Kähler spacetime tangent bundle

Requirements are delineated for the spacetime tangent bundle to be Kählerian. In particlar, an almost complex structure is constructed in the case of a Finsler spacetime, and its covariant derivative in terms of the bundle connection is shown to be vanishing, provided the gauge curvature field is vanishing. The Levi-Civita connection coefficients and the Riemann curvature scalar are also specified for the Kähler spacetime tangent bundle.     

Finsler-spacetime tangent bundle

The Levi-Civita connection coefficients of the spacetime tangent bundle, for the case of a Finsler spacetime, are reduced to the form given by Yano and Davies for a generic tangent bundle of a Finsler manifold. A useful expression is also obtained for the Riemann curvature scalar of a Finsler-spacetime tangent bundle.     

Spacetime tangent bundle with torsion

It is demonstrated explicitly that the bundle connection of the Finslerspacetime tangent bundle can be made compatible with Cartan's theory of Finsler space by the inclusion of bundle torsion, and without the restriction that the gauge curvature field be vanishing. A component of the contorsion is made to cancel the contribution of the gauge curvature field to the relevant component of the bundle connection. Also, it is shown that the bundle manifold remains almost complex, and that the almost complex structure can be made to have a vanishing covariant derivative if additional conditions on the torsion are satisfied. However, the Finsler-spacetime tangent bundle remains complex only if the gauge curvature field vanishes.     

Dust Static Cylindrically Symmetric Solutions in f(R) Gravity

This paper is devoted to investigate non-vacuum solutions of cylindrically symmetric spacetime in the context of metric f(R) gravity. We take dust matter to find energy density of the universe. In particular, we find two exact solutions, which correspond to two f(R) models in each case. The first solution provides constant curvature while the second solution corresponds to non-constant curvature. The functions of the Ricci scalar and energy densities are evaluated in each case.     

Dust Static Cylindrically Symmetric Solutions in f(R) Gravity

This paper is devoted to investigate non-vacuum solutions of cylindrically symmetric spacetime in the context of metric f(R) gravity. We take dust matter to find energy density of the universe. In particular, we find two exact solutions, which correspond to two f(R) models in each case. The first solution provides constant curvature while the second solution corresponds to non-constant curvature. The functions of the Ricci scalar and energy densities are evaluated in each case.     

Complex spacetime tangent bundle

It is demonstrated that the spacetime tangent bundle, in the case of a Finsler spacetime, is complex, provided that the gauge curvature field vanishes. This is accomplished by determining the conditions for the vanishing of the Nijenhuis tensor in the anholonomic frame adapted to the spacetime connection.     

Hamiltonian constructions of Kähler-Einstein metrics and Kähler metrics of constant scalar curvature

Assuming the existence of a real torus acting through holomorphic isometries on a Kähler manifold, we construct an ansatz for Kähler-Einstein metrics and an ansatz for Kähler metrics with constant scalar curvature. Using this Hamiltonian approach we solve the differential equations in special cases and find, in particular, a family of constant scalar curvature Kähler metrics describing a non-linear superposition of the Bergman metric, the Calabi metric and a higher dimensional generalization of the LeBrun Kähler metric. The superposition contains Kähler-Einstein metrics and all the geometries are complete on the open disk bundle of some line bundle over the complex projective spaceP _n. We also build such Kähler geometries on Kähler quotients of higher cohomogeneity.Partially supported by the NSF Under Grant No. DMS 8906809     

Maximal proper acceleration and the structure of spacetime

A limiting proper acceleration in nature follows deductively from known physics and compels the union of spacetime and four-velocity space into a maximal-acceleration invariant phase space having an intrinsic Kaluza-Klein-type fiber-bundle structure with manifest gauge properties. The Riemann curvature scalar of the bundle manifold is determined, and a possible action principle is considered to serve as a basis for the generation of field equations.1. This is an expanded version of an invited paper presented at the Fifth Marcel Grossmann Meeting at the University of Western Australia, 8–12 August 1988.     

Letter: Embedding Diagrams for the Reissner-Nordström Spacetime

We consider embedding diagrams for the Reissner-Nordström spacetime. We embed the r–t and r– 2-surfaces into (2+1)/3-Euclidean space and discuss the relation between the diagrams and the corresponding curvature scalar of the 2-metrics.     

A Note on Algebraically Special,Asymptotically Simple and Empty Spacetime

It is proved that subject to certain physically reasonable restrictions on the curvature of spacetime, an algebraically special, asymptotically simple and empty spacetime is necessarily a copy of the Minkowski space. This imposes further requirements a nontrivial asymptotically simple and empty spacetime has to satisfy.     

Energy Distribution for a Power Model of the Plane Symmetric Spacetime in f(R) Gravity

The study of the energy localization in f(R) theories of gravity has attracted much interest in recent years. In this paper, the vacuum solutions of the modified field equations for a power model of plane symmetric metric are studied in metric f(R) gravity with the assumption of constant Ricci scalar. Next, we determine the energy-momentum complexes in f(R) theories of gravity for this spacetime for some important models. We also show that these models satisfy the stability and constant curvature conditions.     

Regular Electric Field in Electromagnetic Coupled to a Scalar Field

In the framework of an electromagnetic field coupled nonminimally with a scalar field in flat spacetime, the existence of a non-singular electric field is proved for a point electric charge or electric monopole. In analogy with the Maxwell-dilaton system introduced by Gibbons and Wells, first, a Maxwell-anti-dilaton system is constructed where the radial electric field of a static electric monopole is coupled to an anti-dilaton. The field equations are solved analytically for the electric and dilaton fields and observe the nonsingular electric field. Also, the self-energy of the electric monopole is found to be finite. Furthermore, the formalism to a Maxwell-scalar field is generalized where a mechanism is introduced upon which the coupled regular-electric field and scalar field is obtained. The formalism shows that for a given regular electric field there are two supersymmetric coupling functions corresponding to a scalar and a phantom field.     

Do scale-invariant fluctuations imply the breaking of de Sitter invariance?

The quantization of the massless minimally coupled (mmc) scalar field in de Sitter spacetime is known to be a non-trivial problem due to the appearance of strong infrared (IR) effects. In particular, the scale-invariance of the CMB power-spectrum – certainly one of the most successful predictions of modern cosmology – is widely believed to be inconsistent with a de Sitter invariant mmc two-point function. Using a Cesaro-summability technique to properly define an otherwise divergent Fourier transform, we show in this Letter that de Sitter symmetry breaking is not a necessary consequence of the scale-invariant fluctuation spectrum. We also generalize our result to the tachyonic scalar fields, i.e. the discrete series of representations of the de Sitter group, that suffer from similar strong IR effects.     

Massive Scalar Field Evolution in the Dyadosphere Spacetime of Charged Black Hole

Scalar field quasinormal modes in the dyadosphere spacetime of charged black hole are studied by using the third-order WKB approximation. From numerical results obtained, we find that the scalar field mass u plays an important role in studying the quasinormal frequencies. With the scalar field mass increases, the real parts increase and the magnitudes of the imaginary parts decrease. Particulary, these change are almost linearly.     

Complex frequencies of a massless scalar field in loop quantum black hole spacetime

Recently,considerable progress has been made in understanding the early universe by loop quantum cosmology.Modesto et al.investigated the loop quantum black hole(LQBH)using improved semiclassical analysis and they found that the LQBH has two horizons,an event horizon and a Cauchy horizon,just like the Reissner-Nordstr¨om black hole.This paper focuses on the dynamical evolution of a massless scalar wave in the LQBH background.By investigating the relation between the complex frequencies of the massless scalar field and the LQBH parameters using the numerical method,we find that the polymeric parameter P makes the massless scalar field decay more quickly and makes the ground scalar wave oscillate slowly.However,the polymeric parameter P causes the frequency of the high harmonic massless scalar wave to shift according to its value.We also find that the loop quantum gravity area gap parameter a 0 causes the massless scalar field to decay more slowly and makes the period of the massless scalar field wave become longer.In the complex ω plane,the frequency curves move counterclockwise when the polymeric parameter P increases and this spiral effect is more obvious for a higher harmonic scalar wave.     

Polarized Spacetime Foam

An approximate model of a spacetime foam is presented. It is supposed that in the spacetime foam each quantum handle is like to an electric dipole and therefore the spacetime foam is similar to a dielectric. If we neglect of linear sizes of the quantum handle then it can be described with an operator containing a Grassman number and either a scalar or a spinor field. For both fields the Lagrangian is presented. For the scalar field it is the dilaton gravity + electrodynamics and the dilaton field is a dielectric permeability. The spherically symmetric solution in this case give us the screening of a bare electric charge surrounded by a polarized spacetime foam and the energy of the electric field becomes finite one. In the case of the spinor field the spherically symmetric solution give us a ball of the polarized spacetime foam filled with the confined electric field. It is shown that the full energy of the electric field in the ball can be very big.     

Generalized Einstein manifolds

A Finslerian manifold is called a generalized Einstein manifold (GEM) if the Ricci directional curvature R(u,u) is independent of the direction. Let F⁰(M, g_t) be a deformation of a compact n-dimensional Finslerian manifold preserving the volume of the unitary fibre bundle W(M). We prove that the critical points g₀ F⁰(g_t) of the integral I(g_t) on W(M) of the Finslerian scalar curvature (and certain functions of the scalar curvature) define a GEM. We give an estimate of the eigenvalues of Laplacian Δ defined on W(M) operating on the functions coming from the base when (M, g) is of minima fibration with a constant scalar curvature H admitting a conformal infinitesimal deformation (CID). We obtain λ ≥ H/(n − 1) (Δf = λf). If M is simply connected and λ = H/(n − 1), then (M, g) is Riemannian and is isometric to an n-sphere. We first calculate, in the general case, the formula of the second variationals of the integral I (g_t) for G = g₀, then for a CID we show that for certain Finslerian manifolds, I″(g₀) > 0. Applications to the gravitation and electromagnetism in general relativity are given. We prove that the spaces characterizing Einstein-Maxwell equations are GEMs.     

An inequality between intrinsic and extrinsic scalar curvature invariants for codimension 2 embeddings

We study the local and isometric embedding of an m-dimensional Lorentzian manifold in an (m+2)-dimensional pseudo-Euclidean space. An inequality is proven between the basic curvature invariants, i.e. the intrinsic scalar curvature and the extrinsic mean and scalar normal curvature. The inequality becomes an equality if the two components of the second fundamental form have a specified form with respect to some orthonormal basis of the manifold. As an application we look at the space–times embedded in a six-dimensional pseudo-Euclidean space for which the equality holds. They turn out to be Petrov type D models filled with an anisotropic perfect fluid and containing a timelike two-surface of constant curvature.     

Stability problem in Rindler spacetime

The stability problem of the Rindler spacetime is carefully studies by using the scalar wave perturbation. Using two different coordinate systems, the scalar wave equation is investigated. The results are different in the two cases. They are analysed and compared with each other in detail. The following conclusions are obtained: (a) the Rindler spacetime as a whole is not stable; (b) the Rindler spacetime can exist stably only as part of the Minkowski spacetime, and the Minkowski spacetime can be a real entity independently; (c) there are some defects for the scalar wave equation written by the Rindler coordinates, and it is unsuitable for the investigation of the stability properties of the Rindler spacetime. All these results may shed some light on the stability properties of the Schwarzschild black hole. It is natural and reasonable for one to infer that: (a) perhaps the Regge--Wheeler equation is not sufficient to determine the stable properties; (b) the Schwarzschild black hole as a whole might be really unstable; (c) the Kruskal spacetime is stable and can exist as a real physical entity; whereas the Schwarzschild black hole can occur only as part of the Kruskal spacetime.     

On the Ricci Curvature of Compact Spacelike Hypersurfaces in Einstein Conformally Stationary-Closed Spacetimes

In this paper we develop an integral formula involving the Ricci and scalar curvatures of a compact spacelike hypersurface M in a spacetime equipped with a timelike closed conformal vector field K (in short, conformally stationary-closed spacetime), and we apply it, when is Einstein, in order to establish sufficient conditions for M to be a leaf of the foliation determined by K and to obtain some non-existence results. We also get some interesting consequences for the particular case when is a generalized Robertson-Walker spacetime.