Dimensional analysis in relativity and in differential geometry

A few years ago, in my review of Sean Carroll's book in Classical and Quantum Gravity [1], I wrote that while the 1970s was the decade of Weinberg [2] and Misner, Thorne and Wheeler [3], and while the eighties was the decade of Schutz [4] and Wald [5], the 2000s was clearly the decade of Hartle [6] and Carroll [7]. In my opinion, these books continue to stand out in the surprisingly dense crowd of introductory textbooks on general relativity. At the dawn of this new decade I look forward to see what fresh pedagogical insights will be produced next, and who will be revealed as the winners of the 2010s. It is, of course, much too early to tell, but Schutz is back, and he will set the standard just as he did back in 1985.

The spherically symmetric thin shells are the nearest generalizations of the point-like particles. Moreover, they serve as the simple sources of the gravitational fields both in General Relativity and much more complex quadratic gravity theories. We are interested in the special and physically important case when all the quadratic in curvature tensor (Riemann tensor) and its contractions (Ricci tensor and scalar curvature) terms are present in the form of the square of Weyl tensor. By definition, the energy–momentum tensor of the thin shell is proportional to Diracs delta-function. We constructed the theory of the spherically symmetric thin shells for three types of gravitational theories with the shell: (1) General Relativity; (2) Pure conformal (Weyl) gravity where the gravitational part of the total Lagrangian is just the square of the Weyl tensor; (3) Weyl–Einstein gravity. The results are compared with these in General Relativity (Israel equations). We considered in detail the shells immersed in the vacuum. Some peculiar properties of such shells are found. In particular, for the traceless ([Formula: see text] massless) shell, it is shown that their dynamics cannot be derived from the matching conditions and, thus, is completely arbitrary. On the contrary, in the case of the Weyl–Einstein gravity, the trajectory of the same type of shell is completely restored even without knowledge of the outside solution.

In this paper the Feynman Green function for Maxwell’s theory in curved space-time is studied by using the Fock–Schwinger–DeWitt asymptotic expansion; the point-splitting method is then applied, since it is a valuable tool for regularizing divergent observables. Among these, the stress-energy tensor is expressed in terms of second covariant derivatives of the Hadamard Green function, which is also closely linked to the effective action; therefore one obtains a series expansion for the stress-energy tensor. Its divergent part can be isolated, and a concise formula is here obtained: by dimensional analysis and combinatorics, there are two kinds of terms: quadratic in curvature tensors (Riemann, Ricci tensors and scalar curvature) and linear in their second covariant derivatives. This formula holds for every space-time metric; it is made even more explicit in the physically relevant particular cases of Ricci-flat and maximally symmetric spaces, and fully evaluated for some examples of physical interest: Kerr and Schwarzschild metrics and de Sitter space-time.

Considered Schwarzschild black hole solution within General Relativity. The area of black hole event horizon attracts stable and vivid interest. The literature brief review points in Introduction out, that the coordinate singularity on horizon might have physical meaning. In the following two sections this point strengthens. But this contradicts the dogmatic physical worldview, so in Discussion section I disarmed possible critics. I'm signing my faith in Conclusion, that this paper have found the naked (them we can observe) singularities. Even if is showed, that physically reasonable solutions can have the nakedness of singularities, they are far from the actual experimental detection. Contrary to that, recognizing the physical-ness of coordinate singularities makes them automatically experimentally verified. Because the event horizon is in definition of experimentally verified black holes (e.g., 2007 MASTER Team discovered the Black Hole Ergosphere, surrounding event horizon). Context: Natural Sciences: Physics: General Relativity theory: Black holes: event horizon. Aims: To publish original, innovative and novel research article resulting in contradicting results in the field of science. To publish theoretical paper that reports the contradictory finding (coordinate singularity is actually real) in scientific field. To present counter-examples (singular force F and curvature scalar ψ) to usually accepted conjecture, which is “coordinate singularity is unphysical”. Settings and Design: The singularity is defined in Introduction. There is set the function S(r), describing singularities. This is followed by derivation of gravitational attraction F. It turns to be singular with the S(r). Then directly from Curvature Tensor is derived singular scalar tensor ψ, where you can see the S(r). Discussion answers possible critics preceded by Results. Methods and Material: Hardware: a PC; Software: a Web Browser, Maple V Release 5 with GRTensorII Version 1.70; Vector and Tensor analysis. Results: Although the surface gravity g has suffered from Scientific Community the procedure of Renormalization, nevertheless this operation left the gravitational attraction (force F) and scalar ψ to include singular function S(r). Conclusions: Already 11 years I firmly believe. That I found naked (one we can observe) singularity of Curvature, predicted by General Relativity. I called it “Sulo singularity”. It's my personal joy and triumph, which I would like to share through JCRSCI.

A systematic method is presented whereby any compact Lie group of n-real parameters is dealt with from an infinitesimal approach with the representative matrix method based on a group of inner automorphisms suggested in a previous paper. The group manifold, defined in terms of a metric of group parameters, is identified as a Riemannian one in which these parameters play a role of n curvilinear coordinates. Riemannian geometry is thus valid in the group manifold, and geometric quantities are explicitly calculated in terms of the symmetric or (0) connection by a straightforward application of the ordinary procedure of tensor analysis. A new and simpler method of computing the invariant volume element is presented within this framework. Furthermore, we discuss in detail the group of inner automorphisms for the calculation of the matrix element of finite rotations for any irreducible representation (abbreviated MEFRIR). It is found that our method works very well and yields right and left vector fields together with a set of 2n equations to be satisfied by the MEFRIR. The global properties of the group may, therefore, be obtained as a solution to these equations. Moreover, it provides not only the generalized Maurer-Cartan equations, the Lie structure formulas, two parameter groups of point transformations and the adjoint group, but also two additional nonsymmetric (+) and (−) connections with zero curvature, which do not possess any preassigned metric but possess two absolute parallelisms. Thus, our results on differential geometry completely agree with Cartan and Schouten's. A link between differential geometry and representations is presented by the right and left vector fields which are explicitly calculable in terms of the n parameters and through which geometric quantities, e.g., the Riemann tensor, the Ricci tensor, and the scalar curvature, of any connection are explicitly displayed. A theorem relating both the vector fields to the metric tensors is also included. Finally, the l (rank of the group) invariant differential equations to be satisfied by the MEFRIR are cast in the covariant (or Lie derivative) forms in any connection. Examples of the invariant equations are given for SU(2), SO(3), and SU(3). The two invariant equations of the latter can be cast in terms of the eigenvalues of isospin and hypercharge upon carrying out charge and hypercharge quantizations; in this connection, a new nonrelativistic wave equation to be satisfied by SU(3) multiplets as a generalization of the Schrödinger equation of the symmetric top is also proposed.

An important problem in differential geometry and in gravitation is to identify metrics in a fully coordinate independent manner. In fact, the very foundation of Riemannian geometry is based on the existence of a tensor, the Riemann or curvature tensor, which vanishes if and only if the metric is locally flat. Many other such local characterizations of metrics are known. The aim of this article is to present a brief selection of them as an example of the fruitful interplay between differential geometry and gravity.

Introducing quantum space-time into physics by means of the transformation language of noncommuting coordinates gives a simple scheme of generalizing the tensor analysis. The general covariance principle for the quantum space-time case is discussed, within which one can obtain the covariant structure of basic tensor quantities and the motion equation for a particle in a gravitational field. Definitions of covariant derivatives and curvature are also generalized in the given case. It turns out that the covariant structure of the Riemann-Christoffel curvature tensor is not preserved in quantum space-time. However, if the curvature tensor $$\hat R$$ μvλϰ (z) is redetermined up to the value of theL 2 term, then its covariant structure is achieved, and it, in turn, allows us to reconstruct the Einstein equation in quantum space-time.

The metric is an important tensorial object that introduces more structure into a (differential) manifold. The metric coefficients allow for example to determine the length of parameter curves in the manifold and make it possible to relate corresponding co- and contravariant objects defined on the manifold. Thus the inner product of either co- or contravariant quantities and in particular the angle between two vectors or (covectors) may be computed. Adopting the Ricci postulate of vanishing covariant derivative of the metric coefficients results in a decomposition of the fully covariant connection into its Riemann part, that depends exclusively on the metric, and the contortion, that depends exclusively on the torsion. Moreover, in this case the length of vectors and their angle with respect to geodesics are preserved upon parallel transport. Based on the metric coefficients the fully covariant curvature tensor, displaying left and right skew symmetry, is introduced and the corresponding modifications of the Bianchi identities are highlighted. Likewise the metric allows to introduce the mixed-variant Ricci tensor together with corresponding identities. Finally the metric enables to compute the Ricci scalar from the previous curvature tensors. The particular case of a symmetric, metrically connected manifold represents a Riemann geometry with associated Riemann curvature tensor as fundamental for example in the Einstein Theory of General Relativity/Gravitation. A generalization is obtained by allowing for metric manifolds with non-vanishing covariant derivative of its metric coefficients. The presence of the non-metricity is then subsequently reflected by extra terms in the decomposition of the connection and in the explicit representation of the curvature tensor, the Ricci tensor, the Ricci scalar, and the corresponding Bianchi identities for the various curvature quantities.

Purpose With the increasing demand for surveillance and smart transportation, drone technology has become the center of attraction for robotics researchers. This study aims to introduce a new path planning approach to drone navigation based on topology in an uncertain environment. The main objective of this study is to use the Ricci flow evolution equation of metric and curvature tensor over angular Riemannian metric, and manifold for achieving navigational goals such as path length optimization at the minimum required time, collision-free obstacle avoidance in static and dynamic environments and reaching to the static and dynamic goals. The proposed navigational controller performs linearly and nonlinearly both with reduced error-based objective function by Riemannian metric and scalar curvature, respectively. Design/methodology/approach Topology and manifolds application-based methodology establishes the resultant drone. The trajectory planning and its optimization are controlled by the system of evolution equation over Ricci flow entropy. The navigation follows the Riemannian metric-based optimal path with an angular trajectory in the range from 0° to 360°. The obstacle avoidance in static and dynamic environments is controlled by the metric tensor and curvature tensor, respectively. The in-house drone is developed and coded using C++. For comparison of the real-time results and simulation results in static and dynamic environments, the simulation study has been conducted using MATLAB software. The proposed controller follows the topological programming constituted with manifold-based objective function and Riemannian metric, and scalar curvature-based constraints for linear and nonlinear navigation, respectively. Findings This proposed study demonstrates the possibility to develop the new topology-based efficient path planning approach for navigation of drone and provides a unique way to develop an innovative system having characteristics of static and dynamic obstacle avoidance and moving goal chasing in an uncertain environment. From the results obtained in the simulation and real-time environments, satisfactory agreements have been seen in terms of navigational parameters with the minimum error that justifies the significant working of the proposed controller. Additionally, the comparison of the proposed navigational controller with the other artificial intelligent controllers reveals performance improvement. Originality/value In this study, a new topological controller has been proposed for drone navigation. The topological drone navigation comprises the effective speed control and collision-free decisions corresponding to the Ricci flow equation and Ricci curvature over the Riemannian metric, respectively.

Most of us sometimes have to face a student asking: 'What do I need to get started on this'. (In my case 'this' would typically be a topic in general relativity.) After thinking about it for quite a while, and consulting candidate texts again and again, a few days later I usually end up saying: read this chapter in book I (but without going too much detail), then that chapter in book II (but ignore all those comments), then the first few sections of this review paper (but do not try to work out equations NN to NNN), and then come back to see me. In the unlikely event that the student comes back without changing the topic, there follows quite a bit of explaining on a blackboard over the following weeks.

Abstract. In the present paper, we introduce a type of Riemann-ian manifolds (namely, W 4 -birecurrent manifold) and study the sev-eral properties of such a manifold on which some geometric condi-tions are imposed. 1. IntroductionIn [7], Lichnerowicz introduced a recurrent manifold of second or-der (called birecurrent manifold) and obtained some interesting results.Later the weak notion of this manifold was studied by many authors in-cluding Thompson [9], Chaki and Roy Chowdhury [2], Ewert-Krzemieni-ewski [3,4], Garai [5]. In [8], Pokhariyal de ned some curvature tensorswith the help of Weyl’s projective curvature tensor and studied theirphysical and geometrical properties. One of the curvature tensors men-tioned in [8] was the W 4 -curvature tensor de ned by W 4 (X;Y;Z;T) = R(X;Y;Z;T) +1(n 1)[g(X;Z)r(Y;T) g(X;Y)r(Z;T)] or in local coordinates,(1.1) W 4ijkl = R ijkl +1(n 1)[g ik r jl g ij r kl ];where Rand rare the Riemannian curvature tensor and the Ricci tensor,respectively. The W 4

On recurrent Riemannian and Ricci curvatures of Finsler metrics

We study the quantum Riemannian geometry of quantum projective spaces of any dimension. In particular, we compute the Riemann and Ricci tensors using previously introduced quantum metrics and quantum Levi-Civita connections. We show that the Riemann tensor is a bimodule map and derive various consequences of this fact. We prove that the Ricci tensor is proportional to the quantum metric, giving a quantum analogue of the Einstein condition, and compute the corresponding scalar curvature.

This chapter continues the study of calculus on vector manifolds begun in Chapter 4. The emphasis here is on the central object of classical differential geometry, the curvature tensor. We have endeavored to supply simple and systematic derivations of all properties of the curvature tensor including relations to extrinsic geometry, behavior under transformations, and generalization to nonRiemannian curvature. We believe that some of our results are new, but our main objective is to demonstrate the unique advantages of the method and to develop the calculus to the point where application to any problem in differential geometry is straightforward.

In the first eleven chapters of this book we have attempted to introduce the reader to string theory as it is presently understood. Our focus now shifts to making contact with more familiar physics. In this chapter we develop some concepts in differential geometry that are useful in understanding general relativity and Yang-Mills theory even in four dimensions, but which are of particular utility in ten-dimensional physics. Our treatment in this chapter is comparatively elementary and aims mostly to develop the minimum material we require in chapters 13 and 14. In chapter 13, we will discuss supergravity theory in ten dimensions, which at least in perturbation theory is the low-energy limit of ten-dimensional superstring theory. In chapter 14, we will discuss some of the important ideas that arise in compactification from ten to four dimensions. The concluding chapters of this book, chapters 15 and 16, are devoted to more specialized mathematical background and more speculative ideas about compactification. Spinors In General Relativity A good place to start our discussion is to think about the coupling of spinors to a gravitational field. This problem is of great importance in string theories in which both fermions and gravity are present, and this alone would justify its consideration here. In addition, thinking about the coupling of spinors to a gravitational field forces us to examine issues whose analogs for Yang-Mills theory we will wish to consider later. The question of coupling spinors to general relativity was considered briefly in chapter 4, in connection with a discussion of two-dimensional supergravity, but here we will be more extensive.