Diagonal Metric Research Articles

Static cylindrically symmetric dyonic wormholes in six-dimensional Kaluza-Klein theory: Exact solutions

We study cylindrically symmetric Abelian wormholes in ($4+n$)-dimensional Kaluza-Klein theory. It is shown that static, four-dimensional, cylindrically symmetric solutions in ($4+n$)-dimensional Kaluza-Klein theory with maximal Abelian isometry group $\mathrm{U}(1{)}^{n}$ of the internal space with diagonal internal metric can be obtained, as in the case of a supersymmetric static black hole [M. Cveti\ifmmode \check{c}\else \v{c}\fi{} and D. Youm, Phys. Rev. D 52, 2144 (1995)], only if the isometry group of the internal space is broken down to the $\mathrm{U}(1{)}_{e}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1{)}_{m}$ gauge group; they correspond to dyonic configurations with one electric (${Q}_{e}$) and one magnetic (${Q}_{m}$) charge that are related either to the same $\mathrm{U}(1{)}_{e}$ or $\mathrm{U}(1{)}_{m}$ gauge field or to different factors of the $\mathrm{U}(1{)}_{e}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1{)}_{m}$ gauge group of the effective six-dimensional Kaluza-Klein theory. We find new exact solutions of the six-dimensional Kaluza-Klein theory with two Abelian gauge fields, a dilaton field and a scalar field, associated with the internal metric. We obtain new types of cylindrically symmetric wormholes supported by the radial and longitudinal electric and magnetic fields.

Free particle geodesic affine parameter time-space coordinate systems

Taking the time-series t as independent variable, the parameter equations {Xi(t)} of free particle space geodesic can be given. By transforming affine parameter R(t) we achieve homogeneous geodesic differential equations, and derive the first-order differential equations which are satisfied by affine parameter R and the sequence of analytical solutions R marked by rational number Cu. In light of R we define the distance unit of flat four-dimensional coordinate system {t,r,θ,φ}, and then establish a free particle geodesic affine parameter time-space coordinate system {t,ξ,θ,φ}. By the study of the diagonalization process of special relativity time-space interval model metric tensor g in {t,ξ,θ,φ}, we find the spatial and temporal line characteristic quantities t1(t,ξ), τ1(τ,ξ),tt(t,τ,ξ) and ττ1(t,τ,ξ) corresponding to diagonal metric. Derived from these quantities, the dimension of time-space coordinate system is less than 4.

Minimal Lagrangian submanifolds in ℂP n with diagonal metric

We propose a method for constructing minimal Lagrangian submanifolds in ℂPn using the Baker-Akhiezer functions of algebraic spectral curves.

Action of the gravitational field on the dynamical Casimir effect

In this paper, we analyze the action of the gravitational field on the dynamical Casimir effect. We consider a massless scalar field confined in a cuboid cavity placed in a gravitational field described by a static and diagonal metric. With one of the plane mirrors of the cavity allowed to move, we compute the average number of particles created inside the cavity by means of the Bogoliubov coefficients computed through perturbative expansions. We apply our result to the case of an oscillatory motion of the mirror, assuming a weak gravitational field described by the Schwarzschild metric. The regime of parametric amplification is analyzed in detail, demonstrating that our computed result for the mean number of particles created agrees with specific associated cases in the literature. Our results, obtained in the framework of the perturbation theory, are restricted, under resonant conditions, to a short-time limit.

Accelerated expansion of the Universe and multidimensional theory of gravitation

A condition for accelerated expansion of the Universe is derived from multidimensional formulas of gravitation, which is a generalization of the general theory of relativity for n dimensions. The model of a one-component ideal isotropic substance with a power-law diagonal metric is used as initial one. Restrictions on the state equations of our 3D space and accompanying additional dimensions are obtained.

Conformal transformations in scalar-tensor gravitation theories

Conformal transformations in scalar-tensor gravitation theories are investigated in the present paper. It is demonstrated that the field equations of these theories can be expressed in the Vaidya form. A scalar field equation is derived based on the compatibility condition for the field equations. The conformal factor, dilaton, and restrictions on the metric are determined for the diagonal metric of type I in the Bianchi classification.

Matching of spatially homogeneous nonstationary space-times to vacuum in cylindrical symmetry

We study the matching of locally rotationally symmetric spatially homogeneous collapsing dust space-times with nonstationary vacuum exteriors in cylindrical symmetry. Given an interior with diagonal metric we prove existence and uniqueness results for the exterior. The matched solutions contain trapped surfaces and singularities and are determined up to a limit surface $\mathcal{H}$ in the exterior. The solutions cannot be asymptotically flat and we present evidence that they are singular on the limit surface.

Geodesics in the generalized Schwarzschild solution

Since Schwarzshild discovered the point-mass solution to Einstein’s equations that bears his name, many equivalent forms of the metric have been obtained. Using an elementary coordinate transformation, we derive the most general form for the stationary, spherically symmetric vacuum metric, which contains one free function. Different choices for the function correspond to common expressions for the line element. From the general metric, we obtain particle and photon trajectories, and use them to specify several time coordinates adapted to physical situations. The most general form of the metric is only slightly more complicated than the Schwarzschild form, which argues for teaching the general line element in place of the diagonal metric.

Principal chiral models with non-constant metric

Field equations for generalized principal chiral models with non-constant metric and their possible Lax formulation are considered. Ansatz for Lax operators is taken linear in currents. Results of a complete investigation of models allowing Lax formulation with linear ansatz for Lax operators on solvable 2- and 3-dimensional groups are given; all such models appear to be almost linear. Also models on simple groupSU(2) with diagonal metric are considered; it turns out that Lax formulation exists in this case for constant metrics only.

Geodesic flow on SO(4), Kac-Moody Lie algebra and singularities in the complex t-plane

The article studies geometrically the Euler-Arnold equations associated to geodesic flow on $SO(4)$ for a left invariant diagonal metric. Such metric were first introduced by Manakov \cite{17} and extensively studied by Mishchenko-Fomenko \cite{18} and Dikii \cite{6}. An essential contribution into the integrability of this problem was also made by Adler-van Moerbeke \cite{4} and Haine \cite{8}. In this problem there are four invariants of the motion defining in $\Bbb{C}^{4}=\operatorname{Lie}(SO(4)\otimes \Bbb{C})$ an affine Abelian surface as complete intersection of four quadrics. The first section is devoted to a Lie algebra theoretical approach, based on the Kostant-Kirillov coadjoint action. This method allows us to linearizes the problem on a two-dimensional Prym variety $\operatorname{Prym}_{\sigma }(C)$ of a genus 3 Riemann surface $C$. In section 2, the method consists of requiring that the general solutions have the Painleve property, i.e., have no movable singularities other than poles. It was first adopted by Kowalewski \cite{10} and has developed and used more systematically \cite{3}, \cite{4}, \cite{8}, \cite{13}. From the asymptotic analysis of the differential equations, we show that the linearization of the Euler-Arnold equations occurs on a Prym variety $\operatorname{Prym}_{\sigma }(G)$ of an another genus 3 Riemann surface $G$. In the last section the Riemann surfaces are compared explicitly.

Four-dimensional supersymmetric dyonic black holes in eleven-dimensional supergravity

A class of 4-dimensional supersymmetric dyonic black hole solutions that arise in an effective 11-dimensional supergravity compactified on a 7-torus is presented. We give the explicit form of dyonic solutions with diagonal internal metric, associated with the Kaluza-Klein sector as well as the three-form field, and relate them to a class of solutions with a general internal metric by imposing a subset of SO(7) ⊂ E 7 transformations. We also give the field transformations which relate the above configurations to 4-dimensional ground state configurations of Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz sector of type-IIA strings on a 6-torus.

Static four-dimensional Abelian black holes in Kaluza-Klein theory.

Static, four-dimensional (4-d) black holes (BH's) in ($4+n$)-d Kaluza-Klein (KK) theory with Abelian isometry and diagonal internal metric have at most one electric ($Q$) and one magnetic ($P$) charges, which can either come from the same $U(1)$-gauge field (corresponding to BH's in effective 5-d KK theory) or from different ones (corresponding to BH's with $U(1)_M\times U(1)_E$ isometry of an effective 6-d KK theory). In the latter case, explicit non-extreme solutions have the global space-time of Schwarzschild BH's, finite temperature, and non-zero entropy. In the extreme (supersymmetric) limit the singularity becomes null, the temperature saturates the upper bound $T_H=1/4\pi\sqrt{|QP|}$, and entropy is zero. A class of KK BH's with constrained charge configurations, exhibiting a continuous electric-magnetic duality, are generated by global $SO(n)$ transformations on the above classes of the solutions.

Addendum to supersymmetric dyonic black holes in Kaluza-Klein theory

We complete the study of 4-dimensional (4-d), static, spherically symmetric, supersymmetric black holes (BH's) in Abelian (4+ n)-d Kaluza-Klein theory, by showing that for such solutions n electric charges Q ≡ (Q i,…,Q n) and n magnetic charges P ≡ (P 1,…,P n) are subject to the constraint P · Q = 0. All such solutions can be obtained by performing the SO( n) rotations, which do not affect the 4-d space-time metric and the volume of the internal space, on the supersymmetric U(1) M × U(1) E BH's, i.e., supersymmetric BH's with a diagonal internal metric.

The Ricci tensor of a diagonal metric

Formulas in suffix notation are obtained for the general diagonal component and the general off-diagonal component of the Ricci tensor of a pseudo-Riemannian manifold of arbitrary dimension with diagonal metric. Two applications to general relativity are briefly considered.

Neutrino in the presence of gravitational fields: Exact solutions

The present paper is a further development of our previous paper [G. V. Shishkin and V. M. Villalba, J. Math. Phys. 33, 2093 (1992)]. Using the algebraic method of separation of variables, the structure of the metric functions is obtained in the presence of gravitational fields associated with a diagonal metric, which permits separation of variables in the Dirac equation. Applying the results obtained in a previous paper, we study the possibilities of finding exact solutions in the Dirac equation for massless neutrinos.

R2 inflation in anisotropic universes.

The evolution of Bianchi type-I and type-IX universes for a theory of gravity with an {epsilon}{ital R}{sup 2} term added to the usual Lagrangian is considered. As in the spatially flat Robertson-Walker case considered previously by others, inflation is found to occur. For any amount of initial anisotropy, the anisotropy decays quickly relative to the length of the inflationary epoch, and the amount of expansion is enhanced by the anisotropy. The exceptions are Bianchi type-IX universes near or at isotropy. In these cases a wide range of initial parameters causes the universe to recollapse, thus reducing the phase space in which inflation can occur. The diagonal metric is shown to be the most general form in the {ital R}{sup 2} theory for both Bianchi type-I universes with a perfect fluid and vacuum Bianchi type-IX models.

Equivalence between fourth-order theories of gravity and general relativity: Possible implications for the cosmological singularity.

A recent result has shown that the fourth-order theories of gravity with the action S=F \ensuremath{\surd}-g (2\ensuremath{\Lambda}+R+(1/2)${\mathrm{aR}}^{2}$)${d}^{4}$x are conformally equivalent to general relativity with a scalar field. This result may enlighten the validity of timelike and null convergence conditions in spatially homogeneous cosmological models with a diagonal metric. This approach points out that, under particular conditions, the timelike and null geodesics may be complete.

Kerr-Schild geometry, spinors, and instantons

The spin connection is studied for the Kerr-Schild metric. Our choice of tetrad leads to particularly simple results. The associated complex SU(2) gauge fields for many important particular cases, including the axially symmetric stationary Kerr metric, are thus presented in a unified fashion. Static spherically symmetric cases are studied in detail, including a cosmological term. A Lorentz gauge transformation is introduced for this class such that after a further (inverse Finkelstein) coordinate transformation a very simple form is obtained in the diagonal static metric. Using this, we study the passage to the Euclidean section, leading to real (anti) self-dual SU(2) gauge fields for the uncharged case. The role of the cosmological constant concerning the Pontryagin indices is elucidated. Finally a class of solutions of the zero-mass Dirac equation is studied in an appendix. The relation of such solutions possessing stringlike singularities, to similar ones in flat space, in the presence of non-Abelian monopoles and instantons is pointed out.