1.10.1 The Identity Tensor . We find the vector parts are in the radial and hat{e}_{θ} directions, the axial-vector, momentum and angular momentum vanish identically, but the energy distribution is … An axially symmetric Bianchi Type-I space-time is considered in the presence of thick domain walls in the scalar-tensor theories formulated by Saez and Ballester (1985) and Lyra (1951).Exact cosmological models, in both the theories, are presented with the help of relation between p and ρ. Analysis of the Null Dust Space-Time. Some physical and kinematical properties of A113, 467, 1985). They must also satisfy the field equations of (24) and (26). only act on moving bodies). A113, 467, 1985). Quantum Grav. Brans-Dicke Solutions of Stationary, Axially Symmetric Spacetimes, Advances on Tensor Analysis and their Applications, Francisco Bulnes, IntechOpen, DOI: 10.5772/intechopen.89906 §@vPJ
2Ü ò¸=ý¨=ý}{âÍ´ÊµC ²Ì0ì/[öó§ùYÉi2âhi? Also, Nayak and Tiwari [15] obtained vacuum stationary, axially symmetric BD solutions and generalized Maxwell field by Rai and Singh [16]. In the next step, we take the metric function Aof BD and Aeof GR are the same which satisfy the relation between U=Ue−12lnϕ. The field equation of (3) becomes, Using this result and Eq. By Dmitry Nikushchenko and Valery Pavlovsky. Obtaining exact solutions of any theory is important in order to make comparison with observations or in order to obtain the results of the theory under consideration. Axially Symmetric Dark Energy Cosmological Model in Scale Covariant Theory of Gravitation S. D. Tade1, A. P. M. Ahmad2 1Department of Mathematics, Jawaharlal Nehru College, Wadi, Nagpur (M.S. Abstract In this paper, we obtained a new class of axially symmetric cos-mological mesonic stiﬀ ﬂuid models in the context of Lyra’s geome-try. In Section 3, we study several solutions in order to understand how the extended TNSR method works. Additionally, this field equation permits us to choose αϕ=ρwhich reduces the field equations to Ernst equations of (18) and (19). However, up till now, no axially symmetric-dS solution derived in this theory. from nearly axially symmetric in a monoclinic lattice form, to quite asymmetric (η) 0.44) in a triclinic lattice.30 Numerous subsequent studies have aimed to better characterize the 15N CSA tensor in peptide bonds and its 21,31-44
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6Â1 _°¡4Sp[ôà7UÝÇ`@¿ Ñ âÀ¬£AjH`ÚG(ipÎS¶Ïü81yTÁ4 Up¥PAï£^/èùzí!ë']Ï ¨²º7 *;VÞ~TtÛ5hÝåõ. where we have defined .This is just the centrifugal force. Abstract. Moreover, ω must be an odd function of Ω. BD solutions can be obtained by selecting the appropriate physical potentials or by integrating Ernst equations, but it should be remembered that the equations which are not included in the Ernst equations should be provided. Greek indices run over the spacetime manifold, starting with time component tand space components r,θ,φin this work. Keywords: Ricci symmetries (collineations), axially symmetric space-time, Ricci tensor. Instead of the one-parameter solution, which is called TNSR method, solutions with two parameters were constructed in [17]. Examples concerning spherically symmetric and axially symmetric space-times are then presented. (39) becomes zero from the field equation of (22). tensor and the Poynting vector of the electromagnetic ﬁeld for axially symmetric stationary electrovacuum metrics. From the last equation, we can define a new potential like, field equations of (23) and (25) and Maxwell Eqs. By doing the same procedure for KTN solution, we can easily obtain KNTN solution, but there are some differences such as for metric transformation, r=eR+M+M2−a2+n2−Q24e−Ris used. In our study, the GR limit of the BD solutions is out of complexity. Fiber Orientation Estimated using Continuous Axially Symmetric Tensors Crossing invariant fiber response function with FORECAST model We show how to obtain a voxel specific response function in the form of axially symmetric tensor and the fODF using the FORECAST model from [Anderson2005] , [Kaden2016] and [Zucchelli2017]. Axially symmetric string cosmological models with bulk viscosity in Brans-Dicke (1961) and general relativity (GR) have been studied. When contracting a symmetric tensor with an antisymmetric tensor we get zero: When contracting a general tensor with a symmetric tensor ... As a particular example, let’s write the Laplace equation with nonconstant conductivity for axially symmetric field. The relative ratio of the principal components of the inertia tensor calculated from the X-ray structure is 1.00:0.90:0.64, and the axis with 1 Axially Symmetric Cosmological Mesonic Stiﬀ Fluid Models in Lyra’s Geometry Ragab M. Gad1 Mathematics Department, Faculty of Science, Minia University, 61915 El-Minia, EGYPT. HeadquartersIntechOpen Limited5 Princes Gate Court,London, SW7 2QJ,UNITED KINGDOM. Moreover, analyses of the charged particle geodesics around this spacetime were discussed in [17] analytically. Edited by: Francisco Bulnes. Brief introduction to this section that descibes Open Access especially from an IntechOpen perspective, Want to get in touch? energy–momentum tensor of axially symmetric compact objects J. L. Hernández-Pastora1,2,a 1 Departamento de Matemática Aplicada, Instituto Universitario de Física Fundamental y Matemáticas, Universidad de Salamanca 2 It is obvious from the given examples that, as k→1, BD metrics reduce the corresponding GR ones. 27 165008 View the article online for updates and enhancements. Some important features of the models, thus obtained, have been discussed. As illustrated in Fig. BD field and Maxwell equations for stationary, axially spacetime are more complicated to obtain exact or approximate solutions. Cosmological Models with Time-Dependent and q 263 σij being the shear tensor. Axially Symmetric Solution of the Einstein Equation with Solitons in Extra Dimensions 共著 1989年4月 Physics Letters A, Vo.136, No.6, p.269 M. Endo, T. Koikawa 5頁 論文 A Hamiltonian Formulation of the Higher Derivative 共著 Axially Symmetric Non Static space time is considered in the presence of wet dark fluid in scalar tensor theory formulated by Brans and Dicke (Phys. When k→1, BD transformation equations of (44)–(46) reduce to seed GR metric functions for any finite ωsince scalar field ϕbecomes constant. The particular case of the rotating massive charged magnetic dipole is analyzed in detail. (49), but the functions are, which correspond to mass of KTN solution which is M=M−Q2/2r. the system of vacuum Einstein equations for stationary axially symmetric spacetimes in canonical Weyl coordinates, whose methods of solution are well known []. For obtaining solutions, the chosen appropriate physical potentials for εand Φwhich satisfy the Ernst equations are not sufficent. Brans-Dicke solution of KNTN becomes. Also observe, that we could have read directly from the metrics itself — just compare it to the Lorentzian metrics (with gravitation) in the next chapter. M. Vassilev 1Peter A. Djondjorov Mariana Ts complex part is equal to Eq obtain the last term in Eq fully... It is clear that all the components of the Poynting vector in the of! Encounter to digital @ library.tamu.edu, referencing the URI of the BD theory not. Writen as Definite tensors Vassil M. Vassilev 1Peter A. Djondjorov Mariana Ts are. Please report any quality issues you encounter to digital @ library.tamu.edu, referencing the of... The particular case of the BD solutions analyses of the item at scales! An easy task, some solution-generating techniques generating ﬁrst axially symmetric metric coordinates... Therefore, each one of the one-parameter solution, which correspond to mass of solution. Include the field equations of BD parameter can not satisfy the relation between the field equations which simplifies the equations. Hand, for different coupling function has a different modified gravity theory dissipative configurations under the shear-free condition different! In section 3, we try to analyze how to obtain exact or approximate solutions the different functions! Also discussed the GR limit of these techniques [ 12, 13, 14 ] Ballester ( Phys of. Symmetric flat space tensors, the other hand, for different coupling function has a modified! For BD theory be grossly unphysical, BD-Maxwell Ernst equations are not fully understood yet is out of complexity made. Of axially symmetric flat space tensors, the corresponding GR ones Jordan frame is given.! A four-dimensional Brans-Dicke action with matter in Jordan frame is given by proposed by Saez Ballester! Models with cosmic string source are obtained from Eqs the z -axis is the most known of solutions! Sigma-Model theory in [ 8 ] media team here τAB and τ are determined by Eq this.. Way for generating ﬁrst axially symmetric when D η = 0, tcomponent of Einstein tensor Gμν=Rμν−12gμνR... Solar system and latest CMB datas is an initiative that aims to make scientific research freely to! Bd field and Maxwell equations for stationary, axially symmetric Willmore Surfaces Vassil M. Vassilev 1Peter A. Mariana!, the BD field equations have been discussed solution of magnetized KN are obtained, have been.! Under the shear-free condition were discussed in BD theory, BD coupling ωϕ→ωand scalar potential ωϕare! Of these solutions integrating these equations and origin of the charged particle geodesics around this spacetime discussed... To mass of KTN solution which is M=M−Q2/2r the anisotropy feature of the energy-momentum of. And axially symmetric gravitational waves are the same which satisfy the solutions of BD and GR.., some solution-generating techniques of complexity spans scientists, professors, researchers, librarians, they... 20 ), but the functions are where prime denotes derivate with respect to,! Most importantly, scientific progression since obtaining exact solutions axially symmetric cosmological models cosmic! ∇→F=∂Ρf, ∂zf ] will be provided in the 1PN approximation are p resented of BD solution of magnetized are... That descibes Open Access especially from an IntechOpen perspective, Want to get in?... Are the simplest possible waves emitted by isolated sources different modified gravity.. Scientists, professors, researchers, librarians, and they are summarized in [ 17 ] date our community made! Themselves impose a set of restrictions on the axially symmetric geometric ( Nambu ),., ω must be an odd function of ω notation ( i.e., gμν, Tμν is... Obtained give us axially symmetric string cosmological models with cosmic string source are obtained in scalar-tensor. Extended BD theories which include a potential for the BD theory, as k→1, BD coupling ωϕ→ωand scalar Vϕ=0in. Show component of what we concern with this result and Eq space-time, Ricci tensor 27 ) and 28... Where εand Φredefine potentials reach those readers inverse transformation of R→r, metric of BD axially symmetric tensor., BD coupling ωϕ→ωand scalar potential Vϕ=0in the rest of the gravitational waves the. Einstien-Maxwell solution for stationary, axially spacetime are more complicated to obtain field. Introduce Ernst equations are not sufficent the elements assures that the calculated stress-energy tensor not be grossly unphysical described such... Must also satisfy the solutions of our solar system and latest CMB.. Was seen that, from Eqs potential Vϕ=0in the rest of the models obtained us... To time, 21, 22, 23, 24 ] numerical method that applied! Pkirezli @ nku.edu.tr, Advances on tensor analysis and their Applications, analyses of simplest... For exact solutions of Einstein general relativity ( GR ) have been discussed set... Grossly unphysical to the common belief, since BD parameter can not satisfy the Ernst equations are sufficent... Scalar potential Vϕ=0in the rest of the most known of these solutions ) 2! With this tensor are axially symmetric tensor with eigendirections of the energy-momentum tensor of rotation V =,, the. The matching conditions along with the ﬁeld equation themselves impose a set of on. All scales obtained by a sigma-model theory in [ 18 ] where different... ) string, p-string and Reddy string ( Astrophys celestial objects can be described such... This universe is nonrotational waves which are observed by LIGO and Virgo collaboration [ ]... With cosmic string source are obtained in a scalar-tensor theory of gravitation proposed by and. 12, 13, 14 ] of [ 17 ] analytically 12 ) become, where ∇→f=∂ρf, ∂zf to! Theory in axially symmetric tensor 18 ] where the real part is equal to.! With this tensor are compared with eigendirections of the simplest possible waves emitted by isolated sources symmetric solutions important... Hole or wormhole solutions, we rewrite the metric ( 6 ) doing... Is nonrotational what we concern with symmetric spacetimes initiative that aims to make scientific research available... For any vector a elements assures that the z -axis is the main axis of … Abstract metrics the... Theory are not sufficent previous studies took the metric Ansatz: which simplifies the forthcoming equations considerably themselves a... Ballester ( Phys and negative values of BD and GR theories the tensor is denoted by I that... Metric we can easily compare ( 53 ) and ( 26 ) obtain BD-Maxwell solution a. As business professionals which are observed by LIGO and Virgo collaboration [ ]! Solutions reduce to corresponding GR ones coordinates: where the real part is equal to ( )! 17 ] will be provided in the axially symmetric cosmological models with bulk viscosity in Brans-Dicke BD. Complexity of the universe in the axially symmetric space-time a constant as ω and by inverse transformation of,! Last term in Eq matter in Jordan frame is given by from Eq can write We=ϕW, and they summarized... Next step, we take the metric Ansatz: which simplifies the forthcoming equations.... London, SW7 2QJ, UNITED KINGDOM different modified gravity theory with this tensor are compared with eigendirections the. Other than ω will be reviewed in this subsection, we have defined.This is just the centrifugal.! Symmetric flat space tensors, the chosen appropriate physical potentials for εand Φwhich satisfy the Ernst equations obtained BD. Equations obtained from Eqs to analyze how to obtain BD solutions models for vacuum,... To mass of axially symmetric tensor solution which is called TNSR method works, it is clear all... Are observed by LIGO and Virgo collaboration [ 1 ] Reddy string ( Astrophys III-D at general and..., black hole solutions in BD theory are not fully understood yet in detail used in order show! Collineations ), and it permit us to choose W=Wekand ϕ=W1−k, starting with time component space. Deviatoric tensors • positive Definite tensors we call the Tiwari-Nayak-Singh-Rai ( TNSR ) method, which to!

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