Linking Conditions for Models with Geometrical Basis

First Page

In general theory of relativity, Einstein’s field equations relate the geometry of space-time with the distribution of matter within it. These equations were first published by Einstein in the form of a tensor equation which related the local space-time curvature with the local energy and momentum within this space-time. In this article, Einstein’s geometrical field equations interior and exterior to astrophysically real or hypothetical distribution of mass within a spherical geometry were constructed and solved for field whose gravitational potential varies with time, radial distance and polar angle. The exterior solution was obtained using power series. The metric tensors and the solution of the Einstein’s exterior field equations used in this work has only one arbitrary function f(t,r,θ) , and thus put the Einstein’s geometrical theory of gravitation on the same bases with the Newton’s dynamical theory of gravitation. The gravitational scalar potential f(t,r,θ) obtained in this research work to the order of co, c-2 , contains Newton dynamical gravitational scalar potential and post Newtonian additional terms much importance as it can be applied to the study of rotating bodies such as stars. The interior solution was obtained using weak field and slow-motion approximation. The obtained result converges to Newton’s dynamical scalar potential with additional time factor not found in the well-known Newton’s dynamical theory of gravitation which is a profound discovery with the dependency on three arbitrary functions. Our result obeyed the equivalence principle of Physics.

We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be thought as a simple star model: a self-gravitating perfect fluid ball with a differential rotation motion pattern. Using the post--Minkowskian formalism (weak field approximation) and considering rotation as a perturbation (slow rotation approximation), we find approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic coordinates. Interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to get a global solution. The resulting metric depends on four arbitrary constants: mass density, rotational velocity at $r=0$, a parameter which takes into account of the change in the rotational velocity through the star and the star radius in the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined in terms of these four parameters.

In veiw of a possible relevance of the concept of the C-field introduced by Hoyle and Narlikar to the problem of quasi-stellar objects or collapsing stars, an attempt is made to study permissible solutions of their field equations in an empty region with spherical symmetry. Two special, i.e. static and pseudo-staticl, solutions are obtained and their characteristic properties are examined in the light of Hoyle-Narlikar's exterior but non-empty solution. As a result, it is revealed that the problem of boundary conditions has a crucial importance in the decision of superiority of either one among the three exterior solutions.

Approximate Kerr- like interior and exterior solutions for a very slowly rotating star of perfect fluid is presented here which is valid for a very slowly rotating and hence very slightly oblate star of perfect fluid rotating with a very small constant angular velocity. Corresponding approximate exterior metric represents approximate Kerr exterior metric for a perfect fluid. The aim here is to obtain Kerr-like approximate exterior and interior solutions for a very slowly rotating star of perfect fluid simply following standard procedure used to obtain Schwarzschild solutions. Key words: Slowly rotating oblate star, approximate Kerr- like metric, exterior interior solution.

A reformulation of the Schwarzschild solution of the linearized Einstein field equations in post-Riemannian Finsler spacetime is derived. The solution is constructed in three stages: the exterior solution, the event-horizon solution and the interior solution. It is shown that the exterior solution is asymptotically similar to Newtonian gravity at large distances implying that Newtonian gravity is a low energy approximation of the solution. Application of Eddington-Finklestein coordinates is shown to reproduce the results obtained from standard general relativity at the event horizon. Further application of Kruskal-Szekeres coordinates reveals that the interior solution contains maximally extensible geodesics.

A class of solutions of Einstein field equations satisfying Karmarkar embedding condition is presented which could describe static, spherical fluid configurations, and could serve as models for compact stars. The fluid under consideration has unequal principal stresses i.e. fluid is locally anisotropic. A certain physically motivated geometry of metric potential has been chosen and codependency of the metric potentials outlines the formation of the model. The exterior spacetime is assumed as described by the exterior Schwarzschild solution. The smooth matching of the interior to the exterior Schwarzschild spacetime metric across the boundary and the condition that radial pressure is zero across the boundary lead us to determine the model parameters. Physical requirements and stability analysis of the model demanded for a physically realistic star are satisfied. The developed model has been investigated graphically by exploring data from some of the known compact objects. The mass-radius (M-R) relationship that shows the maximum mass admissible for observed pulsars for a given surface density has also been investigated. Moreover, the physical profile of the moment of inertia (I) thus obtained from the solutions is confirmed by the Bejger–Haensel concept.

We investigate the properties of relativistic star spheres made of an electrically charged incompressible fluid, generalizing, thus, the Schwarzschild interior solution. The investigation is carried by integrating numerically the hydrostatic equilibrium equation, i.e., the Tolman-Oppenheimer-Volkoff (TOV) equation, with the hypothesis that the charge distribution is proportional to the energy density. We match the interior to a Reissner-Nordstr\"om exterior, and study some features of these star spheres such as the total mass $M$, the radius $R$, and the total charge $Q$. We also display the pressure profile. For star spheres made of a perfect fluid there is the Buchdahl bound, $R/M\ensuremath{\ge}9/4$, a compactness bound found from generic principles. For the Schwarzschild interior solution there is also the known compactness limit, the interior Schwarzschild limit where the configurations attain infinite central pressure, given by $R/M=9/4$, yielding an instance where the Buchdahl bound is saturated. We study this limit of infinite central pressure for the electrically charged stars and compare it with the Buchdahl-Andr\'easson bound, a limit that, like the Buchdahl bound for the uncharged case, is obtained by imposing some generic physical conditions on charged configurations. We show that the electrical interior Schwarzschild limit of all but two configurations is always below the Buchdahl-Andr\'easson limit, i.e., we find that the electrical interior Schwarzschild limit does not generically saturate the Buchdahl-Andr\'easson bound. We also find that the quasiblack hole limit, i.e., the extremal most compact limit for charged incompressible stars, is reached when the matter is highly charged and the star's central pressure tends to infinity. This is one of the two instances where the Buchdahl-Andr\'easson bound is saturated, the other being the uncharged, interior Schwarzschild solution.

In Chaps. 14 and 15 , we applied Eshelby’s interior solutions and the equivalent inclusion approach. In this chapter, we investigate the development of flanking structures as an example to use Eshelby’s exterior solutions. Flanking structures [Fig. 16.1; Passchier (J Struct Geol 23(6–7):951–962, 2001)] are deflections of linear and planar fabric elements around a cross-cutting element such as a fault, vein, dyke, or any other heterogeneity. Many authors have investigated the development of flanking structures by a combination of field, numerical modeling, analogue modeling, and analytical approaches [Exner and Dabrowski (J Struct Geol 32(12):2009–2021, 2010); Exner et al. (J Struct Geol 26(12):2191–2201, 2004); Grasemann et al. (J Struct Geol 25(1):19–34, 2003), (J Struct Geol 33(11):1650–1661, 2011); Grasemann and Stuwe (J Struct Geol 23(4):715–724, 2001); Kocher and Mancktelow (J Struct Geol 27(8):1346–1354, 2005), (J Struct Geol 28(7):1139–1145, 2006); Wiesmayr and Brasemann (J Struct Geol 27(2):249–264, 2005)]. These studies have greatly improved the understanding of flanking structures. The main conclusions are that the final geometrical features of flanking structures depend on the initial conditions of the cutting element (shape, orientation, relative rheology) and the characteristics of the flow field. One must therefore be cautious when interpreting the kinematics of flanking structures (e.g., Grasemann et al. (J Struct Geol 25(1):19–34, 2003); Wiesmayr and Brasemann (J Struct Geol 27(2):249–264, 2005)]. Because flanking structures result from the progressive deformation around a cutting element heterogeneity, we can regard the cutting element as an Eshelby ellipsoid and apply the exterior solutions (Chap. 11 ) to the development of flanking structures. Compared to previous investigations, the application of Eshelby’s exterior solutions has the following advantages. First, there is no limit to the flow type and the initial geometrical conditions of the cutting element (any ellipsoidal shape, arbitrary orientation with respect to the host element and the bulk flow field). Second, there is no limit to the amount of strain that can be reached because the approach does not rely on meshing. Third, the approach can handle anisotropic viscosity readily, although at a more expensive computational cost (Jiang and Bhandari 2018; Bhandari 2021). Furthermore, although the approach is based on linear viscous materials so far (Chaps. 10 and 11 ), it can be extended to power-law viscous materials by a linearization approach [Jiang (J Struct Geol 68:247–272, 2014), (Tectonophysics 693:116–142, 2016); Chap. 17 ]. This chapter first summarizes the principal equations related to the exterior mechanical fields in the vicinity of a heterogeneous ellipsoid. An outline of the algorithm for the numerical modeling of flanking structures is then presented. The algorithm is implemented in MATLAB. The MATLAB programs are provided in the online resource of the book. The program is applicable to anisotropic viscous materials under any flow field. As examples of using the numerical approach and the MATLAB program, we present numerical results for the development of flanking structures in plane-strain general shearing flows. The cutting element and the host medium are assumed to be isotropic and linear viscous materials. Furthermore, for simplicity, we limit ourselves to cutting elements with one principal axis parallel to the flow vorticity so that the whole deforming system has monoclinic symmetry.

The Einstein-Cartan field equations are solved for a string source with spin-polarisation along the axis of symmetry. The interior solution is matched to an exterior vacuum space-time using Arkuszewski-Kopczynski-Ponomariev junction conditions. The exterior solution is a four dimensional extension of the space-time outside a spinning point particle in three dimensional Einstein theory. It reduces to the geometry outside a conventional straight cosmic string in the case of vanishing spin.

The use of effective distance functions has been explored for many data mining problems including clustering, nearest neighbor search, and indexing. Recent research results show that if the Pearson variation of the distance distribution converges to zero with increasing dimensionality, the distance function will become unstable (or meaningless) in high dimensional space even with the commonly used Lp metric on the Euclidean space. This result has spawned many subsequent studies. We first comment that although the prior work provided the sufficient condition for the unstability of a distance function, the corresponding proof has some defects. Also, the necessary condition for unstability (i.e., the negation of the sufficient condition for the stability) of a distance function, which is required for function design, remains unknown. Consequently, we first provide in this paper a general proof for the sufficient condition of unstability. More importantly, we go further to prove that the rapid degradation of Pearson variation for a distance distribution is in fact a necessary condition of the resulting unstability. With the result, we will then have the necessary and the sufficient conditions for unstability, which in turn imply the sufficient and necessary conditions for stability. This theoretical result derived leads to a powerful means to design a meaningful distance function. Explicitly, in light of our results, we design in this paper a meaningful distance function, called Shrinkage-Divergence Proximity (abbreviated as SDP), based on a given distance function. It is empirically shown that the SDP significantly outperforms prior measures for its being stable in high dimensional data space and robust to noise, and is thus deemed more suitable for distance-based clustering applications than the priorly

The Schwarzschild interior solution, or `Schwarzschild star', which describes a spherically symmetric homogeneous mass with constant energy density, shows a divergence in pressure when the radius of the star reaches the Schwarzschild-Buchdahl bound. Recently Mazur and Mottola showed that this divergence is integrable through the Komar formula, inducing non-isotropic transverse stresses on a surface of some radius $R_{0}$. When this radius approaches the Schwarzschild radius $R_{s}=2M$, the interior solution becomes one of negative pressure evoking a de Sitter spacetime. This gravitational condensate star, or gravastar, is an alternative solution to the idea of a black hole as the ultimate state of gravitational collapse. Using Hartle's model to calculate equilibrium configurations of slowly rotating masses, we report results of surface and integral properties for a Schwarzschild star in the very little studied region $R_{s}<R<(9/8)R_{s}$. We found that in the gravastar limit, the angular velocity of the fluid relative to the local inertial frame tends to zero, indicating rigid rotation. Remarkably, the normalized moment of inertia $I/MR^2$ and the mass quadrupole moment $Q$ approach to the corresponding values for the Kerr metric to second order in $\Omega$. These results provide a solution to the problem of the source of a slowly rotating Kerr black hole.

The geometric basis for the effects of gravitation is traced from the relative orientation structure of pseudo-orthonormal basis vectors on the world line of a material point of interest. The effect of gravitation is specified by a principle of quasi-inertia. The relative basis structure is viewed in terms of a field frame and the effects of gravitation are then expressed by means of a force in this fixed (quasi-inertial) frame. This frame is generalized to a regional space and equations of motion for the nongeodesic particle motion under the influence of gravitation are obtained. The conditions that this regional space actually has the physical significance of an observer space are used to arrive at field equations. These field equations are used to solve for tensor gravitational fields entering in the equations of motion for the case applicable to the gravitational red-shift, the mercury perihelion shift, and the deflection of light by the sun. The usual Einstein-Schwarzschild results are obtained. The unity of the present view of the geometric basis of the effects of gravitation and that of the Lorentz force as described byRiesz are also briefly indicated.

Informative path planning is an important and challenging problem in robotics that remains to be solved in a manner that allows for wide-spread implementation and real-world practical adoption. Among various reasons for this, one is the lack of approaches that allow for informative path planning in high-dimensional spaces and non-trivial sensor constraints. In this work we present a sampling-based approach that allows us to tackle the challenges of large and high-dimensional search spaces. This is done by performing informed sampling in the high-dimensional continuous space and incorporating potential information gain along edges in the reward estimation. This method rapidly generates a global path that maximizes information gain for the given path budget constraints. We discuss the details of our implementation for an example use case of searching for multiple objects of interest in a large search space using a fixed-wing UAV with a forward-facing camera. We compare our approach to a sampling-based planner baseline and demonstrate how our contributions allow our approach to consistently out-perform the baseline by 18.0%. With this we thus present a practical and generalizable informative path planning framework that can be used for very large environments, limited budgets, and high dimensional search spaces, such as robots with motion constraints or high-dimensional configuration spaces. [Code] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> Codebase: https://github.com/castacks/tigris [Video] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sup> Video: https://youtu.be/bMw5nUGL5GQ

The contents of this paper refers to following main subjects: numerical adaptation of our cosmological parameters recently used within the framework of the Projective Unified Field Theory [1] to the new measured values of the Wilkinson Microwave Anisotropy Probe (WMAP) published by Ch.L. Bennett et al. [2]; study of the resulting consequences for our cosmological model of a closed homogeneous isotropic cosmos; approximate treatment of a quasi-static spherically symmetric perfect fluid sphere (interior and exterior solution) being under the adiabatic influence of the expanding cosmos; application of these results to sun-like and earth-like cosmic bodies; time dependence of the “empirical gravitational constant”; treatment of the Einstein effects, particularly of the periastron (perihelion) motion of an orbiting body (mercury) about a central body (sun) with a mass quadrupole moment and calculation of its numerical value for the sun. Further cosmological and astrophysical applications. An appendix points to a different set of parameters.