Energy of the Kerr–Newman-AdS black hole by using approximate Lie symmetries

Using Lie approximate symmetry methods for differential equations second-order approximate symmetries of the geodesic equations for the Kerr–Newman-AdS (KN-AdS) spacetime are investigated. For this purpose the KN-AdS metric is considered as a second perturbation of the AdS metric. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the KN-AdS spacetime has to be rescaled. There is an extra contribution to the energy of the KN-AdS spacetime due to the cosmological constant. This energy expression is compared with that for the Kerr–Newman (KN) spacetime.     

APPROXIMATE PARTIAL NOETHER OPERATORS OF THE SCHWARZSCHILD SPACETIME

The objective of this paper is twofold: (a) to find a natural example of a perturbed Lagrangian that has different partial Noether operators with symmetries different from those of the underlying Lagrangian. First we regard the Schwarzschild spacetime as a perturbation of the Minkowski spacetime and investigate the approximate partial Noether operators for this perturbed spacetime. It is shown that the Minkowski spacetime has 12 partial Noether operators, 10 of which are different from the 17 Noether symmetries for this spacetime. It is found that for the perturbed Schwarzschild spacetime we recover the exact partial Noether operators as trivial first-order approximate partial Noether operators and there is no non-trivial approximate partial Noether operator as for the Noether case. As a consequence we state a conjecture. (b) Then we prove a conjecture that the approximate symmetries of a perturbed Lagrangian form a subalgebra of the approximate symmetries of the corresponding perturbed Euler–Lagrange equations and illustrate it by our examples. This is in contrast to approximate partial Noether operators.     

Effect of dark energy models on the energy content of charged and rotating black holes

The energy content of the charged-Kerr(CK)spacetime surrounded by dark energy(DE)is investigated using approximate Lie symmetry methods for the differential equations.For this,we consider three different DE scenarios:cosmological constant with an equation of state parameter$ω_q=-2/3,quintessence DE with an equation of state parameterω_c=-1,and a frustrated network of cosmic strings with an equation of state parameterω_n=-1/3.To study the gravitational energy of the CK black hole surrounded by the DE,we explore the symmetries of the 2nd-order perturbed geodesic equations.It is noticed,for all the values ofω,the exact symmetries are recovered as 2nd-order approximate trivial symmetries.These trivial approximate symmetries give the rescaling of arc length parameter s in this spacetime which indicates that the energy in the underlying spacetime has to be rescaled by a factor that depends on the black hole parameters and the DE parameter.This rescaling factor is compared with the factor of the CK spacetime found in[Hussain et al.Gen.Relativ.Gravit.(2009)]and the effects of the DE on it are discussed.It is observed that for all the three values of the equation of state parameterω,the effect of DE results in decreased energy content of the black hole spacetime,regardless of values of the charge Q,spin a and the DE parameterα.This reduction in the energy content due to the involvement of the DE favours the idea of mass reduction of black holes by accretion of DE given by[Babichev et al.Phys.Rev.Lett.(2004)].     

Comment on “Classification of Cosmic Scale Factor via Noether Gauge Symmetries” [Int. J. Theor. Phys. 54, 2343 (2015)]

We discuss the relationship between the Noether point symmetries of the geodesic Lagrangian, in a (pseudo)Riemannian manifold, with the elements of the Homothetic algebra of the space. We observe that the classification problem of the Noether symmetries for the geodesic Lagrangian is equivalent with the classification of the Homothetic algebra of the space, which in the case of a Friedmann-Lemaître-Robertson-Walker spacetime is a well-known result in the literature.     

The zero mass limit of Kerr and Kerr-(anti-)de-Sitter space-times: exact solutions and wormholes

Heun-type exact solutions emerge for both the radial and the angular equations for the case of a scalar particle coupled to the zero mass limit of both the Kerr and Kerr-(anti)de-Sitter spacetime. Since any type D metric has Heun-type solutions, it is interesting that this property is retained in the zero mass case. This work further refutes the claims that M going to zero limit of the Kerr metric is both locally and globally the same as the Minkowski metric.     

Killing and Noether Symmetries of Plane Symmetric Spacetime

This paper is devoted to investigate the Killing and Noether symmetries of static plane symmetric spacetime. For this purpose, five different cases have been discussed. The Killing and Noether symmetries of Minkowski spacetime in cartesian coordinates are calculated as a special case and it is found that Lie algebra of the Lagrangian is 10 and 17 dimensional respectively. The symmetries of Taub’s universe, anti-deSitter universe, self similar solutions of infinite kind for parallel perfect fluid case and self similar solutions of infinite kind for parallel dust case are also explored. In all the cases, the Noether generators are calculated in the presence of gauge term. All these examples justify the conjecture that Killing symmetries form a subalgebra of Noether symmetries (Bokhari et al. in Int. J. Theor. Phys. 45:1063, 2006).     

Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.     

n<Superscript><Emphasis Type="Italic">th</Emphasis></Superscript>-Order Approximate Lagrangians Induced by Perturbative Geometries

A family of perturbative Lagrangians that describe approximate and multidimensional Klein-Gordon equations are studied. We probe the existence of approximate Noether symmetries via generalized geometric conditions for a perturbation of any order. The knowledge of the geometric conditions uncovers that unlike exact symmetries, the approximate Noether symmetries of the Lagrangian which describes the motion of a particle in n-dimensional space under the action of an autonomous force, is inequivalent to the Noether symmetries admitted by the Klein-Gordon Lagrangian in general.     

Geometric aspects of some hidden symmetries

Hidden symmetries of two dimensional chiral models are analysed from the geometric point of view. The dual symmetry gives rise to generalized isometries of the metric on the space of dependent variables. The Jacobi equation of geodesic deviation is dual invariant and the generalized isometries lead to generalized symmetries of the field equations. Being variational divergence symmetries they generate families of conservation laws.     

Affine symmetry,geodesics, and homogeneous spacetimes

We show that the conservation laws for the geodesic equation which are associated to affine symmetries can be obtained from symmetries of the Lagrangian for affinely parametrized geodesics according to Noether’s theorem, in contrast to claims found in the literature. In particular, using Aminova’s classification of affine motions of Lorentzian manifolds, we show in detail how affine motions define generalized symmetries of the geodesic Lagrangian. We compute all infinitesimal proper affine symmetries and the corresponding geodesic conservation laws for all homogeneous solutions to the Einstein field equations in four spacetime dimensions with each of the following energy–momentum contents: vacuum, cosmological constant, perfect fluid, pure radiation, and homogeneous electromagnetic fields.     

Classification of local generalized symmetries for the vacuum Einstein equations

A local generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all local generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. To begin, we analyze symmetries that can be built from the metric, curvature, and covariant derivatives of the curvature to any order; these are called natural symmetries and are globally defined on any spacetime manifold. We next classify first-order generalized symmetries, that is, symmetries that depend on the metric and its first derivatives. Finally, using results from the classification of natural symmetries, we reduce the classification of all higher-order generalized symmetries to the first-order case. In each case we find that the local generalized symmetries are infinitesimal generalized diffeomorphisms and constant metric scalings. There are no non-trivial conservation laws associated with these symmetries. A novel feature of our analysis is the use of a fundamental set of spinorial coordinates on the infinite jet space of Ricci-flat metrics, which are derived from Penrose's exact set of fields for the vacuum equations.Dedicated to the memory of H. Rund     

On the Symmetry Structures of the Minkowski Metric and a Weyl Re-Scaled Metric

Symmetries of spacetime manifolds which are given by Killing vectors are compared with the symmetries of a Lagrangian constructed from a Weyl re-scaled metric used in discussing disorder operators in Gauge theories. We find the point generators of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian (Noether symmetries). It is shown that the Noether symmetries obtained by considering the Lagrangian provide additional symmetries which are not provided by the Killing vectors. New conservation law/s are determined.     

Radiation reaction and the self-force for a point mass in general relativity

A point particle of mass mu moving on a geodesic creates a perturbation h(mu), of the spacetime metric g(0), that diverges at the particle. Simple expressions are given for the singular mu/r part of h(mu) and its quadrupole distortion caused by the spacetime. Subtracting these from h(mu) leaves a remainder h(R) that is C1. The self-force on the particle from its own gravitational field corrects the world line at O(mu) to be a geodesic of g(0)+h(R). For the case that the particle is a small nonrotating black hole, an approximate solution to the Einstein equations is given with error of O(mu(2)) as mu-->0.     

耦合Burgers系统的Painlevé性质分析，对称和对称约化

本文给出了耦合Burgers系统的Painlevé性质，逆强对称算子，无穷多对称和李对称约化。通过把强对称和逆强对称算子重复多次作用到耦合Burgers模型的一些平庸对称，如恒等变换，空间平移变换和标度变换上，我们得到了三族无穷多对称。这些对称构成了无穷维李代数。用其中的有限维子代数——点李代数对模型进行对称约化，得到了模型的群不变解。     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

Approximate Symmetry Reduction to the Perturbed One-Dimensional Nonlinear Schrödinger Equation

The one-dimensional nonlinear Schrödinger equation with a perturbation of polynomial type is considered. Using the approximate symmetry perturbation theory, the approximate symmetries and approximate symmetry reduction equations are obtained.     

On quadratic first integrals of the geodesic equations for type {22} spacetimes

It is shown that every type {22} vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries. In particular, the charged Kerr solution (with or without cosmological constant) is shown to admit a Killing tensor of valence 2. The Killing tensor, together with the metric and the two Killing vectors, provides a method of explicitly integrating the geodesics of the (charged) Kerr solution, thus shedding some light on a result due to Carter.     

Geometry of the isotropic oscillator driven by the conformal mode

Geometrization of a Lagrangian conservative system typically amounts to reformulating its equations of motion as the geodesic equations in a properly chosen curved spacetime. The conventional methods include the Jacobi metric and the Eisenhart lift. In this work, a modification of the Eisenhart lift is proposed which describes the isotropic oscillator in arbitrary dimension driven by the one-dimensional conformal mode.     

Hidden symmetries of higher-dimensional rotating black holes

We demonstrate that the rotating black holes in an arbitrary number of dimensions and without any restrictions on their rotation parameters possess the same hidden symmetry as the four-dimensional Kerr metric. Namely, besides the spacetime symmetries generated by the Killing vectors they also admit the (antisymmetric) Killing-Yano and symmetric Killing tensors.