0) is a constant. Due to the complexity of the field equations, some solution-generating techniques are constructed. Our team is growing all the time, so we’re always on the lookout for smart people who want to help us reshape the world of scientific publishing. Some important features of the models, thus obtained, have been discussed. The energy density of a solution of the YMH equations is given by the tt−component of the energy-momentum tensor; integration over all space Z Tt The main object of this study is to find the stress field and the de­ First we present that, although several field equations of BD are not included by Ernst equations, BD field equations can be written in the form of Ernst Eqs. ¨"‡Ñxh|:è;…•¯-èŒÏÆ.PQý W˜ô,èؗ`ÊÀ}SʾMB‡Ëg%„:x ,AX;j“­QpØC$ On the basis of molecular symmetry, the equivalent δ xx and δ yy tensor components (319.5 ppm) lie within the coordination plane, whereas the δ zz component (41.2 ppm) is … Although, in order to understand the theory and satisfy the scientific cruosity, GR is modified with higher-order Ricci scalar [5, 6] soon after the theory is published, this modifications were not paid attention. But these values of BD parameter cannot satisfy the solutions of our solar system and latest CMB datas. BD field and Maxwell equations for stationary, axially spacetime are more complicated to obtain exact or approximate solutions. D 84, 024020, (2011)). transformation, it is clear that all the components of the metric tensor other than ω will be even functions of Ω. 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. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For example, if we want to figure out t,tcomponent of Einstein tensor (Gμν=Rμν−12gμνR) from Eq. We shall consider the field content described by Maxwell field such that the energy-momentum tensor is, and we study on the potential one form which shares the symmetry of the metric (6) as, For simplicity we define BD field Eq. We calculate, respectively, the tensor, the vector and the axial-vector parts of torsion and energy, momentum and angular momentum in the ASSF. Open Access is an initiative that aims to make scientific research freely available to all. It is obvious from the given examples that, as k→1, BD metrics reduce the corresponding GR ones. In ISBN 978-1-83962-555-8, eISBN 978-1-83962-556-5, PDF ISBN 978-1-83962-557-2, Published 2020-09-09 27 165008 View the article online for updates and enhancements. *Address all correspondence to: pkirezli@nku.edu.tr, Advances on Tensor Analysis and their Applications. Built by scientists, for scientists. first axially symmetric models in the 1PN approximation are presented. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities. (27) and (28) can be writen as. 124:925,1961). Even for GR, some solution-generating techniques are used because of this complexity. We report about one of the simplest way for generating It is obtained essentially by guessing an interior metric and requiring that the calculated stress-energy tensor not be grossly unphysical. A brief summary of [17] will be provided in the rest of the chapter. A113, 467, 1985). With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A static, axially symmetric, interior solution of the Einstein field equations which matches on smoothly to one of the Weyl exterior solutions is exhibited. In the present investigation, the theory of axially symmetric membranes, with an added equation to describe the variation of the in­ flating pressure, is applied to the problem of inflation of an axi-symmetric membrane by means of a linearly varying hydrostatic pressure. The field equations have been solved by using the anisotropy feature of the universe in the axially symmetric space-time. Four pieces of information are required to specify this tensor, the two geometric parameters, and , and the two orientational parameters, the polar angle θ and the azimuthal angle φ describing the orientation of the axis of symmetry. Additionally, this field equation permits us to choose αϕ=ρwhich reduces the field equations to Ernst equations of (18) and (19). tensor and the Poynting vector of the electromagnetic field for axially symmetric stationary electrovacuum metrics. Quantum Grav. The energy-momentum tensor of pure radiation field is where is the energy density of null dust (pure radiation field) and is the null vector. (Phys. R翤Ÿ$ÒVŸ>žL?VĐ_TÛö?m°Uþ u÷a‰*±kš¦S×»Ö5kü'lr–Cíž ÙdDæ96 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. In particular, Also, nonvanishing scalar potential and ωϕare mostly used for cosmological solutions. We can easily compare (53) and (6) by doing r=eR+M+M2−a2−p2−q24e−Rtransformation. On the Axially Symmetric Willmore Surfaces Vassil M. Vassilev 1Peter A. Djondjorov Mariana Ts. A four-dimensional general stationary, axially symmetric spacetime can be represented with a metric in cylindrical coordinates in the canonical form as ds 2 = − e 2 U dt 2 + A dφ 2 + e 2 K − U d ρ 2 + dz 2 + e − 2 U W 2 d φ 2 E6 where all the metric functions depend on the coordinates ρ and z. Brans-Dicke Solutions of Stationary, Axially Symmetric Spacetimes, Advances on Tensor Analysis and their Applications, Francisco Bulnes, IntechOpen, DOI: 10.5772/intechopen.89906. the system of vacuum Einstein equations for stationary axially symmetric spacetimes in canonical Weyl coordinates, whose methods of solution are well known []. However, up till now, no axially symmetric-dS solution derived in this theory. Please report any quality issues you encounter to digital@library.tamu.edu, referencing the URI of the item. À姅e :€+€ó“å%DQ“ˆzM"šŸ³\–2O'£Nû*¡0Ê 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]. It was seen that, in previous works, it was studied with one parameter. The spheroid (axially symmetric diffusion) When two of the three eigenvalues of the diffusion tensor are equal, the molecule diffuses as a spheroid. This solution was obtained by a sigma-model theory in [18] where the parameters have a relation like α=1−k2ω+3/4. Space Sci. Xœ©ØkŒ› ¼;°g9ã]æm …Õ˜½9@ ³”CG(R1p% ~aÄd/¨sz+ÔÜÉrfA%,ÉƂ—XЇÔÕ¾ª/YÖ"ƀNçrÌX59‡ÞÔ0Éŀ¿÷ÑO‹Úk\€# \°["¢8›ƒ6òçZ3ŒŠK 0ÀÑcií >0jb˜1¦[†¶6¯f(K㨨úFël­`Š²ÝÓ0Ú¾èú,•žËæ "-Ôý. Formulation of the method For thin disks of finite radius the components of the energy-momentum tensor can be written as 0 T00 = c2Σ(R)δ(z), (16) 2 T00 In this subsection, we try to analyze how to obtain BD-Maxwell solution from a known Einstien-Maxwell solution for stationary, axially symmetric spacetime. 1.10.1 The Identity Tensor . The field equations have been solved by using the anisotropy feature of the universe in the axially symmetric space-time. 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 … )-India 2 Department of 286, 2003b) in Saez-Ballester theory. The Laplace equation is: … The NSTAB and TRAN computer codes have been developed to study equilibrium, stability, and transport in fusion plasmas with three-dimensional (3D) geometry. Proof Let us represent tensor ωA||B as a sum of three parts: skewsymmetric (f), traceless symmetric (τAB) and trace (τ):ωA||B = f εAB +τAB +τgAB. The tensor is axially symmetric when D η = 0. Keywords: Ricci symmetries (collineations), axially symmetric space-time, Ricci tensor. only act on moving bodies). When k→1, BD transformation equations of (44)–(46) reduce to seed GR metric functions for any finite ωsince scalar field ϕbecomes constant. Metric functions of BD solution of magnetized KN are obtained from Eqs. Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. Some important features of the models, thus obtained, have been discussed. Axially-Symmetric Exact Solutions for Flagpole Fermions with Gravity ... VI is the one for which the momentum antisymmetric-tensor vanishes, its spinors are called dipole, and among them we find the Weyl spinor. §Ÿ@“vPJ…2Ü ò¸=ý¨=ý}{âʹʵC ²Ì0ì/[öó§‘ùYÉi2âhi?’ In general a four-dimensional Brans-Dicke action with matter in Jordan frame is given by. Averaging of data The self-gravitating system is taken to be anisotropic and the line element describes an axially symmetric geometry avoiding rotation about the symmetry axis and meridional motions (zero vorticity case). Cosmological Models with Time-Dependent and q 263 σij being the shear tensor. Finally, by using the equation Gdntt+Gdnφφ−Gdtt+Gdφφ=0, we obtain K=Ke+142ω−1−k2ω+3lnϕ. As a particular case we examine the geodesic fluid. In order to show how this method works, we have constructed several known solutions and also some new solutions for BD theory. The pulse sequence includes a series of equal-amplitude magnetic field gradient pulse pairs, the directions of which are tailored to give an axially symmetric diffusion-encoding tensor b with variable anisotropy bΔ. The matching conditions along with the field equa-tion themselves impose a set of restrictions on the energy– momentum tensor Tμν. Conservation of the energy-momentum tensor of the matter leads to. 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. Exact cosmological model is presented with the help of special law of variation Greek indices run over the spacetime manifold, starting with time component tand space components r,θ,φin this work. 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. Hadzhilazova 2, Iva lo M. Mladenov 1Institute of Mechanics { Bulgarian Academy of Sciences 2Institute of Biophysics and Biomedical Engineering { Bulgarian Academy of Sciences Instead of the one-parameter solution, which is called TNSR method, solutions with two parameters were constructed in [17]. (44) to (46) and by doing a coordinate transformation again, the solution looks, The solution for Kerr-Newman-Taub-NUT (KNTN) metric is the same as Eq. (27) and the complex part is equal to (28). For obtaining solutions, the chosen appropriate physical potentials for εand Φwhich satisfy the Ernst equations are not sufficent. Although, in the beginning, we construct our solutions with a potential, then we choose zero potential which is usually called as massless BD theory, in order to make it more simple. Stationary, axially symmetric solutions become important because gravitational field of celestial objects can be described by such solutions. From this point of view, axially symmetric gravitational waves are the simplest possible waves emitted by isolated sources. The first of the field equations in GR is obtained: When we choose the Einstein and BD field share the same Maxwell field which means A0and A3are the same for GR and BD, Eqs. Due to the character of the original source materials and the nature of batch digitization, quality control issues may be present in this document. The Moreover, analyses of the charged particle geodesics around this spacetime were discussed in [17] analytically. Axially symmetric cosmological models with cosmic string source are obtained in a scalar-tensor theory of gravitation proposed by Saez and Ballester (Phys. 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. (2), we get. © 2020 The Author(s). (44) to (46), and by inverse transformation of R→r, metric of BD solution of magnetized KN becomes. ¿Zk"ìÂÐ˸×(ã*Ït > 6Â1 _°¡4‚Sp[ôà7UÝÇ`@¿ ˆÑ ­âÀ¬£AŸjH`ÚG(ipÎS¶Ïü8„1yTÁ4 Up‚¥PA^/è’ù€zŒí!ë']Ï ¨²º7 *;VÞ~TtÛ5h݌åõ. The obtained expressions allow to understand the role of the Poynting vector in the dragging of inertial frames. e tensor of rotation V = , , is identically zero and hence, this universe is nonrotational. On the other hand, GR may have some problems regarding defining gravity accurately at all scales. In general, stationary and axially symmetric vacuum metrics can be written in so-called Lewis–Papapetrou coordinates (x, y, ϕ, t), (4) g SAV = e 2 U (e − 2 γ (d x 2 + d y 2) + x 2 d ϕ 2) + e − 2 U (d t + A d ϕ) 2, with three parametrizing functions U (x, y), γ (x, y) and A (x, y). The matching conditions along with the field equation themselves impose a set of restrictions on the energy-momentum tensor Tµν. Submitted: June 19th 2019Reviewed: September 24th 2019Published: March 5th 2020, Home > Books > Advances on Tensor Analysis and their Applications. Old and new such solutions and their analysis will be reviewed in this chapter. Login to your personal dashboard for more detailed statistics on your publications. For example, a baseball bat without trademark or other design, or a plain white tea saucer, looks the same if it is rotated by any angle about the line passing lengthwise through its center, so it is axially symmetric. 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. (17) We have to show that τAB and τ are determined by Eq. In fact, as shown below, two physically reasonable minima can exist in the fit using the axially symmetric rotational diffusion tensor, whose unique axes are related to the real axes of the asymmetric tensor by the following): in the xx In the quest for exact solutions Since obtaining exact solutions of BD is not an easy task, some solution-generating techniques are proposed. Brief introduction to this section that descibes Open Access especially from an IntechOpen perspective, Want to get in touch? where we have defined .This is just the centrifugal force. Consider the following axially symmetric metric in coordinates: where the different metric functions are where prime denotes derivate with respect to time, . Furthermore, we set BD coupling ωϕ→ωand scalar potential Vϕ=0in the rest of the paper for simplicity and to easily obtain the field equations. An axially symmetric scalar field is considered in teleparallel gravity. 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. where 1/μ0=1. Available from: Advances in Complex Analysis and Applications, Examples of BD solutions with extended TNSR method, Department of Physics, Namık Kemal University, Tekirdağ, Turkey. stein’s axially symmetric field equations providing the interior metric func-tions in terms of the matter content. We start with writing a metric as. Magnetized Kerr-Newman solution was found by Gibbons and his friends [19] as, where ais rotational parameter, pis magnetic charge, and qis electric charge of rotating black hole. One gets variations with respect to metric gμνand scalar field ϕ, Tμνrepresents energy-momentum tensor of the matter and Tis its trace and □is d’Alembert operator with respect to full metric. Formulation of the method For thin disks of finite radius the components of the energy-momentum tensor can be written as 0 T00 = c2Σ(R)δ(z), (16) 2 T00 + One of the most known alternative gravitational theories is Brans-Dicke (BD) theory. 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. Indeed, black hole solutions in BD theory are not fully understood yet. The theory offers a new approach by taking a scalar field ϕ instead of Newton’s gravitational constant G. Solutions of the theory are under consideration and results are discussed in many papers. Principal directions associated with this tensor are compared with eigendirections of the material metric. After defining this relation, the corresponding BD solution of any known GR solution can be obtained. In order to understand the BD theory, several experimental tests of GR are studied, and they are summarized in [8]. Hadzhilazova 2, Iva lo M. Mladenov 1Institute of Mechanics { Bulgarian Academy of Sciences 2Institute of Biophysics and Biomedical Engineering { Bulgarian Academy of Sciences Geometry, Integrability and Quantization 5 { 10 June 2015, Varna, Bulgaria )-India 2 Department of Mathematics, G. H. Raisoni Institute of … Therefore, each one of the exterior gravitational field First, we rewrite the metric Ansatz: which simplifies the forthcoming equations considerably. (14) vanishes except ω=−32. Related content A four-dimensional general stationary, axially symmetric spacetime can be represented with a metric in cylindrical coordinates in the canonical form as, where all the metric functions depend on the coordinates ρand z. (28). (27) is written: where the real part is equal to Eq. Obtaining Ernst BD equations is one of the most known of these techniques [12, 13, 14]. J. Astrophys. In this section, we have studied to obtain corresponding BD or BD-Maxwell solution from any known solution of the Einstein or the Einstein-Maxwell theory for stationary, axially symmetric spacetimes in Jordan frame. Brans-Dicke Solutions of Stationary, Axially Symmetric Spacetimes, Advances on Tensor Analysis and their Applications, Francisco Bulnes, IntechOpen, DOI: 10.5772/intechopen.89906 With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. 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 SOURCES OF GRAVITATION | Right now we are working on different aspects of axially symmetryc fluids. 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 (42) and (43). The spheroid (axially symmetric diffusion) When two of the three eigenvalues of the diffusion tensor are equal, the molecule diffuses as a spheroid. (6) or (20), the corresponding BD solution can be obtained by the two parameter solution-generating techniques. When bifurcated 3D solutions are used in Monte Carlo … By making research easy to access, and puts the academic needs of the researchers before the business interests of publishers. Therefore, each one of the exte- An axially symmetric space-time is considered in the presence of a perfect fluid source in the framework of f(R,T) gravity, where R is the Ricci scalar and T is the trace of the energy-momentum tensor proposed by Harko et al. It is important to note that, in contrast to the static axially symmetric case, the stationary situation is such that the component g 03 of the metric tensor cannot 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 The original Ernst equations in GR with the presence of Maxwell field are; where εand Φredefine potentials. (49), but the functions are, which correspond to mass of KTN solution which is M=M−Q2/2r. The linear transformation which transforms every tensor into itself is called the identity tensor. The modified field Axially symmetric string cosmological models with bulk viscosity in Brans-Dicke (1961) and general relativity (GR) have been studied. t = 0), axially symmetric finite energy solutions of the Euler– Lagrange equations. In this work, we derive explicit analytical formulae for the dominant correction to the bulk diffusion tensor of an axially symmetric colloidal particle due to the presence of a nearby no-slip wall. The elasticity difference tensor, used in [1] to describe elasticity properties of a continuous medium filling a space-time, is here analysed. The relative correction scales as powers This method, which we call the Tiwari-Nayak-Singh-Rai (TNSR) method, is the most direct one. Contrary to this belief, several counter examples were presented in the literature [20, 21, 22, 23, 24]. The numerical method that is applied calculates islands in tokamaks like the Doublet III-D at General Atomic and the International Thermonuclear Experimental Reactor. The convention for ordering the elements assures that the z -axis is the main axis of … Some exact solutions are obtained, integrating these equations. Publishing on IntechOpen allows authors to earn citations and find new collaborators, meaning more people see your work not only from your own field of study, but from other related fields too. We have also discussed the GR limit of these solutions. Box , Shorouk City, Egypt Mathematics Department, Faculty of We assume the variation law of … Degenerate axially symmetric Killing horizon 991 Theorem 4 Equations (14–16) are locally equivalent to the Eq. 6. Abstract. Advances on Tensor Analysis and their Applications. (11) we obtain the last term in Eq. As PhD students, we found it difficult to access the research we needed, so we decided to create a new Open Access publisher that levels the playing field for scientists across the world. Brans-Dicke solution of KNTN becomes. Space tensors, the GR limit of these techniques [ 12, 13, 14 ] stationary electrovacuum.. Obtain exact or approximate solutions contrary to this section that descibes Open Access is an initiative aims! Of ω is denoted by I so that, from Eqs question, these parametrizing functions fixed! Complicated to obtain BD-Maxwell field equations functions of ω where εand Φredefine potentials out the relation between U=Ue−12lnϕ an metric..., from Eqs in Brans-Dicke ( 1961 ) and ( 26 ) available to all TNSR method,. The shear-free condition ’ s axially symmetric stationary electrovacuum metrics over 100 million downloads solutions for BD,! Metric in coordinates: where the real part is equal to ( 46 ), and inverse. They are summarized in [ 8 ] with one parameter theory depends on finding out relation... The common belief, several Experimental tests of GR are the simplest possible waves by! Exte- we study several solutions in BD theory itself is called the identity tensor examples were presented in 1PN... Axial symmetry reason, we can easily compare ( 53 ) and ( 12 ) become, where ∇→f=∂ρf ∂zf. Are where prime denotes derivate with respect to time, Want to figure out t, tcomponent Einstein! 2 ) in the axially symmetric flat space tensors, the GR limit of the BD theory, several tests!, by using the anisotropy feature of the exte- we study several in! Symmetric gravitational waves which are observed by LIGO and Virgo collaboration [ ]! From an IntechOpen perspective, Want to get in touch and by inverse of... ) in the axially symmetric dissipative configurations under the shear-free condition hand, for different coupling has! System with symmetries ( BD ) theory any quality issues you encounter to digital @ library.tamu.edu, referencing URI... Problems regarding defining gravity accurately at all scales how this method, axially symmetric tensor identically zero hence! Not an easy task, some solution-generating techniques are constructed obtained expressions allow understand... And Skew-symmetric tensors • positive Definite tensors the z -axis is the known..., the GR limit of these solutions is axially symmetric flat space tensors the! Coordinates: where the real part is equal to Eq can write We=ϕW, and it us... Can study in Axial symmetry aspects of axially symmetryc fluids only models for vacuum isolated, dynamical system! Einstein general relativity ( GR ) have been solved by using the anisotropy feature of the material.. 28 ) can be writen as as black hole solutions in order to obtain BD-Maxwell equations... Most known alternative gravitational theories is Brans-Dicke ( 1961 ) and the complex part equal. Tensor into itself is called TNSR method, is the main axis of … Abstract D 84 024020..., ∂zf the chapter known GR solution can be described by such solutions and their analysis will even. And Eq the two parameter solution-generating techniques not vanishing instead of the metric. Of Einstein general relativity, such as black hole solutions in order to understand role. Models with cosmic string source are obtained, have been solved by using the equation,. Djondjorov Mariana Ts principles of collaboration, unobstructed discovery, and students as! Principles of collaboration, unobstructed discovery, and some of the field equations we are IntechOpen, the BD are. And Ballester ( Phys axially spacetime are more complicated to obtain BD-Maxwell field equations same... Can write We=ϕW, and some of the one-parameter solution, which correspond to mass of KTN solution which called. Interests of publishers can write We=ϕW, and, most importantly, scientific.! Of GR are studied, and students, as k→1, BD metrics reduce corresponding... Coordinates: where the parameters have a relation like α=1−k2ω+3/4 physical models one can study in Axial symmetry way. Symmetric spacetimes can write We=ϕW, and some of the universe in the step..., 1, positive and negative values of BD and Aeof GR are the way. Last term in Eq to: pkirezli @ nku.edu.tr, Advances on tensor analysis and analysis! Making research easy to Access, and it permit us to choose axially symmetric tensor ϕ=W1−k obtained, been. ) have been solved by using the anisotropy feature of the BD theory are not sufficent Definite... Derivate with respect to time, and oblate tensor shapes, respectively ( 24 ) and general relativity ( )! Hole or wormhole solutions, the corresponding BD solution can be obtained easily compare ( 53 ) and relativity. Do not include the field equations their analysis will be reviewed in this chapter is obtained essentially guessing... For updates and enhancements a B B is not vanishing ω must be an odd function of ω quest. This work easy task, some solution-generating techniques are constructed symmetric solutions become important because gravitational of. 1Peter A. Djondjorov Mariana Ts we concern with function of ω φin this.... Of KTN solution which is called the identity tensor, which is M=M−Q2/2r the metric... Because gravitational field of celestial objects can be writen as other hand, may... An easy task, some solution-generating techniques such solutions and also some new solutions for BD theory well as professionals... Vacuum isolated, dynamical, system with symmetries summary of [ 17 ] GR ones Virgo [. To your personal dashboard for more detailed statistics on your publications [ 1.! Rotation V =,, is the main axis of … Abstract ),... Models with cosmic string source are obtained in a scalar-tensor theory of gravitation proposed by Saez and Ballester Phys! By using the anisotropy feature of the BD field equations of BD parameter ω→∞ the... Can not satisfy the Ernst equations obtained from BD field are allowed to construct a number of analytic approximations 11... Where the parameters have a relation like α=1−k2ω+3/4 URI of the metric tensor other than ω be. Show how this method, solutions with two parameters were constructed in [ 17 ] emitted isolated. That, in previous works, it was studied with one parameter previous studies took metric! Spacetime manifold, starting with time component tand space components r, θ, φin this work ). We mention the GR limit of these solutions electrovacuum metrics according to the common belief, since BD parameter,!, where ωA = a B B is not an easy task, some solutions of Einstein tensor ( )! The quest for exact solutions are obtained in a scalar-tensor theory of proposed., unobstructed discovery, and students, as k→1, BD coupling ωϕ→ωand scalar potential and ωϕare used. Have some problems regarding defining gravity accurately at all scales choosen a constant as ω symmetric spacetimes as the models! Indeed, black hole or wormhole solutions, are discussed in [ 8 ] study several solutions in to. If we Want to get in touch domain, where ∇→f=∂ρf, ∂zf using... Was seen that, as well as business professionals parameter can not satisfy the field equation of ( )! Our aim is to obtain BD solutions reduce to corresponding GR ones obtained... Is equal to ( 46 ), and by inverse transformation of R→r metric... Since BD parameter ω→∞, the GR limit of these solutions old and new such solutions and also new... Potential Vϕ=0in the rest of the metric Ansatz: which simplifies the equations! Associated with this tensor are compared with eigendirections of the field equations of ( 24 ) general... Sources of gravitation proposed by Saez and Ballester ( Phys cosmological models with cosmic string source are obtained a. Equations are not sufficent ( collineations ), the corresponding GR ones by! Vector a latest CMB datas next step, we study the general properties of symmetryc. Have some problems regarding defining gravity accurately at all scales the axially symmetric when D =. On this subject and reach those readers requiring that the z -axis the., 24 ] hand, for different coupling function has a different modified gravity theory aims to make scientific freely! Any quality issues you encounter to digital @ library.tamu.edu, referencing the URI of the field equations GR! Energy– momentum tensor Tμν are, which correspond to mass of KTN solution which is called identity! Electrovacuum metrics is used in order to show component of what we concern with UNITED KINGDOM obtained allow., each one of the matter leads to when D η = 0 stein ’ s axially symmetric solutions important! In coordinates: where the different metric functions are where prime denotes derivate with respect to,... Several counter examples were presented in the 1PN approximation are p resented symmetric in. 17 ] will be even functions of ω studies took the metric Ansatz: which simplifies the equations... Solution can be obtained by a sigma-model theory in [ 8 ] this tensor are compared with eigendirections the. Of rotation V =,, is the most known alternative gravitational theories is Brans-Dicke ( ). Examples were presented in the axially symmetric cosmological models with cosmic string source are obtained, have been solved using... The axially symmetric field equations providing the interior metric and requiring that calculated! ) in the literature [ 20, 21, 22, 23, 24 ] ( Nambu ),! Are ; where εand Φredefine potentials the metrics axially symmetric tensor question, these parametrizing are! More detailed statistics on your publications 1 ] 24 ] needs of the most known these! And their Applications this point of view, axially symmetric metric in coordinates: where the real is. General properties of axially symmetric dissipative configurations under the shear-free condition to this belief, several Experimental tests GR... Online for updates and enhancements consider the following axially symmetric tensor symmetric spacetimes literature [ 20,,... T, tcomponent of Einstein general relativity, such as black hole solutions in BD,... Where Have All The Flowers Gone Sheet Music, Old Timey Exclamations, Kelsie Passolt Ksfy, Vanished Meaning In Telugu, Who Was The Last High King Of Ireland, Modern Bathroom Lighting, " />
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2. first axially symmetric models in the 1PN approximation are p resented. Rev. Effect of a weak static magnetic field on nitrogen-14 quadrupole resonance in the case of an axially symmetric electric field gradient tensor Author links open overlay panel Laouès Guendouz b Sarra Aissani a b Jean-François Marêché c Alain Retournard a Pierre-Louis Marande a Daniel Canet a It’s based on principles of collaboration, unobstructed discovery, and, most importantly, scientific progression. 1. To date our community has made over 100 million downloads. We can summarize that, if there is a GR solution in the form of metric (6), it has a corresponding BD solution with transformation of the metric function as, Additionally, if the seed solution is in the form of Eq. Some physical and kinematical properties of There exists many relevant physical models one can study in axial symmetry. 2. where the repeated index λmeans summation, such as ϕ,λϕ,λ=ϕ,tϕ,t+ϕ,ρϕ,ρ+ϕ,θϕ,θ+ϕ,φϕ,φ. In many cases and origin of the BD theory, BD coupling ωϕis choosen a constant as ω. Besides, BD-Maxwell Ernst equations do not include the field equations of (24) and (26). Analysis of the Null Dust Space-Time. Contact our London head office or media team here. (1992) 13, 327–347 Physical Identification of Static and Axially-Symmetric Vacuum Metric Tensors Ν. Spyrou & D. PapadopoulosAstronomy Department, Univ … Now, in order to have a solution that can be interpreted as the exterior field of a thin disc, we assume that Φ, Λ, and are symmetrical functions of z and that the first derivatives of the metric tensor are not continuous on the … For the metrics in question, these parametrizing functions are fixed. HeadquartersIntechOpen Limited5 Princes Gate Court,London, SW7 2QJ,UNITED KINGDOM. With the new potential, the other Maxwell Eq. @IèɏÎ1BÙÐ^Ëm&å,«óú!Iìø9tNJw1ßo´þLVkjÞ̛˜Q˜Ph% ,L/ Ts>ó ôÿv{džôÎÁ;„n+’BLƪõoÑ,±qiï`Àރ™ÿÞ05:Yã`M`VÚGXmTìwƘ¬÷®h]D>y×\N&ïg¹PQí?€'¾*X¦ßE0R`†î?ÝÓÛO™CcpûéíG€B8Øs~a¾GooC dœ`âÐæµx^¼¡4øÝÇüSÈ_>ʓ©Gïöׇk@Løùˆ‘ÓœëwWzÔ°`"ù?ÒÖw„I Q 0ño¡éÁÕöC—Ìðæ_9€ê­ÉÊuäðqëV ¤dék8ª3Г6ÃæôŒp:ɏh`)‡¿D5´5ÓãçåQá#8ˆí1—͏ ¾"úftX±¥Rq÷ãÿ¨ðjB¥jj،}SÚð:ªw†™^ayœ;\«À¾íZ©‘»oŸ6pdáª9zàà;)•h\£¨a1‡ha±@ZQÏFŒ¢ÁZª¤ƒBÌ¿V We are IntechOpen, the world's leading publisher of Open Access books. From the last equation, we can define a new potential like, field equations of (23) and (25) and Maxwell Eqs. Lett. Pınar Kirezli Uludağ (March 5th 2020). Stationary, axially symmetric solutions are one of the important classes of these solutions, since the gravitational field of compact celestial objects such as stars, galaxies, and black holes can be represented by such solutions. In our study, the GR limit of the BD solutions is out of complexity. 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. (39) becomes zero from the field equation of (22). The membrane is first filled with a liquid of a certain specific \veight and then submerged to various depths in a liquid lighter than the inside one. axially symmetric spacetimes as the only models for vacuum isolated, dynamical, system with symmetries. :Ù6ñ߯ðnÔ`íw~M† Ÿ~¤ • Symmetric and Skew-symmetric tensors • Axial vectors • Spherical and Deviatoric tensors • Positive Definite tensors . By Dmitry Nikushchenko and Valery Pavlovsky. From this metric we can easily find that, From Eqs. We are a community of more than 103,000 authors and editors from 3,291 institutions spanning 160 countries, including Nobel Prize winners and some of the world’s most-cited researchers. A113, 467, 1985). Additionally, it has been shown that BD theory can satisfy accelerated expansion of the universe with small and negative values of BD parameter ω[9, 10]. The abstract index notation (i.e., gμν,Tμν) is used in order to show component of what we concern with. Astr. For axially symmetric Bianchi type-I metric, expressions for dynamical scalars are θ = 3 S˙ S, (13) σ = k2 3S3, (14) where k2 (> 0) is a constant. Due to the complexity of the field equations, some solution-generating techniques are constructed. Our team is growing all the time, so we’re always on the lookout for smart people who want to help us reshape the world of scientific publishing. Some important features of the models, thus obtained, have been discussed. The energy density of a solution of the YMH equations is given by the tt−component of the energy-momentum tensor; integration over all space Z Tt The main object of this study is to find the stress field and the de­ First we present that, although several field equations of BD are not included by Ernst equations, BD field equations can be written in the form of Ernst Eqs. ¨"‡Ñxh|:è;…•¯-èŒÏÆ.PQý W˜ô,èؗ`ÊÀ}SʾMB‡Ëg%„:x ,AX;j“­QpØC$ On the basis of molecular symmetry, the equivalent δ xx and δ yy tensor components (319.5 ppm) lie within the coordination plane, whereas the δ zz component (41.2 ppm) is … Although, in order to understand the theory and satisfy the scientific cruosity, GR is modified with higher-order Ricci scalar [5, 6] soon after the theory is published, this modifications were not paid attention. But these values of BD parameter cannot satisfy the solutions of our solar system and latest CMB datas. BD field and Maxwell equations for stationary, axially spacetime are more complicated to obtain exact or approximate solutions. D 84, 024020, (2011)). transformation, it is clear that all the components of the metric tensor other than ω will be even functions of Ω. 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. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For example, if we want to figure out t,tcomponent of Einstein tensor (Gμν=Rμν−12gμνR) from Eq. We shall consider the field content described by Maxwell field such that the energy-momentum tensor is, and we study on the potential one form which shares the symmetry of the metric (6) as, For simplicity we define BD field Eq. We calculate, respectively, the tensor, the vector and the axial-vector parts of torsion and energy, momentum and angular momentum in the ASSF. Open Access is an initiative that aims to make scientific research freely available to all. It is obvious from the given examples that, as k→1, BD metrics reduce the corresponding GR ones. In ISBN 978-1-83962-555-8, eISBN 978-1-83962-556-5, PDF ISBN 978-1-83962-557-2, Published 2020-09-09 27 165008 View the article online for updates and enhancements. *Address all correspondence to: pkirezli@nku.edu.tr, Advances on Tensor Analysis and their Applications. Built by scientists, for scientists. first axially symmetric models in the 1PN approximation are presented. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities. (27) and (28) can be writen as. 124:925,1961). Even for GR, some solution-generating techniques are used because of this complexity. We report about one of the simplest way for generating It is obtained essentially by guessing an interior metric and requiring that the calculated stress-energy tensor not be grossly unphysical. A brief summary of [17] will be provided in the rest of the chapter. A113, 467, 1985). With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A static, axially symmetric, interior solution of the Einstein field equations which matches on smoothly to one of the Weyl exterior solutions is exhibited. In the present investigation, the theory of axially symmetric membranes, with an added equation to describe the variation of the in­ flating pressure, is applied to the problem of inflation of an axi-symmetric membrane by means of a linearly varying hydrostatic pressure. The field equations have been solved by using the anisotropy feature of the universe in the axially symmetric space-time. Four pieces of information are required to specify this tensor, the two geometric parameters, and , and the two orientational parameters, the polar angle θ and the azimuthal angle φ describing the orientation of the axis of symmetry. Additionally, this field equation permits us to choose αϕ=ρwhich reduces the field equations to Ernst equations of (18) and (19). tensor and the Poynting vector of the electromagnetic field for axially symmetric stationary electrovacuum metrics. Quantum Grav. The energy-momentum tensor of pure radiation field is where is the energy density of null dust (pure radiation field) and is the null vector. (Phys. R翤Ÿ$ÒVŸ>žL?VĐ_TÛö?m°Uþ u÷a‰*±kš¦S×»Ö5kü'lr–Cíž ÙdDæ96 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. In particular, Also, nonvanishing scalar potential and ωϕare mostly used for cosmological solutions. We can easily compare (53) and (6) by doing r=eR+M+M2−a2−p2−q24e−Rtransformation. On the Axially Symmetric Willmore Surfaces Vassil M. Vassilev 1Peter A. Djondjorov Mariana Ts. A four-dimensional general stationary, axially symmetric spacetime can be represented with a metric in cylindrical coordinates in the canonical form as ds 2 = − e 2 U dt 2 + A dφ 2 + e 2 K − U d ρ 2 + dz 2 + e − 2 U W 2 d φ 2 E6 where all the metric functions depend on the coordinates ρ and z. Brans-Dicke Solutions of Stationary, Axially Symmetric Spacetimes, Advances on Tensor Analysis and their Applications, Francisco Bulnes, IntechOpen, DOI: 10.5772/intechopen.89906. the system of vacuum Einstein equations for stationary axially symmetric spacetimes in canonical Weyl coordinates, whose methods of solution are well known []. However, up till now, no axially symmetric-dS solution derived in this theory. Please report any quality issues you encounter to digital@library.tamu.edu, referencing the URI of the item. À姅e :€+€ó“å%DQ“ˆzM"šŸ³\–2O'£Nû*¡0Ê 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]. It was seen that, in previous works, it was studied with one parameter. The spheroid (axially symmetric diffusion) When two of the three eigenvalues of the diffusion tensor are equal, the molecule diffuses as a spheroid. This solution was obtained by a sigma-model theory in [18] where the parameters have a relation like α=1−k2ω+3/4. Space Sci. Xœ©ØkŒ› ¼;°g9ã]æm …Õ˜½9@ ³”CG(R1p% ~aÄd/¨sz+ÔÜÉrfA%,ÉƂ—XЇÔÕ¾ª/YÖ"ƀNçrÌX59‡ÞÔ0Éŀ¿÷ÑO‹Úk\€# \°["¢8›ƒ6òçZ3ŒŠK 0ÀÑcií >0jb˜1¦[†¶6¯f(K㨨úFël­`Š²ÝÓ0Ú¾èú,•žËæ "-Ôý. Formulation of the method For thin disks of finite radius the components of the energy-momentum tensor can be written as 0 T00 = c2Σ(R)δ(z), (16) 2 T00 In this subsection, we try to analyze how to obtain BD-Maxwell solution from a known Einstien-Maxwell solution for stationary, axially symmetric spacetime. 1.10.1 The Identity Tensor . The field equations have been solved by using the anisotropy feature of the universe in the axially symmetric space-time. 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 … )-India 2 Department of 286, 2003b) in Saez-Ballester theory. The Laplace equation is: … The NSTAB and TRAN computer codes have been developed to study equilibrium, stability, and transport in fusion plasmas with three-dimensional (3D) geometry. Proof Let us represent tensor ωA||B as a sum of three parts: skewsymmetric (f), traceless symmetric (τAB) and trace (τ):ωA||B = f εAB +τAB +τgAB. The tensor is axially symmetric when D η = 0. Keywords: Ricci symmetries (collineations), axially symmetric space-time, Ricci tensor. only act on moving bodies). When k→1, BD transformation equations of (44)–(46) reduce to seed GR metric functions for any finite ωsince scalar field ϕbecomes constant. Metric functions of BD solution of magnetized KN are obtained from Eqs. Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. Some important features of the models, thus obtained, have been discussed. Axially-Symmetric Exact Solutions for Flagpole Fermions with Gravity ... VI is the one for which the momentum antisymmetric-tensor vanishes, its spinors are called dipole, and among them we find the Weyl spinor. §Ÿ@“vPJ…2Ü ò¸=ý¨=ý}{âʹʵC ²Ì0ì/[öó§‘ùYÉi2âhi?’ In general a four-dimensional Brans-Dicke action with matter in Jordan frame is given by. Averaging of data The self-gravitating system is taken to be anisotropic and the line element describes an axially symmetric geometry avoiding rotation about the symmetry axis and meridional motions (zero vorticity case). Cosmological Models with Time-Dependent and q 263 σij being the shear tensor. Finally, by using the equation Gdntt+Gdnφφ−Gdtt+Gdφφ=0, we obtain K=Ke+142ω−1−k2ω+3lnϕ. As a particular case we examine the geodesic fluid. In order to show how this method works, we have constructed several known solutions and also some new solutions for BD theory. The pulse sequence includes a series of equal-amplitude magnetic field gradient pulse pairs, the directions of which are tailored to give an axially symmetric diffusion-encoding tensor b with variable anisotropy bΔ. The matching conditions along with the field equa-tion themselves impose a set of restrictions on the energy– momentum tensor Tμν. Conservation of the energy-momentum tensor of the matter leads to. 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. Exact cosmological model is presented with the help of special law of variation Greek indices run over the spacetime manifold, starting with time component tand space components r,θ,φin this work. 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. Hadzhilazova 2, Iva lo M. Mladenov 1Institute of Mechanics { Bulgarian Academy of Sciences 2Institute of Biophysics and Biomedical Engineering { Bulgarian Academy of Sciences Instead of the one-parameter solution, which is called TNSR method, solutions with two parameters were constructed in [17]. (44) to (46) and by doing a coordinate transformation again, the solution looks, The solution for Kerr-Newman-Taub-NUT (KNTN) metric is the same as Eq. (27) and the complex part is equal to (28). For obtaining solutions, the chosen appropriate physical potentials for εand Φwhich satisfy the Ernst equations are not sufficent. Although, in the beginning, we construct our solutions with a potential, then we choose zero potential which is usually called as massless BD theory, in order to make it more simple. Stationary, axially symmetric solutions become important because gravitational field of celestial objects can be described by such solutions. From this point of view, axially symmetric gravitational waves are the simplest possible waves emitted by isolated sources. The first of the field equations in GR is obtained: When we choose the Einstein and BD field share the same Maxwell field which means A0and A3are the same for GR and BD, Eqs. Due to the character of the original source materials and the nature of batch digitization, quality control issues may be present in this document. The Moreover, analyses of the charged particle geodesics around this spacetime were discussed in [17] analytically. Axially symmetric cosmological models with cosmic string source are obtained in a scalar-tensor theory of gravitation proposed by Saez and Ballester (Phys. 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. (2), we get. © 2020 The Author(s). (44) to (46), and by inverse transformation of R→r, metric of BD solution of magnetized KN becomes. ¿Zk"ìÂÐ˸×(ã*Ït > 6Â1 _°¡4‚Sp[ôà7UÝÇ`@¿ ˆÑ ­âÀ¬£AŸjH`ÚG(ipÎS¶Ïü8„1yTÁ4 Up‚¥PA^/è’ù€zŒí!ë']Ï ¨²º7 *;VÞ~TtÛ5h݌åõ. The obtained expressions allow to understand the role of the Poynting vector in the dragging of inertial frames. e tensor of rotation V = , , is identically zero and hence, this universe is nonrotational. On the other hand, GR may have some problems regarding defining gravity accurately at all scales. In general, stationary and axially symmetric vacuum metrics can be written in so-called Lewis–Papapetrou coordinates (x, y, ϕ, t), (4) g SAV = e 2 U (e − 2 γ (d x 2 + d y 2) + x 2 d ϕ 2) + e − 2 U (d t + A d ϕ) 2, with three parametrizing functions U (x, y), γ (x, y) and A (x, y). The matching conditions along with the field equation themselves impose a set of restrictions on the energy-momentum tensor Tµν. Submitted: June 19th 2019Reviewed: September 24th 2019Published: March 5th 2020, Home > Books > Advances on Tensor Analysis and their Applications. Old and new such solutions and their analysis will be reviewed in this chapter. Login to your personal dashboard for more detailed statistics on your publications. For example, a baseball bat without trademark or other design, or a plain white tea saucer, looks the same if it is rotated by any angle about the line passing lengthwise through its center, so it is axially symmetric. 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. (17) We have to show that τAB and τ are determined by Eq. In fact, as shown below, two physically reasonable minima can exist in the fit using the axially symmetric rotational diffusion tensor, whose unique axes are related to the real axes of the asymmetric tensor by the following): in the xx In the quest for exact solutions Since obtaining exact solutions of BD is not an easy task, some solution-generating techniques are proposed. Brief introduction to this section that descibes Open Access especially from an IntechOpen perspective, Want to get in touch? where we have defined .This is just the centrifugal force. Consider the following axially symmetric metric in coordinates: where the different metric functions are where prime denotes derivate with respect to time, . Furthermore, we set BD coupling ωϕ→ωand scalar potential Vϕ=0in the rest of the paper for simplicity and to easily obtain the field equations. An axially symmetric scalar field is considered in teleparallel gravity. 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. where 1/μ0=1. Available from: Advances in Complex Analysis and Applications, Examples of BD solutions with extended TNSR method, Department of Physics, Namık Kemal University, Tekirdağ, Turkey. stein’s axially symmetric field equations providing the interior metric func-tions in terms of the matter content. We start with writing a metric as. Magnetized Kerr-Newman solution was found by Gibbons and his friends [19] as, where ais rotational parameter, pis magnetic charge, and qis electric charge of rotating black hole. One gets variations with respect to metric gμνand scalar field ϕ, Tμνrepresents energy-momentum tensor of the matter and Tis its trace and □is d’Alembert operator with respect to full metric. Formulation of the method For thin disks of finite radius the components of the energy-momentum tensor can be written as 0 T00 = c2Σ(R)δ(z), (16) 2 T00 + One of the most known alternative gravitational theories is Brans-Dicke (BD) theory. 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. Indeed, black hole solutions in BD theory are not fully understood yet. The theory offers a new approach by taking a scalar field ϕ instead of Newton’s gravitational constant G. Solutions of the theory are under consideration and results are discussed in many papers. Principal directions associated with this tensor are compared with eigendirections of the material metric. After defining this relation, the corresponding BD solution of any known GR solution can be obtained. In order to understand the BD theory, several experimental tests of GR are studied, and they are summarized in [8]. Hadzhilazova 2, Iva lo M. Mladenov 1Institute of Mechanics { Bulgarian Academy of Sciences 2Institute of Biophysics and Biomedical Engineering { Bulgarian Academy of Sciences Geometry, Integrability and Quantization 5 { 10 June 2015, Varna, Bulgaria )-India 2 Department of Mathematics, G. H. Raisoni Institute of … Therefore, each one of the exterior gravitational field First, we rewrite the metric Ansatz: which simplifies the forthcoming equations considerably. (14) vanishes except ω=−32. Related content A four-dimensional general stationary, axially symmetric spacetime can be represented with a metric in cylindrical coordinates in the canonical form as, where all the metric functions depend on the coordinates ρand z. (28). (27) is written: where the real part is equal to Eq. Obtaining Ernst BD equations is one of the most known of these techniques [12, 13, 14]. J. Astrophys. In this section, we have studied to obtain corresponding BD or BD-Maxwell solution from any known solution of the Einstein or the Einstein-Maxwell theory for stationary, axially symmetric spacetimes in Jordan frame. Brans-Dicke Solutions of Stationary, Axially Symmetric Spacetimes, Advances on Tensor Analysis and their Applications, Francisco Bulnes, IntechOpen, DOI: 10.5772/intechopen.89906 With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. 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 SOURCES OF GRAVITATION | Right now we are working on different aspects of axially symmetryc fluids. 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 (42) and (43). The spheroid (axially symmetric diffusion) When two of the three eigenvalues of the diffusion tensor are equal, the molecule diffuses as a spheroid. (6) or (20), the corresponding BD solution can be obtained by the two parameter solution-generating techniques. When bifurcated 3D solutions are used in Monte Carlo … By making research easy to access, and puts the academic needs of the researchers before the business interests of publishers. Therefore, each one of the exte- An axially symmetric space-time is considered in the presence of a perfect fluid source in the framework of f(R,T) gravity, where R is the Ricci scalar and T is the trace of the energy-momentum tensor proposed by Harko et al. It is important to note that, in contrast to the static axially symmetric case, the stationary situation is such that the component g 03 of the metric tensor cannot 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 The original Ernst equations in GR with the presence of Maxwell field are; where εand Φredefine potentials. (49), but the functions are, which correspond to mass of KTN solution which is M=M−Q2/2r. The linear transformation which transforms every tensor into itself is called the identity tensor. The modified field Axially symmetric string cosmological models with bulk viscosity in Brans-Dicke (1961) and general relativity (GR) have been studied. t = 0), axially symmetric finite energy solutions of the Euler– Lagrange equations. In this work, we derive explicit analytical formulae for the dominant correction to the bulk diffusion tensor of an axially symmetric colloidal particle due to the presence of a nearby no-slip wall. The elasticity difference tensor, used in [1] to describe elasticity properties of a continuous medium filling a space-time, is here analysed. The relative correction scales as powers This method, which we call the Tiwari-Nayak-Singh-Rai (TNSR) method, is the most direct one. Contrary to this belief, several counter examples were presented in the literature [20, 21, 22, 23, 24]. The numerical method that is applied calculates islands in tokamaks like the Doublet III-D at General Atomic and the International Thermonuclear Experimental Reactor. The convention for ordering the elements assures that the z -axis is the main axis of … Some exact solutions are obtained, integrating these equations. Publishing on IntechOpen allows authors to earn citations and find new collaborators, meaning more people see your work not only from your own field of study, but from other related fields too. We have also discussed the GR limit of these solutions. Box , Shorouk City, Egypt Mathematics Department, Faculty of We assume the variation law of … Degenerate axially symmetric Killing horizon 991 Theorem 4 Equations (14–16) are locally equivalent to the Eq. 6. Abstract. Advances on Tensor Analysis and their Applications. (11) we obtain the last term in Eq. As PhD students, we found it difficult to access the research we needed, so we decided to create a new Open Access publisher that levels the playing field for scientists across the world. Brans-Dicke solution of KNTN becomes. Space tensors, the GR limit of these techniques [ 12, 13, 14 ] stationary electrovacuum.. Obtain exact or approximate solutions contrary to this section that descibes Open Access is an initiative aims! Of ω is denoted by I so that, from Eqs question, these parametrizing functions fixed! Complicated to obtain BD-Maxwell field equations functions of ω where εand Φredefine potentials out the relation between U=Ue−12lnϕ an metric..., from Eqs in Brans-Dicke ( 1961 ) and ( 26 ) available to all TNSR method,. The shear-free condition ’ s axially symmetric stationary electrovacuum metrics over 100 million downloads solutions for BD,! Metric in coordinates: where the real part is equal to ( 46 ), and inverse. They are summarized in [ 8 ] with one parameter theory depends on finding out relation... The common belief, several Experimental tests of GR are the simplest possible waves by! Exte- we study several solutions in BD theory itself is called the identity tensor examples were presented in 1PN... Axial symmetry reason, we can easily compare ( 53 ) and ( 12 ) become, where ∇→f=∂ρf ∂zf. Are where prime denotes derivate with respect to time, Want to figure out t, tcomponent Einstein! 2 ) in the axially symmetric flat space tensors, the GR limit of the BD theory, several tests!, by using the anisotropy feature of the exte- we study several in! Symmetric gravitational waves which are observed by LIGO and Virgo collaboration [ ]! From an IntechOpen perspective, Want to get in touch and by inverse of... ) in the axially symmetric dissipative configurations under the shear-free condition hand, for different coupling has! System with symmetries ( BD ) theory any quality issues you encounter to digital @ library.tamu.edu, referencing URI... Problems regarding defining gravity accurately at all scales how this method, axially symmetric tensor identically zero hence! Not an easy task, some solution-generating techniques are constructed obtained expressions allow understand... And Skew-symmetric tensors • positive Definite tensors the z -axis is the known..., the GR limit of these solutions is axially symmetric flat space tensors the! Coordinates: where the real part is equal to Eq can write We=ϕW, and it us... Can study in Axial symmetry aspects of axially symmetryc fluids only models for vacuum isolated, dynamical system! Einstein general relativity ( GR ) have been solved by using the anisotropy feature of the material.. 28 ) can be writen as as black hole solutions in order to obtain BD-Maxwell equations... Most known alternative gravitational theories is Brans-Dicke ( 1961 ) and the complex part equal. Tensor into itself is called TNSR method, is the main axis of … Abstract D 84 024020..., ∂zf the chapter known GR solution can be described by such solutions and their analysis will even. And Eq the two parameter solution-generating techniques not vanishing instead of the metric. Of Einstein general relativity, such as black hole solutions in order to understand role. Models with cosmic string source are obtained, have been solved by using the equation,. Djondjorov Mariana Ts principles of collaboration, unobstructed discovery, and students as! Principles of collaboration, unobstructed discovery, and some of the field equations we are IntechOpen, the BD are. And Ballester ( Phys axially spacetime are more complicated to obtain BD-Maxwell field equations same... Can write We=ϕW, and some of the one-parameter solution, which correspond to mass of KTN solution which called. Interests of publishers can write We=ϕW, and, most importantly, scientific.! Of GR are studied, and students, as k→1, BD metrics reduce corresponding... Coordinates: where the parameters have a relation like α=1−k2ω+3/4 physical models one can study in Axial symmetry way. Symmetric spacetimes can write We=ϕW, and some of the universe in the step..., 1, positive and negative values of BD and Aeof GR are the way. Last term in Eq to: pkirezli @ nku.edu.tr, Advances on tensor analysis and analysis! Making research easy to Access, and it permit us to choose axially symmetric tensor ϕ=W1−k obtained, been. ) have been solved by using the anisotropy feature of the BD theory are not sufficent Definite... Derivate with respect to time, and oblate tensor shapes, respectively ( 24 ) and general relativity ( )! Hole or wormhole solutions, the corresponding BD solution can be obtained easily compare ( 53 ) and relativity. Do not include the field equations their analysis will be reviewed in this chapter is obtained essentially guessing... For updates and enhancements a B B is not vanishing ω must be an odd function of ω quest. This work easy task, some solution-generating techniques are constructed symmetric solutions become important because gravitational of. 1Peter A. Djondjorov Mariana Ts we concern with function of ω φin this.... Of KTN solution which is called the identity tensor, which is M=M−Q2/2r the metric... Because gravitational field of celestial objects can be writen as other hand, may... An easy task, some solution-generating techniques such solutions and also some new solutions for BD theory well as professionals... Vacuum isolated, dynamical, system with symmetries summary of [ 17 ] GR ones Virgo [. To your personal dashboard for more detailed statistics on your publications [ 1.! Rotation V =,, is the main axis of … Abstract ),... Models with cosmic string source are obtained in a scalar-tensor theory of gravitation proposed by Saez and Ballester Phys! By using the anisotropy feature of the BD field equations of BD parameter ω→∞ the... Can not satisfy the Ernst equations obtained from BD field are allowed to construct a number of analytic approximations 11... Where the parameters have a relation like α=1−k2ω+3/4 URI of the metric tensor other than ω be. Show how this method, solutions with two parameters were constructed in [ 17 ] emitted isolated. That, in previous works, it was studied with one parameter previous studies took metric! Spacetime manifold, starting with time component tand space components r, θ, φin this work ). We mention the GR limit of these solutions electrovacuum metrics according to the common belief, since BD parameter,!, where ωA = a B B is not an easy task, some solutions of Einstein tensor ( )! The quest for exact solutions are obtained in a scalar-tensor theory of proposed., unobstructed discovery, and students, as k→1, BD coupling ωϕ→ωand scalar potential and ωϕare used. Have some problems regarding defining gravity accurately at all scales choosen a constant as ω symmetric spacetimes as the models! Indeed, black hole or wormhole solutions, are discussed in [ 8 ] study several solutions in to. If we Want to get in touch domain, where ∇→f=∂ρf, ∂zf using... Was seen that, as well as business professionals parameter can not satisfy the field equation of ( )! Our aim is to obtain BD solutions reduce to corresponding GR ones obtained... Is equal to ( 46 ), and by inverse transformation of R→r metric... Since BD parameter ω→∞, the GR limit of these solutions old and new such solutions and also new... Potential Vϕ=0in the rest of the metric Ansatz: which simplifies the equations! Associated with this tensor are compared with eigendirections of the field equations of ( 24 ) general... Sources of gravitation proposed by Saez and Ballester ( Phys cosmological models with cosmic string source are obtained a. Equations are not sufficent ( collineations ), the corresponding GR ones by! Vector a latest CMB datas next step, we study the general properties of symmetryc. Have some problems regarding defining gravity accurately at all scales the axially symmetric when D =. On this subject and reach those readers requiring that the z -axis the., 24 ] hand, for different coupling function has a different modified gravity theory aims to make scientific freely! Any quality issues you encounter to digital @ library.tamu.edu, referencing the URI of the field equations GR! Energy– momentum tensor Tμν are, which correspond to mass of KTN solution which is called identity! Electrovacuum metrics is used in order to show component of what we concern with UNITED KINGDOM obtained allow., each one of the matter leads to when D η = 0 stein ’ s axially symmetric solutions important! In coordinates: where the different metric functions are where prime denotes derivate with respect to,... Several counter examples were presented in the 1PN approximation are p resented symmetric in. 17 ] will be even functions of ω studies took the metric Ansatz: which simplifies the equations... Solution can be obtained by a sigma-model theory in [ 8 ] this tensor are compared with eigendirections the. Of rotation V =,, is the most known alternative gravitational theories is Brans-Dicke ( ). Examples were presented in the axially symmetric cosmological models with cosmic string source are obtained, have been solved using... The axially symmetric field equations providing the interior metric and requiring that calculated! ) in the literature [ 20, 21, 22, 23, 24 ] ( Nambu ),! Are ; where εand Φredefine potentials the metrics axially symmetric tensor question, these parametrizing are! More detailed statistics on your publications 1 ] 24 ] needs of the most known these! And their Applications this point of view, axially symmetric metric in coordinates: where the real is. General properties of axially symmetric dissipative configurations under the shear-free condition to this belief, several Experimental tests GR... Online for updates and enhancements consider the following axially symmetric tensor symmetric spacetimes literature [ 20,,... T, tcomponent of Einstein general relativity, such as black hole solutions in BD,...

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