Exact Interior Solutions for Charged Fluid Spheres

A new method is discussed to obtain the interior solution of Einstein-Maxwell equations for a charged static sphere from a known particular solutions of a similar kind. Beginning with a charged fluid interior solution reported by Patel and Pandya [11], a new interior Reissner-Nördstrom metric is obtained using this method and physical aspects of it are extensively discussed.     

Exact Solutions of Covariant Wave Equations with a Multipole Source Term on Curved Spacetimes

A formalism is presented for calculating exactsolutions of covariant inhomogeneous scalar and tensorwave equations whose source terms are arbitrary ordermultipoles on a curved background spacetime. The developed formalism is based on the theory ofthe higher-order fundamental solutions for wave equationwhich are the distributions that satisfy theinhomogeneous wave equation with the corresponding order covariant derivatives of the Dirac deltafunction on the right-hand side. Like the classicalGreen's function for a scalar wave equation, thehigher-order fundamental solutions contain a direct termwhich has support on the light cone as well as a tailterm which has support inside the light cone. Knowinghow to compute the fundamental solutions of arbitraryorder, one can find exact multipole solutions of wave equations on curved spacetimes. Wepresent complete recurrent algorithms for calculatingthe arbitrary-order fundamental solutions and the exactmultipole solutions in a form convenient for practical computations. As an example we apply thealgorithm to a massless scalar wave field on aparticular Robertson-Walker spacetime.     

Perfect Fluid Spacetimes with a Purely Magnetic Weyl Tensor

We present a solution of the gravitational fieldequations which is similar in form to that given byWainwright. Several cases are considered, in particularwe find a general algebraic perfect fluid solution with equation of state p = whose Weyl tensor is of the purely magnetic type within a finiteregion of the spacetime. It is shown, for an observerwith four-velocity, u_mag say, that themetric's Weyl tensor is purely magnetic within the finiteregion while it is purely electric, as read by anotherobserver with four-velocity u_ele, elsewhere.Another observer, independent of the observers whomeasure the Weyl tensor to be purely electric ormagnetic, interprets the perfect fluid to have anequation of state p = . The Petrov type of themetric, in this case, is I(M) by theArianrhod-McIntosh classification and therefore there exists noconformally related metric which is vacuum. The vacuumseed metrics are derived for the perfect fluidsolutions.     

O(d,d)-invariance in Inhomogeneous String Cosmologies with Perfect Fluid

In the first part of the present paper, we showthat O(d,d)-invariance usually known in a homogeneouscosmological background written in terms of proper timecan be extended to backgrounds depending on one or several coordinates [which may be anyspace-like or time-like coordinate(s)]. In all cases,the presence of a perfect fluid is taken into accountand the equivalent duality transformation in Einstein frame is explicitly given. In the second part,we present several concrete applications to somefour-dimensional metrics, including inhomogeneous ones,which illustrate the different duality transformations discussed in the first part. Note that most ofthe dual solutions given here do not seem to be known inthe literature.     

LETTER TO THE EDITOR: General Non-Rotating Perfect-Fluid Solution with an Abelian Spacelike C3 Including Only One Isometry

The general solution for non-rotating perfect-fluid spacetimes admitting one Killing vector and two conformal (non-isometric) Killing vectors spanning an abelian three-dimensional conformal algebra (C₃) acting on spacelike hypersurfaces is presented. It is of Petrov type D; some properties of the family such as matter contents are given. This family turns out to be an extension of a solution recently given in [9] using completely different methods. The family contains Friedman-Lemaître-Robertson-Walker particular cases and could be useful as a test for the different FLRW perturbation schemes. There are two very interesting limiting cases, one with a non-abelian G₂ and another with an abelian G₂ acting non-orthogonally transitively on spacelike surfaces and with the fluid velocity non-orthogonal to the group orbits. No examples are known to the authors in these classes.     

Comment on the Roberts Solution for the Spherically-Symmetric Einstein-Scalar Field Equations

We critically examine the Roberts homothetic solution for the spherically symmetric Einstein-scalar field equations in double null coordinates, and show that the Roberts solution indeed solves the field equations only for one non-trivial case. We generalize this solution and discuss its relations with other known exact solutions.     

Invariance of the Massless Field Equations under Changes of the Metric

It is shown that the Maxwell equations with sources, expressed in terms of the covariant tensor field F_ijand the current density four-vector Jⁱ, are invariant under the change of the metric g_ijby g_ij = g_ij+ l_il_jif l_iis a principal null direction of F_ijand that an analogous result holds in the case of the massless Klein-Gordon equation if l_iis null and orthogonal to the gradient of the field and in the case of the null dust equations if l_iis parallel to the dust four-velocity. An elementary proof of the following generalization of the Xanthopoulos theorem is also given: Let (g_ij, F_ij) be an exact solution of the Einstein-Maxwell equations and let l_ibe a principal null direction of F_ij, then (g_ij+ l_il_j, F_ij) is also an exact solution of the Einstein-Maxwell equations if and only if (l_il_j, 0) satisfies the Einstein-Maxwell equations linearized about the background solution (g_ij, F_ij). Furthermore, analogous theorems, where the source of the gravitational field is a massless Klein-Gordon field or null dust, are presented.     

The GHP Formalism and a Type N Twisting Vacuum Metric with Killing Vector

It is shown that the formalism introduced byGeroch, Held and Penrose has a geometrical basis. Withthe help of the resulting insight a canonical splittingof the complex function which appears in the standard form of the Algebraically Special metrics isrealized. The results of this splitting are applied tothe problem of a (special) Type N vacuum metric with atwisting principle null direction. It is demonstrated that it is possible (but not feasable) to findthe metric without the use of differential equations. Anestimate of the size of the metric is given.     

On a Static Solution of the Einstein Equation with Incoming and Outgoing Radiation

The Einstein equation with T = k k + where k, l are null is considered with spherical symmetry and staticity. The solution has a naked singularity and is not asymptotically flat. However, it may be interpreted as an envelope for any static spherical body making it more massive. Such an interpretation and some of its implications are detailed.     

General Behaviour of Bianchi VI0 Solutions with an Exponential-Potential Scalar Field

The solutions to the Einstein-Klein-Gordon equations without a cosmological constant are investigated for an exponential potential in a Bianchi VI₀ metric. There exists a two-parameter family of solutions which have a power-law inflationary behaviour when the exponent of the potential, k, satisfies k² < 2. In addition, there exists a two-parameter family of singular solutions for all k² values. A simple anisotropic exact solution is found to be stable when 2 < k².     

The Relation of Generalized Scalar-Tensor Theory with the 5D Space-Time-Matter Theory

The relationship between cosmological solutionsof five-dimensional Space-Time-Matter (STM) theory anda Generalized Scalar-Tensor (GST) theory is investigatedin which the cosmological term Lambda depends not only on a scalar field but also onits time derivative .Identification of these solutions allows us to solve forthe functional form of the cosmological term, and mayhave relevance for the early Universe.     

Toda Chains with Type Am Lie Algebra for Multidimensional m-component Perfect Fluid Cosmology

We consider a D-dimensional cosmological modeldescribing an evolution of Ricci-flat factor spaces,M₁,..., M_n (n 3), in thepresence of an m-component perfect fluid source (n– 1 m 2). We find characteristicvectors, related to the matter constants in thebarotropic equations of state for fluid components ofall factor spaces. We show that, in the case where wecan interpret these vectors as the root vectorsof a Lie algebra of Cartan type A _m = sl(m + 1, i), the model reduces tothe classical open m-body Toda chain. Using an eleganttechnique by Anderson for solving this system, weintegrate the Einstein equationsfor the model and present the metric in aKasner-like form.     

Massive, Non-ghost Solutions for the Dirac Field Coupled Self-consistently to Gravity

We present new, massive, non-ghost solutions for the Dirac field coupled self-consistently to gravity. We employ a gauge-theoretic formulation of gravity which automatically identifies the spin of the Dirac field with the torsion of the gauge fields. Homogeneity of the field observables requires that the spatial sections be flat. Expanding and collapsing singular solutions are given, as well as a solution which expands from a singularity before recollapsing. Torsion effects are only important while the Compton wavelength of the Dirac field is larger than the Hubble radius. We study the motion of spinning point-particles in the background of the expanding solution. The anisotropy due to the torsion is manifest in the particle trajectories.     

Killing Pairs in Flat Space

In this paper the integrability conditions for Killing pairs in flat spaces are investigated and it is shown that only trivial Killing pairs exist.     

Singularity-Free Cosmological Solutions with Non-Rotating Perfect Fluids

The paper establishes the result that solutions of the type described in the title of the article are in essence only those that have been already presented in the literature provided the acceleration vector is hypersurface orthogonal. The procedure adopted in the paper is somewhat novel - while the usual practice is to display an exact solution and then to examine whether it is singularity free, the present paper discovers the conditions which a singularity free solution of the desired type must satisfy. There is no attempt to obtain exact solutions. Simply, the conditions that were ad-hoc introduced in the deduction of singularity free solutions are here shown to follow from the requirement of non-singularity.     

Mixmaster Universe Models with Matter in Jordan-Brans-Dicke Theory

We discuss the dynamics of anisotropic Bianchitype IX models in Jordan-Brans-Dicke cosmological theoryrendering the evolution of a universe model with closedspace near its beginning before inflation sets in. This paper displays how, when writtenin terms of reduced variables, the field equations allowstraightforward partial integration. The mean expansionH, the scalar field, and the three scale factors aregiven in terms of the volume expansion.     

Application of Halpin–Tsai equation to microfibril reinforced polypropylene/poly(ethylene terephthalate) composites

While the main goal of the first part of this study was to check the applicability of the Tsai–Hill equation to a polymer-polymer microfibril reinforced composite (MFC), in which the reinforcing elements represented microfibrils with a diameter around 1–3 μm and aspect ratio of about 100, in the present paper a similar goal involves the Halpin–Tsai equation. In addition, using compatibilised blends an attempt is made to shed light on the mechanism of the microfibril formation during drawing of isotropic polymer blends. For this purpose, injection moulded dog-bone test samples of blends of polypropylene/poly(ethylene terephthalate) (PP/PET) (60/40 wt%) have been prepared starting from highly drawn bristles, also from blends containing 0–9 wt% compatibiliser (ethylene-glycidyl methacrylate). The MFC structure of the test sample is established by X-ray and scanning electron microscope (SEM) analyses. The tensile data are compared with those predicted according to the Halpin–Tsai equation. It is shown that the predicted values are slightly higher than the measured ones and this finding is explained by the presence of a compatibiliser resulting in much lower aspect ratios of the microfibrils. The suggested mechanism of the microfibril formation is based on the coalescence of the individual elongated spheres during drawing. The SEM observations also show that the compatibilised blends contain much shorter microfibrils because the compatibiliser prevents the coalescence process.     

On Average Properties of Inhomogeneous Fluids in General Relativity: Perfect Fluid Cosmologies

For general relativistic spacetimes filled with an irrotational perfect fluid a generalized form of Friedmann's equations governing the expansion factor of spatially averaged portions of inhomogeneous cosmologies is derived. The averaging problem for scalar quantities is condensed into the problem of finding an "effective equation of state" including kinematical as well as dynamical "backreaction" terms that measure the departure from a standard FLRW cosmology. Applications of the averaged models are outlined including radiation-dominated and scalar field cosmologies (inflationary and dilaton/string cosmologies). In particular, the averaged equations show that the averaged scalar curvature must generically change in the course of structure formation, that an averaged inhomogeneous radiation cosmos does not follow the evolution of the standard homogeneous-isotropic model, and that an averaged inhomogeneous perfect fluid features kinematical "backreaction" terms that, in some cases, act like a free scalar field source. The free scalar field (dilaton) itself, modelled by a "stiff" fluid, is singled out as a special inhomogeneous case where the averaged equations assume a simple form.     

Dissipative Nonlinear Schrödinger Equation with External Forcing in Rotational Stratified Fluids and Its Solution

The dissipative nonlinear Schrödinger equation with a forcing item is derived by using of multiple scales analysis and perturbation method as a mathematical model of describing envelope solitary Rossby waves with dissipation effect and external forcing in rotational stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt-Vaisala frequency and β effect are important factors in forming the envelope solitary Rossby waves. By employing Jacobi elliptic function expansion method and Hirota's direct method, the analytic solutions of dissipative nonlinear Schrödinger equation and forced nonlinear Schrödinger equation are derived, respectively. With the help of these solutions, the effects of dissipation and external forcing on the evolution of envelope solitary Rossby wave are also discussed in detail. The results show that dissipation causes slowly decrease of amplitude of envelope solitary Rossby waves and slowly increase of width, while it has no effect on the propagation speed and different types of external forcing can excite the same envelope solitary Rossby waves. It is notable that dissipation and different types of external forcing have certain influence on the carrier frequency of envelope solitary Rossby waves.     

An Exact Solution of Perfect Fluid in Isotropic Coordinates, Compatible with Relativistic Modeling of Star

We present a spherically symmetric solution of the general relativistic field equations in isotropic coordinates for perfect fluid, compatible with a super dense star modeling. The solution is well behaved for all the values of u lying in the range 0<u≤0.12. Further, we have constructed a super-dense star model with all degree of suitability and by assuming the surface density ρ _b =2×10¹⁴ g/cm³. Corresponding to u=0.12, the resulting well behaved model has maximum mass M=0.912M _Θ with radius R _b ≈11.27 km and Moment of inertia 0.97×10⁴⁵ gm?cm². The good matching of our results for Vela pulsars show the stoutness of our model.