting curvature theory of gravity and used them to analyse properties of black holes which are created by the collapse of null fluid. We also apply these solutions to study interesting features of a black hole evaporation.

Improved gravitational-wave constraints on higher-order curvature theories of gravity

Gravitational wave observations of compact binaries allow us to test general relativity (and modifications thereof) in the strong and highly dynamical field regime of gravity. Here, we confront two extensions to general relativity, dynamical Chern-Simons, and Einstein-dilaton-Gauss-Bonnet theories, against the gravitational wave sources from the GWTC-1 and GWTC-2 catalogs by the LIGO-Virgo Collaboration. By stacking the posterior of individual events, we strengthen the constraint on the square root of the coupling parameter in Einstein-dilaton-Gauss-Bonnet gravity to $\sqrt{{\ensuremath{\alpha}}_{\mathrm{EdGB}}}<1.7\text{ }\text{ }\mathrm{km}$, but we are unable to place meaningful constraints on dynamical Chern-Simons gravity. Importantly, we also show that our bounds are robust to (i) the choice of general-relativity base waveform model, upon which we add modifications, (ii) unknown higher post-Newtonian order terms in the modifications to general relativity, (iii) the small-coupling approximation, and (iv) uncertainties on the nature of the constituent compact objects.

Analytic charged BHs in f(R) gravity

In this article, we seek exact charged spherically symmetric black holes (BHs) with considering f(R) gravitational theory. This BH is characterized by convolution and error functions. Those two functions depend on a constant of integration which is responsible to make such a solution deviate from Einstein's general relativity (GR). The error function which constitutes the charge potential of the Maxwell field depends on the constant of integration and when this constant is vanishing we can not reproduce Reissner-Nordström BH in the lower order of f(R). This means that we can not reproduce Reissner-Nordström BH in lower-order–curvature theory, i.e., in GR limit f(R)=R, we can not get the well known charged BH. We study the physical properties of these BHs and show that it is asymptotically approached as a flat spacetime or approach AdS/dS spacetime. Also, we calculate the invariants of the BHS and show that the singularities are milder than those of BH's of GR. Additionally, we derive the stability condition through the use of geodesic deviation. Moreover, we study the thermodynamics of our BH and show the effect of the higher-order–curvature theory. Finally, we study the stability analysis using the odd-type mode and show that all the BHs are stable and have radial speed equal to one.

A study of curvature theory for different symmetry classes of Hamiltonian

We study and present the results of curvature for different symmetry classes (BDI, AIII and A) of model Hamiltonians and also present the transformation of model Hamiltonian from one distinct symmetry class to the other based on the curvature property. We observe the mirror symmetric curvature for the Hamiltonian with BDI symmetry class but there is no evidence of such behaviour for Hamiltonians of AIII symmetry class. We show the origin of torsion and its consequences on the parameter space of topological phase of the system. We find the evidence of torsion for the Hamiltonian of A symmetry class. We present Serret–Frenet equations for all model Hamiltonians in \(\mathbf {R}^3\) space. To the best of our knowledge, this is the first application of curvature theory to the model Hamiltonian of different symmetry classes which belong to the topological state of matter.

The kinematics of curved profiles mating with a caged idle roller - higher-path curvature analysis

This investigation focuses on the relative motion of a force-closure higher pair mechanism, herein termed caged-roller joint and composed of a cylindrical roller meshing with two slot profiles, carved on different bodies. A distinctive feature of such a mechanism is the pure rolling velocity constraint imposed between its components to limit friction losses. This makes the caged-roller joint an ideal candidate to control the kinematics of high-speed operating devices, such as centrifugal pendulum vibration absorbers.In this paper, a theoretical framework for the motion of caged-roller joint is proposed, employing intrinsic geometry, higher-path curvature and envelope theory. For design-analysis purposes, the correlations between instantaneous invariants, slots profiles curvature properties and pure rolling conditions within the higher pairs have been established. Eventually, the obtained results have been verified for three particular curves profiles, to test their consistency and offer an immediate application.

Anisotropic compact stars in higher-order curvature theory

In this paper we shall consider spherically symmetric spacetime solutions describing the interior of stellar compact objects, in the context of higher-order curvature theory of the {{mathrm {f(R)}}} type. We shall derive the non-vacuum field equations of the higher-order curvature theory, without assuming any specific form of the {{mathrm {f(R)}}} theory, specifying the analysis for a spherically symmetric spacetime with two unknown functions. We obtain a system of highly non-linear differential equations, which consists of four differential equations with six unknown functions. To solve such a system, we assume a specific form of metric potentials, using the Krori–Barua ansatz. We successfully solve the system of differential equations, and we derive all the components of the energy–momentum tensor. Moreover, we derive the non-trivial general form of {{mathrm {f(R)}}} that may generate such solutions and calculate the dynamic Ricci scalar of the anisotropic star. Accordingly, we calculate the asymptotic form of the function {mathrm {f(R)}}, which is a polynomial function. We match the derived interior solution with the exterior one, which was derived in [1], with the latter also resulting to a non-trivial form of the Ricci scalar. Notably but rather expected, the exterior solution differs from the Schwarzschild one in the context of general relativity. The matching procedure will eventually relate two constants with the mass and radius of the compact stellar object. We list the necessary conditions that any compact anisotropic star must satisfy and explain in detail that our model bypasses all of these conditions for a special compact star textit{Her X--1}, which has an estimated mass and radius (mass = 0.85 pm 0.15M_{circledcirc } and radius = 8.1 pm 0.41~text {km}). Moreover, we study the stability of this model by using the Tolman–Oppenheimer–Volkoff equation and adiabatic index, and we show that the considered model is different and more stable compared to the corresponding models in the context of general relativity.

Perturbative S-matrix unitarity (S†S = 1) in Rμν2 gravity

We show that in the quadratic curvature theory of gravity, or simply [Formula: see text] gravity, the tree-level unitarity bound (tree unitarity) is violated in the UV region but an analog for [Formula: see text]-matrix unitarity [Formula: see text] is satisfied. This theory is renormalizable, and hence the failure of tree unitarity is a counter example of Llewellyn Smith’s conjecture on the relation between them. We have recently proposed a new conjecture that [Formula: see text]-matrix unitarity gives the same conditions as renormalizability. We verify that [Formula: see text]-matrix unitarity holds in the matter-graviton scattering at the tree level in the [Formula: see text] gravity, demonstrating our new conjecture.

Measuring curvature of trajectory traced by coupler of an optimal four-link spherical mechanism

This research paper presents design of a gripper finger mechanism and a new curvature measurement method of fingertip trajectory. Firstly, the conditions of a grasp provided by gripper, which was designed from a spherical mechanism, was explained. Therefore, trajectory length of coupler point was modeled as objective function which is maximized. Ball-Burmester theory was used to form constraints which can provide a spherical curve with curvature close to zero. A plausible dimension range of mechanism, transmission angle, instantaneous invariants were arranged as constraint function. A meta-heuristic algorithm previously developed by one of authors was used to search the best instantaneous invariants based on the type of mechanism. The optimal solution was prototyped, and three-fingered gripper assembled. Secondly, an endoscopic camera peculiarly was used to measure curvature of the trajectory traced by coupler point of an optimal four-link spherical mechanism which was supposed to compose a finger of a robotic gripper. The coupler of the mechanism was expected to trace approximately a great circle arc to provide a grasp. Therefore, the technique to measure curvature was based on image processing and geodetic curvature theory. A calibrated endoscopic camera was used for acquisition of video record above the prototype which was centered in the center of the camera screen. The curvature of coupler point of optimal spherical mechanism design and prototype were compared. According to performed regression analysis, results of the measurement were %97 reliable for the useful part of the trajectory.

Reconstructing wormhole solutions in curvature based Extended Theories of Gravity

Static and spherically symmetric wormhole solutions can be reconstructed in the framework of curvature based Extended Theories of Gravity. In particular, extensions of the General Relativity, in metric and curvature formalism give rise to modified gravitational potentials, constituted by the classical Newtonian potential and Yukawa-like corrections, whose parameters can be, in turn, gauged by the observations. Such an approach allows to reconstruct the spacetime out of the wormhole throat considering the asymptotic flatness as a physical property for the related gravitational field. Such an argument can be applied for a large class of curvature theories characterising the wormholes through the parameters of the potentials. According to this procedure, possible wormhole solutions could be observationally constrained. On the other hand, stable and traversable wormholes could be a direct probe for this class of Extended Theories of Gravity.

Quantum extremal islands made easy. Part III. Complexity on the brane

We examine holographic complexity in the doubly holographic model introduced in [1, 2] to study quantum extremal islands. We focus on the holographic complexity=volume (CV) proposal for boundary subregions in the island phase. Exploiting the Fefferman-Graham expansion of the metric and other geometric quantities near the brane, we derive the leading contributions to the complexity and interpret these in terms of the generalized volume of the island derived from the induced higher-curvature gravity action on the brane. Motivated by these results, we propose a generalization of the CV proposal for higher curvature theories of gravity. Further, we provide two consistency checks of our proposal by studying Gauss-Bonnet gravity and f(ℛ) gravity in the bulk.