Metric Tensor

The intermediate stage of the development of general relativity is inseparable of Marcel Grossmann's mathematical assistance. Einstein acknowledges Grossmann's help during 1912-1914 to the development of general relativity. In fact, as with special relativity so was it with General relativity, Einstein received assistance only from his old friends, Marcel Grossmann and Michele Besso. However, he continued to consider Besso as his eternal "sounding board"…

The article discusses some of the mathematical results widely used in practice which contain the Riemann ζ-function, and, at first glance, are in contradiction with common sense. A geometric approach is suggested, based on the concept of the curvature of space, in which is calculated an algorithm that specifies the representation of ζ-function as an infinite diverging series. The analysis is based on the use of Einstein equations to calculate the metric of curved space-time. The solution of the Einstein equations is a metric that has a singularity, like the metric in the vicinity of the black hole. The result can be interpreted in the spirit of a Turing machine that performs the proposed algorithm for calculating the sum of a divergent series.

I discuss Albert Einstein's 1914 review paper, "The Formal Foundation of the General Theory of Relativity" from two points of view: the main elements in the paper that appear to have led to the downfall of the Einstein-Grossman theory; and the elements that seem to have inspired Einstein during October 1915 to reformulate the 1914 Einstein-Grossmann theory in the form of the November 1915 and the 1916 General Theory of Relativity. First paper among three papers.

I discuss Einstein's path-breaking November 1915 General Relativity papers. I show that Einstein's field equations of November 25, 1915 with an additional term on the right hand side involving the trace of the energy-momentum tensor appear to have sprung from his first November 1915 paper: the November 4, 1915 equations. Second paper among three papers.

I discuss Albert Einstein's 1916 General Theory of Relativity. I show that in Einstein's 1916 review paper, "the Foundation of the General Theory of Relativity", he derived his November 25, 1915 field equations with an additional term on the right hand side involving the trace of the energy-momentum tensor (he posed the condition square root -g=1) using the equations he presented on November 4, 1915. Series of papers: Final paper.

Abstract: In this paper I explore the dialectics underlying the choice between a geometrical and a field interpretation of the metric tensor in general relativity. I highlight the role that analogical reasoning plays in determining whether the metric represents spacetime or just a physical field. This question took center stage in the philosophical literature on general relativity after the publication of Earman and Norton’s seminal discussion of the hole argument. More recently, though, various theorists have come to regard this problem as a verbal dispute in which one can opt for any of the two rival interpretations of the metric according to one’s own preferences. I critically examine the key arguments that have been offered in the literature for the view that the metric tensor represents a substantivalist spacetime. I show that such arguments rely on similarity assessments and then argue that the relevant similarity facts underdetermine the application of the concepts of spacetime and physical field in the context of general relativity. This form of underdetermination, I suggest, vindicates the idea that there is more than one reasonable way of understanding the ontological import of the metric tensor.

Keywords: analogical reasoning, general relativity, metric tensor, substantivalism

On the basis of an exact solution for the field of a charge in a uniformly accelerated noninertial frame of reference (NFR) and formulated "Equivalent Situation Postulate" the nonlinear electrostatic theory of bound charges has been constructed. Proposed method is outside of the flat space-time, however the curvature is not directly connected with the Einstein gravitational theory. The method proposed eliminates divergence of the proper energy and makes classical electrodynamics consistent at any sufficiently small distances.

Altough the Metric Tensor is notoriously difficult to fully grasp, in this paper I plan to explain this fundamental object in an intuitive way.We will take a look at two situations where the Metric Tensor arises naturally, almost as a requirement for generalizing certain ideas, firstly in the 1-D case, secondly the 2-D case, and then the more general n-D case, as we will generalize the way to think about the tensor.

An extended advancing front technique (AFT) with shift operations and Riemann metric named as shifting-AFT is presented for finite element mesh generation on 3D surfaces, especially 3D closed surfaces. Riemann metric is used to govern the size and shape of the triangles in the parametric space. The shift operators are employed to insert a floating space between real space and parametric space during the 2D parametric space mesh generation. In the previous work of closed surface mesh generation, the virtual boundaries are adopted when mapping the closed surfaces into 2D open parametric domains. However, it may cause the mesh quality-worsening problem. In order to overcome this problem, the AFT kernel is combined with the shift operator in this paper. The shifting-AFT can generate high-quality meshes and guarantee convergence in both open and closed surfaces. For the shifting-AFT, it is not necessary to introduce virtual boundaries while meshing a closed surface; hence, the boundary discretization procedure is largely simplified, and moreover, better-shaped triangles will be generated because there are no additional interior constraints yielded by virtual boundaries. Comparing with direct methods, the shifting-AFT avoids costly and unstable 3D geometrical computations in the real space. Some examples presented in this paper have demonstrated the advantages of shift-AFT in 3D surface mesh generation, especially for the closed surfaces.

In this short paper, I will derive-by defining several concepts. First I will simply define the term, then introduce you to. By the way of Minkowski metric you ′ ll be able to apply to the vectors In. Finally I ′ ll apply Minkowski inner product to four-momentum and after doing several mathematical manipulations the energy-momentum relationship will eventually be derived.

This paper describes a new computational method of fully automated anisotropic triangulation of a trimmed parametric surface. Given as input: (1) a domain geometry and (2) a 3 x 3 tensor field that specifies a desired anisotropic node-spacing, this new approach first packs ellipsoids closely in the domain by defining proximity-based interacting forces among the ellipsoids and finding a force-balancing configuration using dynamic simulation. The centers of the ellipsoids are then connected by anisotropic Delaunay triangulation for a complete mesh topology. Since a specified tensor field controls the directions and the lengths of the ellipsoids' principal axes, the method generates a graded anisotropic mesh whose elements conform precisely to the given tensor field.