Tensor In General Relativity Research Articles

Imprints of cloud of strings and quintessence field on compact stellar configurations

This paper looks into how stellar configurations with quintessence and a cloud of string fields behave in general relativity in spherical symmetric spacetime. The goal is to find a stable, stellar model. We compute the unknown constants in the metric using the masses and radii of PSRJ1614 − 2230 stars. We employ the effective energy-momentum tensor in general relativity to analyze field equations. Additionally, we also demonstrate the anisotropic behavior, energy conditions, energy density, quintessence density, pressure profiles, gradients, anisotropic factor, energy conditions, sound speeds, compactification, mass function, EoS components, redshift, and stability of the compact stars. We demonstrate that according to the theory of general relativity, these compact stars exhibit physically plausible structures in the quintessence field with clouds of string.

On the existence of the electromagnetic energy tensor in second-order Lovelock gravities

In this work, we prove that there are no electromagnetic energy tensors in second-order Lovelock gravities that verify properties equivalent to those of the Maxwell electromagnetic energy tensor in general relativity.

A critique on some aspects of GUP effective metric

The generalized uncertainty principle (GUP) is a gravitational correction of Heisenberg’s uncertainty principle, which allows us to probe some features of quantum gravity even without the full theory. We are used to working with metric tensors in general relativity; they are convenient to have available when we wish to calculate physical quantities like Hawking temperature and black hole shadow. Various authors have tried to incorporate GUP into an effective metric that allows such calculations. In this note, I point out that some of these results are not correct due to working with series truncation rather than the full GUP-corrected expressions. Perhaps more importantly, we rely too much on heuristic arguments and lack a guiding principle in constructing the correct effective metric.

Kerr–Schild Tetrads and the Nijenhuis Tensor

We write the Kerr–Schild tetrads in terms of the flat space–time tetrads and of a (1, 1) tensor Sμλ. This tensor can be considered as a projection operator, since it transforms (i) flat space–time tetrads into non-flat tetrads, and vice-versa, and (ii) the Minkowski space–time metric tensor into a non-flat metric tensor, and vice-versa. The Sμλ tensor and its inverse are constructed in terms of the standard null vector field lμ that defines the Kerr–Schild form of the metric tensor in general relativity, and that yields black holes and non-linear gravitational waves as solutions of the vacuum Einstein’s field equations. We demonstrate that the condition for the vanishing of the Ricci tensor obtained by Kerr and Schild, in empty space–time, is also a condition for the vanishing of the Nijenhuis tensor constructed out of Sμλ. Thus, a theory based on the Nijenhuis tensor yields an important class of solutions of the Einstein’s field equations, namely, black holes and non-linear gravitational waves. We also demonstrate that the present mathematical framework can easily admit modifications of the Newtonian potential that may explain the long range gravitational effects related to galaxy rotation curves.

Quantum Field Theory and its Anomalies for Topological Matter

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.

Van der Waals anomaly: Analog of dark energy with ultracold atoms

In inhomogeneous dielectric media the divergence of the electromagnetic stress is related to the gradients of \varepsilon and \mu, which is a consequence of Maxwell's equations. Investigating spherically symmetric media we show that this seemingly universal relationship is violated for electromagnetic vacuum forces such as the generalized van der Waals and Casimir forces. The stress needs to acquire an additional anomalous pressure. The anomaly is a result of renormalization, the need to subtract infinities in the stress for getting a finite, physical force. The anomalous pressure appears in the stress in media like dark energy appears in the energy-momentum tensor in general relativity. We propose and analyse an experiment to probe the van der Waals anomaly with ultracold atoms. The experiment may not only test an unusual phenomenon of quantum forces, but also an analogue of dark energy, shedding light where nothing is known empirically.

Dynamic wormhole solutions in Einstein-Cartan gravity

In the present work we investigate evolving wormhole configurations described by a constant redshift function in Einstein-Cartan theory ({{\sf ECT}}). The matter content consists of a Weyssenhoff fluid along with an anisotropic matter which together generalize the anisotropic energy momentum tensor in general relativity ({{\sf GR}}) in order to include the effects of intrinsic angular momentum (spin) of particles. Using a generalized Friedmann-Robertson-Walker ({\sf FRW}) spacetime, we derive analytical evolving wormhole geometries by assuming a particular equation of state ({{\sf EoS}}) for energy density and pressure profiles. We introduce exact asymptotically flat and anti-de Sitter spacetimes that admit traversable wormholes and respect energy conditions throughout the spacetime. The rate of expansion of these evolving wormholes is determined only by the Friedmann equation in the presence of spin effects.

Einstein-Cartan wormhole solutions

In the present work we investigate wormhole structures and the energy conditions supporting them, in Einstein-Cartan theory ({\sf ECT}). The matter content consists of a Weyssenhoff fluid along with an anisotropic matter which together generalize the anisotropic energy momentum tensor in general relativity ({\sf GR}) to include spin effects. Assuming that the radial pressure and energy density obey a linear equation of state, we introduce exact asymptotically flat and anti-de-Sitter spacetimes that admit traversable wormholes and respect energy conditions. Such wormhole solutions are studied in detail for two specific forms for the redshift function, namely a constant redshift function and the one with power law dependency.

Electric and magnetic parts of the Weyl tensor and spin coefficients

It is known that electric and magnetic parts of the Weyl tensor in General Relativity are analogous to that occur in the theory of electromagnetism. On the other hand, Newman-Penrose formalism provides a new dimension to the solutions and interpretations for the theory of relativity. Its significance in understanding spacetimes with some particular Petrov types, solutions with different material contents is well known. This paper provides spin coefficient form of electric and magnetic parts of the Weyl tensor, which leads to a new and efficient way for the computation of electric and magnetic parts of the Weyl tensor. It is observed that the computational time is greatly reduced when done using computer algebra system in comparison to that is done manually. This technique is elaborated in the example of Pure Radiation metric.

Solution of conharmonic curvature tensor in General Relativity

Conharmonically at space time is studied in the frame work of FRW cosmological model. Einstein’s field equations with variable cosmological constant are solved by using the law of variation of cosmological constant that scales as a-m (where a is scale factor and m is a positive constant). A new class of exact solutions of the field equations has been obtained in which cosmological constant decreases with cosmic time. Physical and geometrical properties are also discussed in detail.

Construction of energy–momentum tensor of gravitation

We construct the gravitational energy–momentum tensor in general relativity through the Noether theorem. In particular, we explicitly demonstrate that the constructed quantity can vary as a tensor under the general coordinate transformation. Furthermore, we verify that the energy–momentum conservation is satisfied because one of the two indices of the energy–momentum tensor should be in the local Lorentz frame. It is also shown that the gravitational energy and the matter one cancel out in certain space-times.

TensorPack: a Maple-based software package for the manipulation of algebraic expressions of tensors in general relativity

In this paper we: (1) introduce TensorPack, a software package for the algebraic manipulation of tensors in covariant index format in Maple; (2) briefly demonstrate the use of the package with an orthonormal tensor proof of the shearfree conjecture for dust. TensorPack is based on the Riemann and Canon tensor software packages and uses their functions to express tensors in an indexed covariant format. TensorPack uses a string representation as input and provides functions for output in index form. It extends the functionality to basic algebra of tensors, substitution, covariant differentiation, contraction, raising/lowering indices, symmetry functions and other accessory functions. The output can be merged with text in the Maple environment to create a full working document with embedded dynamic functionality. The package offers potential for manipulation of indexed algebraic tensor expressions in a flexible software environment.

Stability of regular energy density in Palatini $$f(R)$$ f ( R ) gravity

The present work explores the effects of the three-parametric $$f(R)$$ model on the stability of the regular energy density of planar fluid configurations with the Palatini $$f(R)$$ formalism. For this purpose, we develop a link between the Weyl scalar and structural properties of the system by evaluating a couple of differential equations. We also see the effects of Palatini $$f(R)$$ terms in the formulation of structure scalars obtained by orthogonal splitting of the Riemann tensor in general relativity. We then identify the parameters which produce energy density irregularities in expansive and expansion-free dissipative as well as non-dissipative matter distributions. It is found that particular combinations of the matter variables lead to irregularities in an initially homogeneous fluid distribution. We conclude that Palatini $$f(R)$$ extra corrections tend to decrease the inhomogeneity, thereby imparting stability to the self-gravitating system.

Energy Transfer and Fluid Flow around a Massive Astrophysical Object

In this work it is presented the modeling and simulation of energy transfer and fluid flow of a stationary spherical arrangement of particles surrounding a gravitational body such as an astrophysical object that carries the curvature of space-time continuum in general relativity, taking into account the thermodynamics of the second law. This model also predicts the drag of space and time around an astrophysical object as it rotates, with results close to the experimental data reported by other authors. To model the energy transfer of the mass and the fluid flow in the space-time, it is used a 4-dimensional system. In order to make measurements of entropy in the arrow of time (past-present), tensors in General Relativity were used to calculate this thermodynamic quantity and with this, the big bang ́s low entropy condition in phase space of coarse graining (Hawking ́s box), according to Weyl curvature hypothesis (WCH) of Roger Penrose. Contribution of this paper is presented by tensors which carry information that has to do with something as non-distortion effect in fluid flow around the astrophysical object and the low entropy condition that is believed to exist in the past, in the big bang; what leads us to search for a new physical-mathematical science to continue. At this point, the Einstein field equations are out of context, which leads us to conclude that it is necessary a mathematical science that allows us to make calculations to rescue lost information due to collapse of matter to a black hole. This math should allow us to clear up physical phenomena (like origin of the universe) and their relationship, with the objective of unifying theories that lead to a physical science without uncertainties, as at the present time. In this regard, we propose a metric in hyperbolic coodinates to build a physical wormhole shaped object where gravitational bodies can be housed that allow us to link the past entropy with the present entropy according to the second law of thermodynamics, as a kind of mathematical space or alternative model to compensate in some way, the link between WCH and the phase space volume of the Hawking's box, and the link between WCH and the quantum-mechanical state-vector reduction, , proposed by Penrose which still have not been determined by any author. Nomenclature

Bernhard Riemann, a(rche)typical mathematical-physicist?

The work of Bernhard Riemann is discussed under the perspective of present day mathematics and physics, and with a prospective view towards 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: (i) 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; (ii) 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 ps

Spherically Symmetric Fluid Cosmological Model with Anisotropic Stress Tensor in General Relativity

This paper deals with the cosmological models for the static spherically symmetric spacetime for perfect fluid with anisotropic stress energy tensor in general relativity by introducing the generating functions g(r) and w(r) and also discussing their physical and geometric properties.

PSEUDO Z SYMMETRIC RIEMANNIAN MANIFOLDS WITH HARMONIC CURVATURE TENSORS

In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1–34]). Finally, we study some properties of (PZS)4 spacetime manifolds.

The role of special conformal Killing tensors in general relativity

Special conformal Killing tensors have a number of interesting properties, an overview of which is given in the Riemannian case. Some of these properties remain valid in pseudo-Riemannian manifolds. Presently, we are investigating the latter problem, where, in order to obtain a better insight, we first examine some specific cases, such as special conformal Killing tensors in conformally flat or Petrov type D space-times.

Dynamical laws of superenergy in general relativity

The Bel and Bel–Robinson tensors were introduced nearly 50 years ago in an attempt to generalize to gravitation the energy–momentum tensor of electromagnetism. This generalization was successful from the mathematical point of view because these tensors share mathematical properties which are remarkably similar to those of the energy–momentum tensor of electromagnetism. However, the physical role of these tensors in general relativity has remained obscure and no interpretation has achieved wide acceptance. In principle, they cannot represent energy and the term superenergy has been coined for the hypothetical physical magnitude lying behind them. In this work, we try to shed light on the true physical meaning of superenergy by following the same procedure which enables us to give an interpretation of the electromagnetic energy. This procedure consists in performing an orthogonal splitting of the Bel and Bel–Robinson tensors and analyzing the different parts resulting from the splitting. In the electromagnetic case such splitting gives rise to the electromagnetic energy density, the Poynting vector and the electromagnetic stress tensor, each of them having a precise physical interpretation which is deduced from the dynamical laws of electromagnetism (Poynting theorem). The full orthogonal splitting of the Bel and Bel–Robinson tensors is more complex but, as expected, similarities with electromagnetism are present. Also the covariant divergence of the Bel tensor is analogous to the covariant divergence of the electromagnetic energy–momentum tensor and the orthogonal splitting of the former is found. The ensuing equations are to the superenergy what the Poynting theorem is to electromagnetism. Some consequences of these dynamical laws of superenergy are explored, among them the possibility of defining superenergy radiative states for the gravitational field.

Non-localizability of electric coupling and gravitational binding of charged objects

The energy–momentum tensor in general relativity contains only localized contributions to the total energy–momentum. Here, we consider a static, spherically symmetric object consisting of a charged perfect fluid. For this object, the total gravitational mass contains a non-localizable contribution of electric coupling (ordinarily associated with electromagnetic mass). We derive an explicit expression for the total mass which implies that the non-localizable contribution of electric coupling is not bound together by gravity, thus ruling out the existence of objects with pure Lorentz electromagnetic mass in general relativity.