Curvature Coordinates Research Articles

A new class of traversable wormhole metrics

In this work, we have formulated a new class of traversable wormhole metrics. Initially, we have considered a wormhole metric in which the temporal component is an exponential function of r but the spatial components of the metrics are fixed. Following that, we have again constructed a generalized wormhole metric in which the spatial component is an exponential function of r, but the temporal component is fixed. Finally, we have considered the generalized wormhole metric in which both the temporal and spatial components are generalized exponential functions of r. We have also studied some of their properties including throat radius, stability, and energy conditions, examined singularity, the metric in curvature coordinates, effective refractive index, innermost stable circular orbit (ISCO) and photon sphere, Regge–Wheeler potential and their quasinormal modes, gravitational entropy, and determined the curvature tensor. The radius of the throat is found to be consistent with the properties of wormholes and does not contain any types of singularities. Most interestingly, we find that their throat radius is the same for the same spatial component and the same range of values of m. In addition to these, they also violate the Null Energy Condition (NEC) near the throat. These newly constructed metrics form a new class of traversable wormholes.

NEW SPHERICALLY SYMMETRIC ANISOTROPIC NEUTRON STAR MODEL USING FIELD EQUATIONS

In this paper, we obtain a new static spherically symmetric anisotropic fluid model of a neutron star in curvature coordinates. We consider , where , by taking the value of parameter in Durgapal solutions (Durgapal, 1982). The energy density, radial pressure, tangential pressure, and redshift are positive finite, and decreasing with respect to the increasing radius in this model. The pressure and density also decrease from the inner core to the crust of the object. We obtain this model by the specific choice of the constants and the anisotropy factor . The central value of the anisotropy factor is zero due to the perfect fluid nature at the center of the anisotropic model and it is increasing with the increasing radius. In our analysis we take and The model well behaves for the anisotropic neutron stars, which is shown analytically and graphically. The solution is free from any central singularity. We take realistic objects such as EXO 1785-248, SMC X-1, Her X-1, 4U 1538-52, LMC X-4, RX J1856-37, Cen X-3, PSR J1903-327, and Vela X-1 to represent our solutions graphically and numerically. We obtain this solution for for the first time. Therefore, the solution is quite new and derived for the first time.

Nonlinear vibrations of doubly curved micropanels reinforced by graphene nanoplates in length direction under non-uniform thermal loading

Structures made of composites are quite common in different industries entailing automotive, marine, and aircraft industries, thanks to their favorable lightweight and strength. This study investigated the nonlinear vibrations of the various shapes of size-dependent graphene nanoplatelets reinforced micropanels under non-uniform thermal loading. In diverse ways in the axial direction, the material characteristics of graphene nanoplates (GPLs) alter. Taylor’s series expansion terms in curvature coordinates are used to illustrate the displacement field. Nonlinear von Karman theory and Hamiltonian principle work together to derive the nonlinear form of the pertinent equations and boundary conditions. Strain gradient theory with consideration for inherent length scale characteristics is investigated to account for size effects. By discretizing the spatial domain equations, deriving Duffing-type equations, and using Kronecker and Hadamard products, the governing equations and numerous nonlinear boundary edges are resolved. Using the techniques and answers outlined above, the results section is produced. There are four sections for this research. The findings are cross-checked with those from other studies that have been published in the first section. The effects of geometric and physical factors are shown in Parts 2, 3, and 4 on the nonlinear to linear frequency curve, the nonlinear frequency curve, and the sensitivity of the nonlinear frequency-to-size effect (SNFSE).

Calculation of harrington function (desirability function) values under interval determination of its arguments

Purpose of work. Development of proposals for desirability function calculation by interval arithmetic method. Obtained results. A connection was shown between multiple-criterial (vector) optimization problem and calculation of Harrington function (desirability function) values. Generalized desirability function was shown to be treated as a multiplicative convolution of partial desirabilities. Using golden section properties, a connection was discerned between hierarchy analysis methods and calculation of desirability function values. In this case the distribution of probable desirability function values among intervals determined in accordance with golden mean rule does coincide with values obtained by expert method. Application of golden proportion enables increasing the number of intervals; thus, it may prove convenient in improvement of Harrington function sensitivity. Conclusions. For partial desirability function in its explicit form such geometric characteristics were determined as values of tangent, curvature of curve and center of curvature coordinates. For a necessary averaged desirability system obtaining process control, such characteristics were determined in general and explicit form as partial desirability elasticity, generalized desirability elasticity relative to partial desirability, generalized desirability elasticity relative to a variable that has physical sense and respective dimension depending on particular subject area and affects partial desirability value. To determine the boundary value of one partial desirability being substituted for another partial desirability, the rate of substitution function and boundary rate of substitution elasticity function were established. Due to errors emerging in determination of desirability function values application of interval analysis methods was proposed. Application of interval numbers as presented in hyperbolic form was shown for calcutaion of partial and generalized desirability, A partial power series sum was determined relative to a variable presented in hyperbolic form. Suggestions. The method of calculation of Harrington function (desirability function) values at interval determination of arguments was shown to be eventually used in development of specifications and preliminary design of complicated engineering аnd organizational systems.

New exact models of ideal gas in 5D EGB using curvature coordinates

Exploiting the use of curvature coordinates (also called Schwarzschild coordinates), new classes of exact solutions are discovered. By prescribing the timelike potential two new solutions are found one of which displays all the necessary qualitative features demanded of closed compact astrophysical objects. Since the analysis reduces to a single first order nonlinear differential equation, the single integration constant is obtained in terms of the mass and radius through matching of the interior and exterior metrics. Invoking the second matching condition results in constraining the value of the Gauss–Bonnet coupling constant in terms of the mass and radius of the star. Stability, causality and energy conditions are satisfied for a suitable choice of parameter space. In attempting to specify the radially oriented potential it was only possible to find a defective spacetime and to regain the interior Schwarzschild metric for an incompressible fluid sphere.

Local reparametrization by approximating lines of curvature

Although a surface parametrization by orthogonal curvilinear coordinates offers many benefits, ways to construct them are not immediate. The utility of lines of curvature for constructing such representations globally is severely limited by the presence of umbilical points, but a collection of local parametrizations may suffice in applications. Here we develop a practical method to obtain a local reparametrization of a piecewise parametric surface model by consistently accurate lines of curvature coordinates. Our analysis of possible instabilities occurring at an isolated umbilical point indicates that a slight modification of the original patches may suffice to ensure that the configuration of the reparametrized surface always stays regular. We give explicit representations of differential geometric quantities which are particularly useful in treating finite element formulations defined on surfaces.

Applying spline interpolation to increase accuracy of correlation-emergency navigation systems

Relevance. Spline interpolation is used to improve the accuracy of correlation-extreme navigation systems. A two-stage algorithm for combining images in correlation-extreme navigation systems is proposed. At the first stage, the surface of the decision function of the algorithm is constructed in the vicinity of its extremum using a quadratic interpolator by six points and its Gaussian curvature and extremum coordinates are estimated. These parameters are used to determine the optimal value of the parameter of the cubic spline interpolator used in the second stage in order to refine the rough estimate of the coordinates and improve the positioning accuracy of the navigation system. Purpose of the work: The purpose of the work is to develop an algorithm for aligning images in correlation-extreme navigation systems, which makes it possible to realize a cubic spline parameter close to the optimal value for each of the possible shifts of the current image relative to the reference image and, as a result, to increase the accuracy of determining the coordinates. Materials and methods. In correlation-extreme navigation systems, the coordinates of the aircraft are determined by calculating the mutual shift of the current image obtained using the sensor of the Earth's physical field and the reference image, which is known in advance. At the same time, the alignment accuracy of discrete current and reference images, which are usually used in practice, does not exceed half a pixel. Therefore, the problem of improving the accuracy of navigation systems is of great importance. One of the possible ways to solve this problem is to use methods for approximating the decision function of the image alignment algorithm in the vicinity of its global maximum.Results: To illustrate the gain in the accuracy of the positioning of navigation systems, statistical tests of the algorithm with a 6-point interpolator and the above-described two-stage procedure for minimizing the decision function containing spline interpolation at the second stage were carried out. A typical image was used as a reference image. The coordinates of the center of the current and reference images were played randomly in accordance with the two-dimensional normal distribution law, the average value of which coincided with the center of the reference image; the standard deviation is also found. Then the current image was formed. The constructed current image was noisy with additive white Gaussian noise with zero mean value and the same standard deviation for each element . Image alignment was assumed to be correct if the following conditions were met: , where – is the shift estimate generated by the algorithm. Then, the algorithms were repeatedly run with different realizations of the noise component of the current image, and the dependences of the root-mean-square error in each direction on the mean-square value were plotted . The figures in the article show the dependencies for the algorithm with a 6-point interpolator (upper curve) and for a two-stage algorithm (lower curve). Analysis of the graphs allows us to conclude that the second algorithm wins in the accuracy of determining the coordinates of the shift by about 5 times. The dependencies for both algorithms practically coincide with those shown in the figure. It should be noted the weak dependence of the positioning accuracy on the change in the parameter in the area . Conclusions: It is shown that the optimal value of the parameter of the cubic spline interpolator depends to a lesser extent on the magnitude of the local shift of the images and, to a greater extent, on the correlation interval of the reference image in the vicinity of the image alignment point, which is proposed to be estimated using the Gaussian curvature parameter.

Singularities in Static Spherically Symmetric Configurations of General Relativity with Strongly Nonlinear Scalar Fields

There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly nonlinear scalar field, which allow the appearance of singularities of a new type (“spherical singularities”) outside the center of curvature coordinates. As the example, we consider a scalar field potential ∼sinh(ϕ2n),n>2, which grows rapidly for large field values. The space-time is assumed to be asymptotically flat. We fulfill a numerical investigation of solutions with different n for different parameters, which define asymptotic properties at spatial infinity. Depending on the configuration parameters, we show that the distribution of the stable circular orbits of test bodies around the configuration is either similar to that in the case of the Schwarzschild solution (thus mimicking an ordinary black hole), or it contains additional rings of unstable orbits.

Construction of Developable Surface with Geodesic or Line of Curvature Coordinates

In this paper, a developable surface with geodesic or line of curvature coordinates is constructed in the Euclidean 3-space. A developable surface is coordinated by two families of parametric curves, base curves (directrices) and lines (rulings). Since any part of a straight line on a developable surface is geodesic and line of curvature, we only need to show that the directrices curves are geodesics or lines of curvature to ensure that the developable surface is parameterized by geodesic or line of curvature coordinates. The necessary and sufficient conditions for the directrices curves to be geodesics or lines of curvature are studied. The main results of this paper show that the developable surface with geodesic coordinates is a generalized cylinder, and the developable surface with line of curvature coordinates is a tangent surface.

Spherically symmetric, static black holes with scalar hair, and naked singularities in nonminimally coupled k -essence

We apply a recently developed 2+1+1 decomposition of spacetime, based on a nonorthogonal double foliation for the study of spherically symmetric, static black hole solutions of Horndeski scalar-tensor theory. Our discussion proceeds in an effective field theory (EFT) of modified gravity approach, with the action depending on metric and embedding scalars adapted to the nonorthogonal 2+1+1 decomposition. We prove that the most generic class of Horndeski Lagrangians compatible with observations can be expressed in this EFT form. By studying the first order perturbation of the EFT action we derive three equations of motion, which reduce to those derived earlier in an orthogonal 2+1+1 decomposition, and a fourth equation for the metric parameter N related to the nonorthogonality of the foliation. For the Horndeski class of theories with vanishing $G_3$ and $G_5$, but generic functions $G_2(\phi,X)$ (k-essence) and $G_4(\phi)$ (nonminimal coupling to the metric) we prove the unicity theorem that no action beyond Einstein--Hilbert allows for the Schwarzschild solution. Next we integrate the EFT field equations for the case with only one independent metric function obtaining new solutions characterized by a parameter interpreted as either mass or tidal charge, the cosmological constant and a third parameter. These solutions represent naked singularities, black holes with scalar hair or have the double horizon structure of the Schwarzschild--de Sitter spacetime. Solutions with homogeneous Kantowski--Sachs type regions also emerge. Finally, one of the solutions obtained for the function $G_4$ linear in the curvature coordinate, in certain parameter range exhibits an intriguing logarithmic singularity lying outside the horizon. The newly derived hairy black hole solutions evade previously known unicity theorems by being asymptotically nonflat, even in the absence of the cosmological constant.

On Discrete Conjugate Semi-Geodesic Nets

Abstract We introduce two canonical discretizations of nets on generic surfaces, which consist of an one-parameter family of geodesics and its associated family of conjugate lines. The two types of “discrete conjugate semi-geodesic nets” constitute discrete focal nets of circular nets, which mimics the classical connection between surfaces parametrized in terms of curvature coordinates and their focal surfaces on which one corresponding family of coordinate lines are geodesics. The discrete surfaces constructed in this manner are termed circular-geodesic and conical-geodesic nets, respectively, and may be characterized by compact angle conditions. Geometrically, circular-geodesic nets are constructed via normal lines of circular nets, while conical-geodesic nets inherit their name from their intimately related conical nets, which also discretize curvature line parametrized surfaces. We establish a variety of properties of these discrete nets, including their algebraic and geometric 3D consistency, with the latter playing an important role in (integrable) discrete differential geometry.

Coordinate effect: Vaidya solutions without integrating the field equations

We extend Vaidya’s algorithm for the description of a central mass losing or gaining energy due to electromagnetic-type radiation (‘null dust’) to the case of arbitrary radial corpuscular radiation. We also demonstrate the remarkable possibility of purely algebraic deduction of the Vaidya solution without integrating the field equations, and interpret this possibility as an artifact of curvature coordinates. Since Vaidya’s approach by itself cannot lead to certain dependence of mass on spacetime coordinates, the search for a corresponding mass-function represents an independent issue. In this regard, as a perspective, we discuss an outlook on the problem of variable masses as a whole.

Path Planning of Mobile Robot With Improved Ant Colony Algorithm and MDP to Produce Smooth Trajectory in Grid-Based Environment.

This approach has been derived mainly to improve quality and efficiency of global path planning for a mobile robot with unknown static obstacle avoidance features in grid-based environment. The quality of the global path in terms of smoothness, path consistency and safety can affect the autonomous behavior of a robot. In this paper, the efficiency of Ant Colony Optimization (ACO) algorithm has improved with additional assistance of A* Multi-Directional algorithm. In the first part, A* Multi-directional algorithm starts to search in map and stores the best nodes area between start and destination with optimal heuristic value and that area of nodes has been chosen for path search by ACO to avoid blind search at initial iterations. The path obtained in grid-based environment consist of points in Cartesian coordinates connected through line segments with sharp bends. Therefore, Markov Decision Process (MDP) trajectory evaluation model is introduced with a novel reward policy to filter and reduce the sharpness in global path generated in grid environment. With arc-length parameterization, a curvilinear smooth route has been generated among filtered waypoints and produces consistency and smoothness in the global path. To achieve a comfort drive and safety for robot, lateral and longitudinal control has been utilized to form a set of optimal trajectories along the reference route, as well as, minimizing total cost. The total cost includes curvature, lateral and longitudinal coordinates constraints. Additionally, for collision detection, at every step the set of optimal local trajectories have been checked for any unexpected obstacle. The results have been verified through simulations in MATLAB compared with previous global path planning algorithms to differentiate the efficiency and quality of derived approach in different constraint environments.

Relativistic compact stars in the Kuchowicz space-time

We present an anisotropic charged analogue of Kuchowicz (1971) solution of the general relativistic field equations in curvature coordinates by using simple form of electric intensity $E$ and pressure anisotropy factor $\Delta$ that involve charge parameter $K$ and anisotropy parameter $\alpha$ respectively. Our solution is well behaved in all respects for all values of $X$ ( $X$ is related to the radius of the star ) lying in the range $0< X \le 0.6$, $\alpha$ lying in the range $0 \le \alpha \le 1.3$, $K$ lying in the range $0< K \le 1.75$ and Schwarzschild compactness parameter "$u$" lying in the range $0< u \le 0.338$. Since our solution is well behaved for a wide range of the parameters, we can model many different types of ultra-cold compact stars like quark stars and neutron stars. We present some models of super dense quark stars and neutron stars corresponding to $X=0.2,~\alpha=0.2$ and $K=0.5$ for which $u_{max}=0.15$. By assuming surface density $\rho_b=4.6888\times 10^{14}~ g/cc$ the mass and radius are $0.955 M_\odot$ and $9.439 km$ respectively. For $\rho_b=2.7\times 10^{14}~ g/cc$ the mass and radius are $1.259 M_\odot$ and $12.439 km$ respectively and for $\rho_b=2\times 10^{14}~ g/cc$ the mass and radius are $1.463 M_\odot$ and $14.453 km$ respectively. It is also shown that inclusion of more electric charge and anisotropy enhances the static stable configuration under radial perturbations. The $M-R$ graph suggests that the maximum mass of the configuration depends on the surface density {\bf i.e. with the increase of surface density} the maximum mass and corresponding radius decrease. This may be because of existence of exotic matters at higher densities that soften the EoSs.

Static spherically symmetric configurations with N nonlinear scalar fields: Global and asymptotic properties

In case of a spherically symmetric non-linear scalar field (SF) in flat space, besides singularity at the center, spherical singularities can occur for non-zero values of radial variable $r>0$. We show that in the General Relativity the gravitational field suppresses the occurrence of the spherical singularities under some generic conditions. Our consideration deals with asymptotically flat space-times around static spherically symmetric configurations in presence of $N$ non-linear SFs, which are minimally coupled to gravity. Constraints are imposed on the SF potentials, which guarantee a monotonicity of the fields as functions of radial variable; also the potentials are assumed to be exponentially bounded. We give direct proof that solutions of the joint system of Einstein -- SF equations satisfying the conditions of asymptotic flatness are regular for all values of $r$, except for naked singularities in the center $r=0$ in the Schwarzschild (curvature) coordinates. Asymptotic relations for SF and metric near the center are derived, which appear to be remarkably similar to the case of the Fisher solution for free SF. These relations determine two main types of the corresponding geodesic structure when photons can be captured by the singularity or not depending on the existence of the photon sphere. To illustrate, the case of one SF with monomial potential is analyzed in detail numerically. We show that the image of the accretion disk around the singularity, observed from infinity, can take the form of a bright ring with a dark spot in the center, like the case of an ordinary black hole.

Aero-heating modelling on the ablative noses during flight trajectory

PurposeThe purpose of this paper is to present the more accurate estimation of aero-heating for the ablative 3-D noses by using the viscous shock layers and similarity of viscous boundary layer methods.Design/methodology/approachThe combination of viscous shock layer, similarity of viscous boundary layer (SVBL) methods, Park ablation and Baldwin–Lomax turbulent models is presented in this paper. The proposed method reduces computational memory and run time as compared to the time marching algorithms during flight trajectory. Therefore, the space marching algorithm and finite difference method is used, and the governing equations are transferred into curvature coordinate by using the mapping terms.FindingsThe solving for an ogive nose during flight trajectory shows that the convergence of this technique is fast as compared to the user defined function based on the fluent solvers, program to axisymmetric regular geometry code and other research. The results of this research are validated by the mentioned research studies. The relative error for the aero-heating, species concentration of the shock layer gas mixture because of dissociation/ionization of air and surface ablation results is less than 6, 5 and 11 per cent, respectively.Research limitations/implicationsThe required time for an aerodynamic design of hypersonic noses reduces as the induced aero-heating is one of the principal design parameters in standpoint aerodynamic, structural and other terms. The magnitude of this parameter, surface temperature and surface recess because of ablation should be corrected during flight trajectory.Social implicationsThe results of this research are applicable for aerospace industries.Originality/valueThe originality of this paper is 90 per cent.

A Comparative Study of Generic Visual Components of Two-Dimensional Versus Three-Dimensional Laparoscopic Images

AimsThere is a strong evidence to suggest that 3D imaging improves the laparoscopic task performance when compared against 2D. However, to date, no study has explained why that might be. We identified six generic visual components during laparoscopic imaging and aimed to study each component in both 2D and 3D environments for comparison.MethodsTwenty-four consented laparoscopic novices performed specific isolated tasks in a laparoscopic Endo Trainer in 2D and 3D separately. The six endpoints were the accuracy in detecting changes in the laparoscopic images in the following components: distance, area, angle, curvature, volume and spatial coordinates. All the components except the spatial coordinates were assessed by creation, measurement and comparison. Each component was analysed between 2D and 3D groups and within each group at different values. Tests of spatial coordinates were video-recorded and analysed for error number and error types by human reliability analysis technique. Errors types included past-pointing, not reaching the object and touching the wrong object. The results were statistically analysed with independent T test.ResultsThere was no statistically significant difference between 2D and 3D accuracy in the angle, area, distance and curvature. 3D performed more accurately in comparing volumes (p = 0.05). In spatial coordinates, there were a statistically significant higher number of errors in 2D as compared to 3D (p < 0.001). Past-pointing and touching the wrong objects were significantly higher in 2D (p < 0.05).ConclusionBetween all the visual components, detecting change in volume and the spatial coordinates showed significant improvement in 3D environment when compared to 2D.

Canonical Chern-Simons gravity

We study the canonical description of the axisymmetric vacuum in 2+1 dimensional gravity, treating Einstein's gravity as a Chern Simons gauge theory on a manifold with the restriction that the dreibein is invertible. Our treatment is in the spirit of Kucha\v r's description of the Schwarzschild black hole in 3+1 dimensions, where the mass and angular momentum are expressed in terms of the canonical variables and a series of canonical transformations are performed that turn the curvature coordinates and their conjugate momenta into new canonical variables. In their final form, the constraints are seen to require that the momenta conjugate to the Killing time and curvature radius vanish and what remains are the mass, the angular momentum and their conjugate momenta, which we derive. The Wheeler-DeWitt equation is trivial and describes time independent systems with wave functions described only by the total mass and total angular momentum.

Curvature coordinates to describe the explosion of Chernobyl’s reactor core in April 1986, using the tensor equations

This research analyzes the process of the explosion of the reactor core of Chernobyl nuclear plant in April 1986, using the tensor equations. These tensor equations show a movement of a vector in the three dimensional curvature coordinates of time, water flow, and void. The equations shows that this vector moves along the geodesic in the curvature coordinates, which is described by fundamental tensor ( g µν ), Christoffel symbol (Γ α µνσ ) and Ricci tensor ( R µν ), where µ, ν, σ, α are suffixes that indicate the coordinates. The solution of the tensor equations shows that the geodesic of the vector has a singular point, which describes a turning point of the reactor core from the normal operation to the explosion, which we reported in our previous articles [1, 2].

All static charged dust spheres in general relativity

Exact solutions for distributions of static charged dust in general relativity are sought in a systematic way in curvature coordinates. An algorithm is proposed to identify all dust spheres subject to carrying out a single integration after specifying a gravitational potential. It is argued that charged dust distributions are described by metrics that are inherently singular. The algorithm is illustrated for a simple case that generalises several known cases treated historically. Further new dust models are generated by specifying the other gravitational potential and well known metric Ansätze, namely, the generalised Schwarzschild and Finch–Skea prescriptions are treated.