Tensors in general relativity

Chapter 12 - Curvature, Strong Gravity, and Gravitational Waves

The de Rham Laplacian Δ (dR) for differential forms is a geometric generalization of the usual covariant Laplacian Δ, and it may be extended naturally to tensor-valued p-forms using the exterior covariant derivative associated with a metric connection. Using it the wave equation satisfied by the curvature tensors in general relativity takes its most compact form. This wave equation leads to the Teukolsky equations describing integral spin perturbations of black hole spacetimes.

In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.

The covariant derivative is a generalization of differentiating vectors. The Euclidean derivative is a special case of the covariant derivative in Euclidean space. The covariant derivative gathers broad attention, particularly when computing vector derivatives on curved surfaces and volumes in various applications. Covariant derivatives have been computed using the metric tensor from the analytically known curved axes. However, deriving the global axis for the domain has been mathematically and computationally challenging for an arbitrary two-dimensional (2D) surface. Consequently, computing the covariant derivative has been difficult or even impossible. A novel high-order numerical scheme is proposed for computing the covariant derivative on any 2D curved surface. A set of orthonormal vectors, known as moving frames, expand vectors to compute accurately covariant derivatives on 2D curved surfaces. The proposed scheme does not require the construction of curved axes for the metric tensor or the Christoffel symbols. The connectivity given by the Christoffel symbols is equivalently provided by the attitude matrix of orthonormal moving frames. Consequently, the proposed scheme can be extended to the general 2D curved surface. As an application, the Helmholtz‐Hodge decomposition is considered for a realistic atrium and a bunny.

The topological space of a bound conservative quantum system is studied in reference to the corresponding integrable one with the concept of parallel transport. A scheme of iterative perturbation is proposed. The destruction of invariant subspaces due to the singular behavior at avoided crossings is carefully investigated. It is shown that before the first avoided crossing the invariant subspaces are merely distorted, and after the first avoided crossing the invariant subspaces begin to be destroyed.

A limiting diagram for the Segre classification in 5-dimensional space–times is obtained, extending a recent work on limits of the energy–momentum tensor in general relativity. Some of Geroch’s results on limits of space–times in general relativity are also extended to the context of five-dimensional Kaluza–Klein space–times.

The first chapter in Part II contains basic information on general relativity and all necessary notations and formulas. The consideration is concise but rather brief; it is not meant to replace a textbook on general relativity. Covariant derivatives, curvature tensors, Bianchi identities, covariant equations, the classical limit and Einstein equations are covered. The fact that the Schwarzschild solution and the cosmological solution contain singularities, which can be interpreted as indicating a need for quantum gravitational theory, is addressed. In addition, the applicability of general relativity and Planck units are discussed.

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.

PurposeIn this paper, we give some properties of the α-connections on statistical manifolds and we study the α-conformal equivalence where we develop an expression of curvature R¯ for ∇¯ in relation to those for ∇ and ∇^.Design/methodology/approachIn the first section of this paper, we prove some results about the α-connections of a statistical manifold where we give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds treated in [1, 3], and we construct some examples.FindingsWe give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.Originality/valueWe give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.

A list is given of a canonical set of the Newman–Penrose quantities ΦAB, the tetrad components of the trace-free Ricci tensor, for each Plebański class according to Plebański’s classification of this tensor. This comparative list can easily be extended to cover the classification in tetrad language of any second-order, trace-free, symmetric tensor in a space-time. A fourth-order tensor which is the product of two such tensors was defined by Plebański and used in his classification. This has the same symmetries as the Weyl tensor. The Petrov classification of this tensor, here called the Plebański tensor, is discussed along with the classification of the Ricci tensor. The use of the Plebański tensor in a couple of areas of general relativity is also briefly discussed.

Based on Newtonian mechanics, in this article, we present a heuristic derivation of the Friedmann equations, providing an intuitive foundation for these fundamental relations in cosmology. Additionally, using the first law of thermodynamics and Euler’s equation, we derive a set of equations that, at linear order, coincide with those obtained from the conservation of the stress-energy tensor in general relativity. This approach not only highlights the consistency between Newtonian and relativistic frameworks in certain limits, but also serves as a pedagogical bridge, offering insights into the physical principles underlying the dynamics of the universe.

What has dark energy to do with van der Waals forces? Why does the stress from van der Waals forces appear to violate Maxwell's equations? Why are ultracold atoms good candidates for testing this analog of dark energy? Find out in this paper.

The work of Bernhard Riemann is discussed under the perspective of present day mathematics and physics, and with a prospective view toward the future, too. Against the (unfortunately rather widespread) trend—which predominantly dominated national scientific societies in Europe during the last century—of strictly classifying the work of scientists with the aim to constrain them to separated domains of knowledge, without any possible interaction among those and often even fighting against each other (and which, no doubt, was in part responsible for the decline of European in favor of American science), it will be here argued, using Riemann as a model, archetypical example, that good research transcends any classification. Its uses and applications arguably permeate all domains, subjects and disciplines one can possibly define, to the point that it can be considered to be universally useful. After providing a very concise review of the main publications of Bernhard Riemann on physical problems, some connections between Riemann's papers and contemporary physics will be considered: (1) the uses of Riemann's work on the zeta function for devising applications to the regularization of quantum field theories in curved space-time, in particular, of quantum vacuum fluctuations; (2) the uses of the Riemann tensor in general relativity and in recent generalizations of this theory, which aim at understanding the presently observed acceleration of the universe expansion (the dark energy issue). Finally, it will be argued that mathematical physics, which was yet not long ago a model paradigm for interdisciplinary activity—and had a very important pioneering role in this sense—is now quickly being surpassed by the extraordinarily fruitful interconnections which seem to pop up from nothing every day and simultaneously involve several disciplines, in the classical sense, including genetics, combinatorics, nanoelectronics, biochemistry, medicine, and even psychology, with such intriguing issues that include, e.g., artificial life and the modelization of the process of consciousness.

Topology enters in quantum field theory (qft) in multiple forms: one of the most important, in non-abelian gauge theories, being in the identification of the θ vacuum in QCD. A very relevant aspect of this connection is through the phenomenon of chiral and conformal qft anomalies. It has been realized that a class of materials, comprising topological insulators and Weyl semimetals, also exhibit the phenomenon of anomalies, which are responsible for several exotic phenomena, such as the presence of edge currents, resilient under perturbations and scattering by impurities. Another example comes from the response functions of these materials under thermal and mechanical stresses, that may be performed using correlation functions of stress energy tensors in General Relativity. In this case the conformal anomaly plays an important role. We briefly illustrate some salient features of this correspondence, and the effective action describing the long-range interactions that may account for such topological effects.

This paper investigates the properties and behaviour of generalized βP-recurrent Finsler spaces with an emphasis on the covariant derivatives of Weyl's projective tensors, which include torsion, curvature, and deviation tensors. We define a GβP-recurrent space as a Finsler space where the second curvature tensor satisfies a specific recurrence condition involving non-zero covariant vector fields. Through a series of mathematical derivations, the paper explores the equivalence of different characterizations of the GβP-recurrent space, proving that the torsion tensor, its associate tensor, the P-Ricci tensor, and the curvature vector are non-vanishing. Further, we analyze the relationship between the covariant vector fields showing their dependence or independence on the direction argument. The study also includes the application of Berwald’s covariant derivative to the projective curvature tensor, concluding that these tensors exhibit generalized recurrence under certain conditions. The results are presented in a series of theorems that contribute to the deeper understanding of the geometric structure of - spaces.