Levi-Civita Metric Research Articles

Geometrical and physical interpretation of the Levi-Civita spacetime in terms of the Komar mass density

We revisit the interpretation of the cylindrically symmetric, static vacuum Levi-Civita metric, known in either Weyl, Einstein–Rosen, or Kasner-like coordinates. The Komar mass density of the infinite axis source arises through a suitable compactification procedure. The Komar mass density mu _{K} calculated in Einstein–Rosen coordinates, when employed as the metric parameter, leads to a number of advantages. It eliminates double coverages of the parameter space, vanishes in flat spacetime and when small, it corresponds to the mass density of an infinite string. After a comprehensive analysis of the local and global geometry, we proceed with the physical interpretation of the Levi-Civita spacetime. First we show that the Newtonian gravitational force is attractive and its magnitude increases monotonically with all positive mu _{K}, asymptoting to the inverse of the proper distance in the radial direction. Second, we reveal that the tidal force between nearby geodesics (hence gravity in the Einsteinian sense) attains a maximum at mu _{K}=1/2 and then decreases asymptotically to zero. Hence, from a physical point of view the Komar mass density of the Levi-Civita spacetime encompasses two contributions: Newtonian gravity and acceleration effects. An increase in mu _{K} strengthens Newtonian gravity but also drags the field lines increasingly parallel, eventually transforming Newtonian gravity through the equivalence principle into a pure acceleration field and the Levi-Civita spacetime into a flat Rindler-like spacetime. In a geometric picture the increase of mu _{K} from zero to infty deforms the planar sections of the spacetime into ever deepening funnels, eventually degenerating into cylindrical topology in an appropriately chosen embedding.

3-dimensional Levi-Civita metrics with projective vector fields

Projective vector fields are the infinitesimal transformations whose local flow preserves geodesics up to reparametrisation. In 1882, Sophus Lie posed the problem of describing 2-dimensional metrics admitting a non-trivial projective vector field, which was solved in recent years. In the present paper, we solve the analog of Lie's problem in dimension 3, for Riemannian metrics and, more generally, for Levi-Civita metrics of arbitrary signature.

ON THE STRUCTURE OF COHOMOLOGICAL MODELS OF ELECTRODYNAMICS AND GENERAL RELATIVITY

In the present paper, we take case of a complex scalar field on a Riemannian manifold and study diff erential geometry and cohomological way to construct field theory Lagrangians. The total Lagrangian of the model is proposed as 4-form on Riemannian manifold. To this end, we use inner product of differential (p, q)-forms and Hodge star operators. It is shown that actions, including that for gravity, can be represented in quadratic forms of fields of matter and basic tetrad fields. Our study is limited to the case of the Levi-Civita metric. We stress some features arisen within the approach regarding nil potency property. Within the model, Klein-Gordon, Maxwell and general relativity actions have been reproduced.

Landau levels in a gravitational field: the Levi-Civita and Kerr spacetimes case

We have recently found that the gravitational field of a static spherical mass removes the Landau degeneracy of the energy levels of a particle moving around the mass inside a magnetic field by splitting the energy of the Landau orbitals. In this paper we present the second part of our investigation of the effect of gravity on Landau levels. We examine the effect of the gravitational fields created by an infinitely long massive cylinder and a rotating spherical mass. In both cases, we show that the degeneracy is again removed thanks to the splitting of the particle's orbitals. The first case would constitute an experimental test - which is quantum mechanical in nature - of the gravitational field of a cylinder. The approach relies on the Newtonian approximation of the gravitational potential created by a cylinder but, in view of self-consistency and for future higher-order approximations, the formalism is based on the full Levi-Civita metric. The second case opens up the possibility for a novel quantum mechanical test of the well-known rotational frame-dragging effect of general relativity.

Linear stability of the Linet–Tian solution with negative cosmological constant

In this paper we analyze the linear stability of the Linet–Tian solution with negative cosmological constant. In the limit of vanishing cosmological constant the Linet–Tian metric reduces to a form of the Levi–Civita metric, and, therefore, it can be considered as a generalization of the former to include a cosmological constant. The gravitational instability of the Levi–Civita metric was recently established, and the purpose of this paper is to investigate what changes result from the introduction of a cosmological constant. A fundamental difference brought about by a (negative) cosmological constant is in the structure at infinity. This introduces an added problem in attempting to define an evolution for the perturbations because the constant time hypersurfaces are not Cauchy surfaces. In this paper we show that under a large set of boundary conditions that lead to a unique evolution of the perturbations, we always find unstable modes, that would generically be present in the evolution of arbitrary initial data, leading to the conclusion that the Linet–Tian space times with negative cosmological constant are linearly unstable under gravitational perturbations.

Axial gravitational perturbations of an infinite static line source

The Levi-Civita metric, which contains a naked singularity that has been interpreted as an infinite static line source, appears, for instance, as the possible end point in the collapse of cylindrically symmetric objects such as shells of dust. The analysis of its gravitational stability should therefore be relevant in the contexts of the cosmic censorship and hoop conjectures. In this paper we study axial gravitational perturbations of the Levi-Civita metric. The perturbations are restricted to axial symmetry but break the cylindrical symmetry of the background metric. We analyze the gauge issues that arise in setting up the appropriate form of the perturbed metric and show that it is possible to restrict the perturbations to diagonal terms but that this does not fix the gauge completely. We derive and solve the perturbation equations. The solutions contain gauge-trivial parts, and we show how to extract the gauge-nontrivial components. We impose appropriate boundary conditions on the solutions and show that these lead to a boundary value problem that determines the allowed functional forms of the perturbation modes. The associated eigenvalues determine a sort of ‘dispersion relation’ for the frequencies and corresponding ‘wave vector’ components. The central result of this analysis is that the spectrum of allowed frequencies contains one unstable (imaginary frequency) mode for every possible choice of the background metric. The completeness of the mode expansion in relation to the initial value problem and to the gauge problem is discussed in detail, and we show that the perturbations contain an unstable component for generic initial data and therefore that the Levi-Civita space times are gravitationally unstable. We also include, for completeness, a set of approximate eigenvalues and examples of the functional form of the solutions.

Geodesics dynamics in the Linet–Tian spacetime with $$\Lambda <0$$ Λ < 0

We investigate the geodesics’ kinematics and dynamics in the Linet–Tian metric with \(\Lambda <0\) and compare with the results for the Levi–Civita metric, when \(\Lambda =0\). This is used to derive new stability results about the geodesics’ dynamics in static vacuum cylindrically symmetric spacetimes with respect to the introduction of \(\Lambda <0\). In particular, we find that increasing \(|\Lambda |\) always increases the minimum and maximum radial distances to the axis of any spatially confined planar null geodesic. Furthermore, we show that, in some cases, the inclusion of any \(\Lambda <0\) breaks the geodesics’ orbit confinement of the \(\Lambda =0\) metric, for both planar and non-planar null geodesics, which are therefore unstable. Using the full system of geodesics’ equations, we provide numerical examples which illustrate our results.

Centrifugal force induced by relativistically rotating spheroids and cylinders

Starting from the gravitational potential of a Newtonian spheroidal shell we discuss electrically charged rotating prolate spheroidal shells in the Maxwell theory. In particular we consider two confocal charged shells which rotate oppositely in such a way that there is no magnetic field outside the outer shell. In the Einstein theory we solve the Ernst equations in the region where the long prolate spheroids are almost cylindrical; in equatorial regions the exact Lewis ‘rotating cylindrical’ solution is so derived by a limiting procedure from a spatially bound system. In the second part we analyze two cylindrical shells rotating in opposite directions in such a way that the static Levi-Civita metric is produced outside and no angular momentum flux escapes to infinity. The rotation of the local inertial frames in flat space inside the inner cylinder is thus exhibited without any approximation or interpretational difficulties within this model. A test particle within the inner cylinder kept at rest with respect to axes that do not rotate as seen from infinity experiences a centrifugal force. Although in suitably chosen axes the spacetime there is exactly Minkowskian out to the inner cylinder, nevertheless, those inertial frame axes rotate with respect to infinity, so relative to the inertial frame inside the inner cylinder a test particle is traversing a circular orbit.

Perfect-Fluid Sources for the Levi-Civita Metric II

A regular static interior solution of Einstein’s field equations representing a perfect fluid cylinder of finite radius is presented. The solution is matched to the Levi-Civita vacuum solution at a boundary where the pressure vanishes. The density and pressure are finite and positive inside the cylinder for a specific range of the mass parameter. The solution could thus represent a reasonable source for the Levi-Civita metric.

Energy-momentum distribution of the Weyl-Lewis-Papapetrou and the Levi-Civita metrics

This paper is devoted to compute the energy-momentum densities for two exact solutions of the Einstein field equations by using the prescriptions of Einstein, Landau-Lifshitz, Papapetrou and M\"{o}ller. The spacetimes under consideration are the Weyl-Lewis-Papapetrou and the Levi-Civita metrics. The Weyl metric becomes the special case of the Weyl-Lewis-Papapetrou solution. The Levi-Civita metric provides constant momentum in each prescription with different energy density. The Weyl-Lewis-Papapetrou metric yields all the quantities different in each prescription. These differences support the well-defined proposal developed by Cooperstock and from the energy-momentum tensor itself.

Perfect-fluid sources for the Levi-Civita metric

Cylindrically symmetric perfect fluid solutions are derived for the Levi-Civita metric. The pressure P is finite. The matter density is greater than the stresses in the material. The solutions are inside cylinders of bounded radius at which the pressure vanishes. The range of σ, for which the sources have been matched to the Levi-Civeta metric is ∞>σ>0. The solutions are regular and satisfy energy conditions

Notes on static cylindrical shells

Static cylindrical shells made of various types of matter are studied as sources of the vacuum Levi-Civita metrics. Their internal physical properties are related to the two essential parameters of the metrics outside. The total mass per unit length of the cylinders is always less than ¼. The results are illustrated by a number of figures.

Gyroscope precession in cylindrically symmetric spacetimes

We present calculations of gyroscope precession in spacetimes described by Levi-Civita and Lewis metrics, under different circumstances. By doing so we are able to establish a link between the parameters of the metrics and observable quantities, thereby providing a physical interpretation for those parameters, without specifying the source of the field.

An energy principle for relativistic fluid cylinders

The spacetime exterior to a static fluid cylinder is known to belong to a family of metrics of generalized Levi-Civita type. The metric due to an infinite straight cosmic string belongs to a one-parameter subclass of this family, and it has been suggested that this metric may be unstable to perturbations in the effective equation of state of the source fluid, leading among other things to measurably different predictions for gravitational lensing. In this paper I construct a general conservation principle for non-static fluid cylinders and show that the transition from a string metric to a non-flat Levi-Civita metric is forbidden energetically. The proposed instability is therefore unlikely.

The Levi-Civita space–time as a limiting case of the γ space–time

It is shown that the Levi-Civita metric can be obtained from a family of the Weyl metric, the γ metric, by taking the limit when the length of its Newtonian image source tends to infinity. In this process a relationship appears between two fundamental parameters of both metrics.

Nonstationary Einstein-Maxwell fields interacting with a superconducting cosmic string

Non stationary cylindrically symmetric exact solutions of the Einstein-Maxwell equations are derived as single soliton perturbations of a Levi-Civita metric, by an application of Alekseev inverse scattering method. We show that the metric derived by L. Witten, interpreted as describing the electrogravitational field of a straight, stationary, conducting wire may be recovered in the limit of a `wide' soliton. This leads to the possibility of interpreting the solitonic solutions as representing a non stationary electrogravitational field exterior to, and interacting with, a thin, straight, superconducting cosmic string. We give a detailed discussion of the restrictions that arise when appropiate energy and regularity conditions are imposed on the matter and fields comprising the string, considered as `source', the most important being that this `source' must necessarily have a non- vanishing minimum radius. We show that as a consequence, it is not possible, except in the stationary case, to assign uniquely a current to the source from a knowledge of the electrogravitational fields outside the source. A discussion of the asymptotic properties of the metrics, the physical meaning of their curvature singularities, as well as that of some of the metric parameters, is also included.

Perfect-fluid sources for the Levi-Civita metric

A class of four static cylindrically symmetric perfect-fluid solutions is derived using a formulation given by Kramer. The solutions are regular and satisfy energy conditions inside cylinders of bounded radii at which the pressure vanishes. Matching to the exterior Levi-Civita metric is achieved and thus the solutions could represent cylindrical sources of gravitational fields.

Perfect-fluid cylinders and walls - sources for the Levi-Civita spacetime

The diagonal metric tensor whose components are functions of one spatial coordinate is considered. Einstein's field equations for a perfect-fluid source are reduced to quadratures once a generating function, equal to the product of two of the metric components, is chosen. The solutions are either static fluid cylinders or walls depending on whether or not one of the spatial coordinates is periodic. Cylinder and wall sources are generated and matched to the vacuum (Levi-Civita) spacetime. A match to a cylinder source is achieved for , where is the mass per unit length in the Newtonian limit , and a match to a wall source is possible for , this case being without a Newtonian limit; the positive (negative) values of correspond to a positive (negative) fluid density. The range of for which a source has previously been matched to the Levi-Civita metric is for a cylinder source.

The parameters of the Lewis metric for the Weyl class

We provide physical interpretation for the four parameters of the stationary Lewis metric restricted to the Weyl class. Matching this spacetime to a completely anisotropic, rigidly rotating, fluid cilinder, we obtain from the junction conditions that one of these parameters is proportional to the vorticity of the source. From the Newtonian approximation a second parameter is found to be proportional to the energy per unit of length. The remaining two parameters may be associated to a gravitational analog of the Aharanov-Bohm effect. We prove, using the Cartan scalars, that the Weyl class metric and static Levi-Civita metric are locally equivalent, i.e., indistinguishable in terms of its curvature.

On the parameters of the Lewis metric for the Lewis class

The physical and geometrical meaning of the four parameters of the Lewis metric for the Lewis class are investigated. Matching this spacetime to a completely anisotropic, rigidly rotating, fluid cylinder, we find from the junction conditions that the four parameters are related to the vorticity of the source. Furthermore, it is shown that one of the parameters must vanish if one wishes to reduce the Lewis class to a locally static spacetime. Using the Cartan scalars it is shown that the Lewis class does not include Minkowski as a special class globally, and that it is not locally equivalent to the Levi-Civita metric. It is also shown that, in contrast with the Weyl class, the parameter responsible for the vorticity appears explicitly in the expression for the Cartan scalars. Finally, to enhance our understanding of the Lewis class, we analyse the van Stockum metric.