Geometry In General Relativity Research Articles

Different Aspects of Spin in Quantum Mechanics and General Relativity

In this paper, different aspects of the concept of spin are studied. The most well-established one is, of course, the quantum mechanical aspect: spin is a broken symmetry in the sense that the solutions of the Dirac equation tend to have directional properties that cannot be seen in the equation itself. It has been clear since the early days of quantum mechanics that this has something to do with the indefinite metric in Lorentz geometry, but the mechanism behind this connection is elusive. Although spin is not the same as rotation in the usual sense, there must certainly be a close relationship between these concepts. And, a possible way to investigate this connection is to instead start from the underlying geometry in general relativity. Is there a reason why rotating motion in Lorentz geometry should be more natural than non-rotating motion? In a certain sense, the answer turns out to be yes. But, it is by no means easy to see what this should correspond to in the usual quantum mechanical picture. On the other hand, it seems very unlikely that the similarities should be just coincidental. The interpretation of the author is that this can be a golden opportunity to investigate the interplay between these two theories.

ISCOs and OSCOs in the Presence of a Positive Cosmological Constant in Massive Gravity

We study the impact of a non-vanishing (positive) cosmological constant on the innermost and outermost stable circular orbits (ISCOs and OSCOs, respectively) within massive gravity in four dimensions. The gravitational field generated by a point-like object within this theory is known, generalizing the usual Schwarzschild–de Sitter geometry of General Relativity. In the non-relativistic limit, the gravitational potential differs by the one corresponding to the Schwarzschild–de Sitter geometry by a term that is linear in the radial coordinate with some prefactor γ, which is the only free parameter. Starting from the geodesic equations for massive test particles and the corresponding effective potential, we obtain a polynomial of fifth order that allows us to compute the innermost and outermost stable circular orbits. Next, we numerically compute the real and positive roots of the polynomial for several different structures (from the hydrogen atom to stars and globular clusters to galaxies and galaxy clusters) considering three distinct values of the parameter γ, determined using physical considerations, such as galaxy rotation curves and orbital precession. Similarly to the Kottler spacetime, both ISCOs and OSCOs appear. Their astrophysical relevance as well as the comparison with the Kottler spacetime are briefly discussed.

Model of compact star with ordinary and dark matter

We study compact stars formed by dark and ordinary matter, with attributes of both neutron star matter and quark star matter. We assume an equation of state for dark matter which is consistent with the rotational curves of galaxies and for color-flavor-locked (CFL) distributions for ordinary matter. This is done in the curved Krori-Barua spacetime geometry in general relativity. We find new exact solutions of dark matter admixed compact objects, with maximum mass 2.67 \(M_{\odot }\) and radius around 11.16 \(\mbox{km}\), with \(52.40 \%\) of dark matter content. By varying the dark matter ratio, we obtain the masses of dark compact stars with masses less than 2.67 \(M_{\odot }\).

Nonextensive Entropy and Geometry in General Relativity

Con-formally invariant metrics can be obtained from the smooth deformation of otherwise flat spacetime. We obtain these metrics from the maximization of the non-extensive entropy of Tsallis with 2nd moments, quadratics of the coordinates as constraints. Evolution equations are Fokker-Planck diffusion type and the usual diffusion tensor (inverse) relations to fundamental metric tensors hold. It is found that an nD deformed q-metric corresponds to an embedding in an (n+1)D flat spacetime, and secondly that a q-non-extensive statistics derived FRW metric and Friedman equations and Hubble term may be obtained and connected to thermo-statistics.

Aspects of gravitational wave/particle duality: Bulk torsion ↔boundary gravity correspondence

A geometric torsion (GT) underlying a [Formula: see text]-form in a [Formula: see text]-dimensional [Formula: see text] gauge theory is revisited with a renewed perspective for a nonperturbation (NP) gravity in [Formula: see text]. In this context, we provide evidences to a holographic correspondence between a bulk GT and a boundary NP gravity. Interestingly the Killing symmetries in General Relativity (GR) are shown to provide a subtle clue to the quantum gravity. The NP gravity is shown to incorporate a [Formula: see text] coupling, sourced by a non-Newtonian potential, to an exact geometry in GR. Remarkably the NP correction is identified as a mass dipole and is shown to be sourced by a propagating GT. A detailed analysis is performed in a bulk GT to show a modification to the precession of perihelion in a boundary NP gravity. The perspective of an electromagnetic (EM) wave in the bulk is investigated to reveal a spin [Formula: see text] (mass-less) quantum sourced by an apparent 2-form. A Goldstone scalar is absorbed by the apparent 2-form to describe a massive [Formula: see text]-form in the coulomb gauge. Alternately a Goldstone scalar together with a local degree of GT and 2-form is argued to govern a composite (mass-less) spin [Formula: see text] particle in Lorentz gauge. Both the scenarios, further ensure a graviton in a boundary NP gravity. A qualitative analysis reveals a (noninteracting) graviton underlying a plausible gravitational wave/particle duality in NP gravity.

Relaxations of perturbations of spacetimes in general relativity coupled to nonlinear electrodynamics

Three well known exact regular solutions of general relativity (GR) coupled to nonlinear electrodynamics (NED), namely the Maxwellian, Bardeen and Hayward regular spacetimes, which can describe either a regular black hole or a geometry without horizons, have been considered. Relaxation times for the scalar, electromagnetic (EM) and gravitational perturbations of black holes (BHs) and no-horizon spacetimes have been estimated in comparison with the ones of the Schwarzschild and Reissner-Nordstr\"{o}m (RN) spacetimes. It has been shown that the considered geometries in GR coupled to the NED have never vanishing circular photon orbits and on account of this fact these spacetimes always oscillate the EM perturbations with quasinormal frequencies (QNFs). Moreover we have shown that the EM perturbations in the eikonal regime can be a powerful tool to confirm that (i) the light rays do not follow null geodesics in the NED by the relaxation rates; (ii) if the underlying solution has a correct weak field limit to the Maxwell electrodynamics (LED) by the angular velocity of the circular photon orbit.

Soliton Resolution for Equivariant Wave Maps on a Wormhole

In this paper, we initiate the study of finite energy equivariant wave maps from the (1+3)-dimensional spacetime $\mathbb R \times (\mathbb R \times \mathbb{S}^2) \rightarrow \mathbb{S}^3$ where the metric on $\mathbb R \times (\mathbb R \times \mathbb{S}^2)$ is given by ds^2 = -dt^2 + dr^2 + (r^2 + 1) \left ( d \theta^2 + \sin^2 \theta d \varphi^2 \right ), \quad t,r \in \mathbb{R}, (\theta,\varphi) \in \mathbb{S}^2. The constant time slices are each given by the Riemannian manifold $\mathcal M := \mathbb R \times \mathbb{S}^2$ with metric ds^2 = dr^2 + (r^2 + 1) \left ( d \theta^2 + \sin^2 \theta d \varphi^2 \right ). The Riemannian manifold $\mathcal M$ contains two asymptotically Euclidean ends at $r \rightarrow \pm \infty$ that are connected by a spherical throat of area $4 \pi^2$ at $r = 0$. The spacetime $\mathbb R \times \mathcal M$ is a simple example of a wormhole geometry in general relativity. In this work we will consider 1--equivariant or corotational wave maps. Each corotational wave map can be indexed by its topological degree $n$. For each $n$, there exists a unique energy minimizing corotational harmonic map $Q_{n} : \mathcal M \rightarrow \mathbb{S}^3$ of degree $n$. In this work, we show that modulo a free radiation term, every corotational wave map of degree $n$ converges strongly to $Q_{n}$. This resolves a conjecture made by Bizon and Kahl in the corotational case.

On the Physics inside a Closed, Static, Rotating Einsteinian Hypersphere in Due Consideration of the Galaxy

Einstein’s weak equivalence principle suggests that gravity and acceleration (centrifugal force) are indistinguishable from each other and, therefore, equivalent. We maintain that they are not only equivalent, but even identical, or to rephrase the main statement of this work: A gravitational force does not exist. Rather, gravity is a fictitious force, or, more pointedly: Gravity is the centrifugal force which acts upon material bodies within the rotating S3-hypersphere of the Universe. These in turn warp the adjacent space-fabric, shaping it to the well-known field geometry of general relativity.

Isotropic extensions of the vacuum solutions in general relativity

In this work, we obtain isotropic extensions of the usual spherically symmetric vacuum geometries in general relativity. Exact and perturbative solutions are derived. The classes of geometries obtained include black holes in compact and noncompact universes, wormholes in the interior region of cosmological horizons, and anti-de Sitter geometries with excess/deficit solid angle. The tools developed here are applicable in more general contexts.

Complementarity of kinematics and geometry in general relativity

Relations between kinematics, geometry and law of reference frame motion are considered. We show, that kinematical tensors define geometry up to a space functional arbitrariness when integrability condition for spin tensor is satisfied. Some aspects of geometrization principle and geometrical conventionalism of Poincare are discussed in a light of the obtained results.

Distributional geometry in general relativity

In this article we look at the extent to which one can use classical linear distributional geometry in general relativity. We then go on to look at a non-linear theory of distributional geometry based on Colombeau algebras and show that this is compatible with the linear theory in situations where both may be used. For both the linear and non-linear theories of distributional geometry we will use a geometric coordinate free description. We conclude by looking at the example of the thin string limit of solutions of the field equations for an infinite length gravitating straight cosmic string described by a complex scalar field coupled to a U(1) gauge field. We show that within the Colombeau algebra this has a well-defined energy–momentum tensor and curvature which are associated to classical distributions.

Classifying geometries in general relativity:III. Classification in practice

This is the third in a series of papers concerning a project to set up acomputer database of exact solutions in general relativity which can beaccessed and updated by the user community. In this paper, we describehow the Cartan-Karlhede method for classifying a geometry isaccomplished in practice. We give as an example the classification ofthe Edgar-Lugwig conformally flat pure radiation metrics.

Classifying geometries in general relativity:II. Spinor tools

This is the second in a series of papers concerning a project to set up acomputer database of exact solutions in general relativity which can beaccessed and updated by the user community. In this paper, we discussthe choice of computer algebra platform and the general relativityapplication package. The derivative operators needed in the Cartan-Karlhede classification algorithm and the behaviour of symmetric spinors under frame rotations are then presented.

Classifying geometries in general relativity: I. Standard forms for symmetric spinors

This is the first in a series of papers concerning a project to set up a computer database of exact solutions in general relativity which can be accessed and updated by the user community. In this paper, we briefly discuss the Cartan-Karlhede invariant classification of geometries and the significance of the standard form of a spinor. We then present algorithms for putting the Weyl spinor, Ricci spinor and general spinors into standard form.

Symmetries and geometry in general relativity

This paper provides a geometrical discussion of affine (including isometric and homothetic), conformal and curvature symmetries in space-time. Emphasis is laid more on a systematic presentation of the material rather than the accumulation of exact solutions of Einstein's equations. A review of recent work is given and some new results presented.

A computer-aided complete classification of geometries in general relativity. First results

We report on a computer-aided classification of geometries in terms of the curvature tensor and its covariant derivatives. This extends the Petrov classification to a complete classification of the local geometry and the dimensions of the group of motions and its isotropy subgroup are obtained. As a simple example of the usefulness of the method three of the Harrison metrics are shown to be equivalent.

Methods of Local and Global Differential Geometry in General Relativity. Proceedings of the Regional Conference on Relativity held at the University of Pittsburgh, Pennsylvania, July 13–17, 1970 (Lecture Notes in Physics, Vol. 14). VI + 188 S. Berlin/Heidelberg/New York 1972. Springer‐Verlag. Preis brosch. DM 18,—

Methods of Local and Global Differential Geometry in General Relativity. Proceedings of the Regional Conference on Relativity held at the University of Pittsburgh, Pennsylvania, July 13–17, 1970 (Lecture Notes in Physics, Vol. 14). VI + 188 S. Berlin/Heidelberg/New York 1972. Springer‐Verlag. Preis brosch. DM 18,—