On General Solutions for Field Equations in Einstein and?Higher Dimension Gravity

We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing exact solutions in gravity. The main idea of this method is to introduce on (pseudo) Riemannian manifolds an alternative (to the Levi-Civita connection) metric compatible linear connection which is also completely defined by the same metric structure. Such a canonically distinguished connection is with nontrivial torsion which is induced by some nonholonomy frame coefficients and generic off-diagonal terms of metrics. It is possible to define certain classes of adapted frames of reference when the Einstein equations for such an alternative connection transform into a system of partial differential equations which can be integrated in very general forms. Imposing nonholonomic constraints on generalized metrics and connections and adapted frames (selecting Levi-Civita configurations), we generate exact solutions in Einstein gravity and extra dimension generalizations.     

A Possible Reinterpretation of Einstein’s Equations

In this paper, we first review Huei’s formulation in which it is shown that the linearized Einstein equations can be written in the same form as the Maxwell equations. We eliminate some imperfections like the scalar potential which is ill linked to the electric-type field, the Lorentz-type force which is obtained with a time independence restriction and the undesired factor 4 which appears in the magnetic-type part. Second, from these results and in the light of a recent work by C.C. Barros, we propose an extension of the equivalence principle and we suggest a new interpretation for Einstein’s equations by showing that the electromagnetic Maxwell equations can be derived from a new version of Einstein’s ones.     

Approximate Analytical Solutions for a Class of Laminar Boundary-Layer Equations

A simple and efficient approximate analytical technique is presented to obtain solutions to a class of two-point boundary value similarity problems in fluid mechanics. This technique is based on the decomposition method which yields a genera/analytic solution in the form of a convergent infinite series with easily computable terms. Comparative study is carried out to show the accuracy and effectiveness of the technique.     

Well Behaved Charged Generalization of Buchdahl’s Fluid Spheres

In the present article, we have obtained a class of well behaved charged analogues of Buchdahl (Phys. Rev. 116:1027–1034, 1959) neutral perfect fluid solution, which reduces to its neutral counter part in the absence of charge. The solutions so obtained are utilized to depict the super-dense stars models such as models for neutron stars and strange star. It is observed that the models are well behaved for restricted range of the parameter K (1<K≤1.64). Over all the maximum mass and corresponding radius is 2.4495M _Θ and 16.7289 respectively and moment of inertia . Also the pulsars character of the super-dense stars so obtained and has been analyzed with the help of moment of inertia. The analysis of the models reveals both vela and crab pulsars.     

Invariant Solutions of the Yang–Mills Field Equations Coupled to a Scalar Doublet

The Yang–Mills system of field equations which includes coupling to an SU(2) scalar matter doublet is developed. It is shown that an SU(2) current for a scalar matter doublet can be developed. The basic structure which fits the Yang–Mills system is somewhat different from the case of the scalar triplet. Using this form for the scalar current, it is possible to write down the Yang–Mills system which couples to the scalar matter doublet. It is shown that several sets of solutions to this system of equations can be obtained.     

Lie Symmetries of Einstein’s Vacuum Equations in N Dimensions

Abstract

We investigate Lie symmetries of Einstein’s vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein’s equations. Instead of setting to zero the coefficients of all independent partial derivatives (which involves a very complicated substitution of Einstein’s equations), we set to zero the coefficients of derivatives that do not appear in Einstein’s equations. This considerably constrains the coefficients of symmetry generating vector fields. Using the Lie algebra property of generators of symmetries and the fact that general coordinate transformations are symmetries of Einstein’s equations, we are then able to obtain all the Lie symmetries. The method we have used can likely be applied to other types of equations.     

Exact Solutions for Einstein＇s Hyperbolic Geometric Flow

In this paper we investigate the Einstein＇s hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed and we believe that these properties of Einstein＇s hyperbolic geometric flow are very helpful to understanding the Einstein equations and the hyperbolic geometric flow.     

General Dynamical Equations for Dingle’s Space-Times Filled with a Charged Non-perfect Fluid

In this paper we have assumed charged non-perfect fluid as the material content of the space-time. The expression for the “mass function-M(r,y,z,t)” is obtained for the general situation and the contributions from the Ricci tensor in the form of material energy density ρ, pressure anisotropy [\fracp₂+p₃2-p₁][\frac{p_{2}+p_{3}}{2}-p_{1}] , electromagnetic field energy ℰ and the conformal Weyl tensor, viz. energy density of the free gravitational field ε (=\frac-3Y₂4p)(=\frac{-3\Psi_{2}}{4\pi}) are made explicit. This work is an extension of the work obtained earlier by Rao and Hasmani (Math. Today XIIA:71, 1993; New Directions in Relativity and Cosmology, Hadronic Press, Nonantum, 1997) for deriving general dynamical equations for Dingle’s space-times described by this most general orthogonal metric,

Generating Techniques and Analytically Extended Solutions of the Einstein–Maxwell Equations

The correspondence of arbitrary parameters inexact axisymmetric solutions of the Einstein-Maxwellequations constructed with the aid of differentgenerating methods to the analytically extendedparameter sets is discussed and examples of the extendedsolutions are given.     

Uniqueness of Self-similar Solutions to Smoluchowski’s Coagulation Equations for Kernels that are Close to Constant

We consider self-similar solutions to Smoluchowski’s coagulation equation for kernels \(K=K(x,y)\) that are homogeneous of degree zero and close to constant in the sense that $$\begin{aligned} -\varepsilon \le K(x,y)-2 \le \varepsilon \Big ( \Big (\frac{x}{y}\Big )^{\alpha } + \Big (\frac{y}{x}\Big )^{\alpha }\Big ) \end{aligned}$$ for \(\alpha \in [0,1)\) . We prove that self-similar solutions with given mass are unique if \(\varepsilon \) is sufficiently small which is the first such uniqueness result for kernels that are not solvable. Our proof relies on a contraction argument in a norm that measures the distance of solutions with respect to the weak topology of measures.     

A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations

For the Noyes-Fields equations, two-dimenslonal hyperbolic equations of conversation laww and the Burgers-KdV equation, a class of travellng wave solutions has been obtained by constructhag appropriate function transformations. The main idea of solving the equations is that nonlinear partial differential equations are changed into solving algebraic equations. This method has a wide-ranging practicability.     

Dispersion of Singularities of Solutions for Schrödinger Equations

We consider the singularities of solutions for the Schrödinger evolution equation associated with where Q is a d×d real symmetric matrix with the eigenvalues ₁,,_d, and WC(R^d,R) satisfies W(x)=o(|x|²) as |x|. Under additional conditions, we show the dispersion of microlocal singularities of solutions due to the principal symbol in all directions at time and in the nondegenerate directions at t. We also show the weaker dispersion of microlocal singularities of solutions due to the subprincipal symbol W in the degenerate directions at t if W satisfies W(x)=O(|x|¹⁺) as |x| for some 0<<1 and additional conditions. In particular, we prove the dispersion of microlocal singularities of solutions at resonant times when H is a perturbed harmonic oscillator.Partly supported by Grand-in-Aid for Young Scientists (B) 14740110, Japan Society of the Promotion of Science; and Mathematical Sciences Research Institute in BerkeleyDedicated to Professor Mitsuru Ikawa on his sixtieth birthday     

A New Class of Inhomogeneous Cosmological Models with Electromagnetic Field in Normal Gauge for Lyra’s Manifold

A new class of exact solutions of Einstein’s modified field equations in inhomogeneous space-time for perfect fluid distribution with electromagnetic field is obtained in the context of normal gauge for Lyra’s manifold. We have obtained solutions by considering the time dependent displacement field. The source of the magnetic field is due to an electric current produced along the z-axis. Only F ₁₂ is a non-vanishing component of the electromagnetic field tensor. It has been found that the displacement vector β(t) behaves like the cosmological constant Λ in the normal gauge treatment and the solutions are consistent with the recent observations of Type Ia supernovae. Physical and geometric aspects of the models are also discussed in the presence of magnetic field.     

Celebrity Physicist: How the Press Sensationalized Einstein’s Search for a Unified Field Theory

In Einstein’s later years, from the late 1920s onward, his reputation in the physics community as an innovator had faded as he pursued increasingly unrealistic unified field theories. Yet from the perspective of the press, his image and ideas were still marketable. We will see how his various attempts to craft a unified field theory generated numerous headlines, despite their lack of experimental evidence or even realistic solutions. We will examine how Einstein’s “latest theory,” became a much sought-after commodity used to generate interest in books, magazines, and newspapers.     

A Variational Analysis of Einstein–Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds

We establish new existence and non-existence results for positive solutions of the Einstein–scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein–scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.     

Development of the Earth’s Field NMR Spectrometer for Liquid Screening

Earth??s field nuclear magnetic resonance (NMR) can be used for non-destructive in-bottle liquid screening by evaluating the spin?Clattice relaxation times of protons in various liquids. An Earth??s field NMR apparatus has been developed and optimized to measure the spin?Clattice relaxation times of various liquids contained in 500?ml PET bottles. Two methods to generate 90-degree readout pulses using transient oscillating signals generated in a resonator are reported. The lower detection limit of pure water was 1?ml. The minimum measurable spin?Clattice relaxation time was 50?ms.     

Asymptotic Lower Bounds for a Class of Schrödinger Equations

We shall study the following initial value problem:

Existence of Noncompact Static Spherically Symmetric Solutions of Einstein SU(2)-Yang–Mills Equations

We consider static spherically symmetric solutions of the Einstein equations with cosmological constant Λ coupled to the SU(2)-Yang–Mills equations that are smooth at the origin r=0. We prove that all such solutions have a radius r _c at which the solution in Schwarzschild coordinates becomes singular. However, for any positive integer N, there exists a small positive Λ _N such that whenever 0<Λ<Λ _N , there exist at least N distinct solutions for which the singularity is only a coordinate singularity and the solution can be extended to r≥r _c . Received: 5 June 2000 / Accepted: 13 March 2001