Conformal Killing Vector Research Articles

Existence of gradient CKV and gradient conformally stationary LRS spacetimes

In this work, we study the existence of gradient (proper) CKVs in locally rotationally symmetric spacetimes (LRS), those CKVs in the space spanned by the tangent to observers’ congruence and the preferred spatial direction, allowing us to provide a (partial) characterization of gradient conformally static (GCSt) LRS solutions. Irrrotational solutions with non-zero spatial twist admit an irrotational timelike gradient conformal Killing vector field and hence are GCSt. In the case that both the vorticity and twist vanish, that is, restricting to the LRS II subclass, we obtain the necessary and sufficient condition for the spacetime to admit a gradient CKV. This is given by a single wave-like PDE, whose solutions are in bijection to the gradient CKVs on the spacetime. We then introduce a characterization of these spacetimes as GCSt using the character of the divergence of the CKV, provided that the metric functions of the spacetimes obey certain inequalities.

A study of Yukawa–Casimir wormholes in some Rastall frameworks via conformal killing vectors approach

Present work explores the possibility of obtaining analytical wormhole solutions in Rastall theory of gravity and its modified torsion based version where the conformal killing vectors are introduced. To achieve this target, we assume spherically symmetric wormhole metric with anisotropic background matter and find the analytical solutions by taking Casimir as well as Yukawa corrected energy densities into account. The formulated solutions are then examined graphically for checking the essential properties of wormhole configuration and also, the impact of Rastall parameter, in this respect, is explored. It is found that primary wormhole properties hold in all cases while the obtained wormhole solutions in torsion based Rastall theory do not fulfill asymptotic flatness criteria for some certain choices of Rastall parameter. Further, behavior of null energy constraint is explored and the volume integral quantifier is computed for each case. We also discuss the equilibrium of proposed wormhole configurations by checking the behavior of four different forces using generalized Tolman–Oppenheimer–Volkoff equation in Rastall background (as the equilibrium condition gets modified) and conclude that these forces refer to the stability of the system in all cases, especially when one moves away from the wormhole throat.

Effect of Rastall parameter on wormhole spacetime

In this paper, we investigate wormhole geometries within Rastall gravity. The field equations have been solved by assuming that spacetime admits conformal killing vectors. By assuming two separate scenarios, namely anisotropic and isotropic EoS, we are able to find the wormhole solutions. We note that the shape function for both the isotropic and anisotropic cases fails to meet the asymptotic flatness conditions, but both the solutions obey the flaring-out condition. The TOV equation has been used to verify the stability of wormhole solutions. Additionally, we discover that for the anisotropic case, the null energy condition is violated which is graphically analyzed and proves the presence of exotic matter. Finally, we use the volume integral quantifier to determine how much exotic matter is needed near the wormhole throat in the anisotropic scenario.

Efficient Computation of Null Affine Parameters

Finding affine parameters for null geodesics is often of considerable physical importance, especially when studying null geodesics or dealing with conservation laws and/or averaged energy conditions. But explicitly finding null affine parameters is also often quite tedious and can sometimes even be somewhat tricky. Herein we shall demonstrate that the existence of a conformally related spacetime containing a conformal Killing vector, timelike in the domain of outer communication, is quite sufficient to define a preferred set of spatial three-slices—on which a well-defined “affine” three-metric can be introduced to capture the notion of affine null parameter—before explicitly finding the null geodesics. The construction depends on the properties of conformal transformations and on the conserved quantity associated with the conformal Killing vector. Having the affine null parameter in hand before attempting to find the actual null geodesics often quite radically simplifies other parts of the analysis. We emphasize that the successful identification of affine null parameters is a general-purpose tool of wide applicability in both general relativistic and astrophysical settings.

A comprehensive study of Bardeen stars with conformal motion in f(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f({\\mathcal {G}})$$\\end{document} gravity

In this manuscript, we use the conformal killing vectors to solve the Einstein–Maxwell field equations in the background of modified f(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f({\\mathcal {G}})$$\\end{document} gravity. We consider the observational data for four different stars namely PSRJ1614-2230\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$PSR~J1614-2230$$\\end{document}, PSRJ1930+327\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$PSR ~J1930+327$$\\end{document}, CenX-3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Cen~X-3$$\\end{document} and VelaX-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Vela~X-1$$\\end{document} respectively. By using the data, we derive the physical parameters from the matching criteria using conformal killing vectors and the exterior Bardeen geometry. Furthermore, several physical properties of compact stellar structures are examined in order to achieve the physical validity and stability for considered models. The graphical analysis including energy density, pressure components, equation of state parameters, and energy conditions are examined. Moreover, the equilibrium conditions through Tolman–Oppenheimer–Volkoff equation and stability criteria through adiabatic index for the charged stellar structure study are investigated. It is worth noting that all of these compact objects are physically viable and stable under conformal motion in context of Bardeen geometry for two different models of f(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f({\\mathcal {G}})$$\\end{document} theory of gravity. Conclusively, the results show that f(G) gravity with Bardeen model may provide more massive stellar objects as compared to general relativity.

Entanglement entropy in conformal quantum mechanics

We consider sets of states in conformal quantum mechanics associated to generators of time evolution whose orbits cover different regions of the time domain. States labelled by a continuous global time variable define the two-point correlation functions of the theory seen as a one-dimensional conformal field theory. Such states exhibit the structure of a thermofield double built on bipartite eigenstates of generators of non-global time evolution. In terms of the correspondence between radial conformal symmetries in Minkowski space-time and time evolution in conformal quantum mechanics proposed in [1, 2] these generators coincide with conformal Killing vectors tangent to worldlines of Milne and diamond observers at constant radius. The temperature of the thermofield double states in conformal quantum mechanics reproduces the temperatures perceived by such diamond and Milne observers. We calculate the entanglement entropy associated to the thermofield double states and obtain a UV divergent logarithmic behaviour analogous to known results in two-dimensional conformal field theory in which the entangling boundary is point-like.

New Casimir wormholes in f(R, T) gravity admitting conformal killing vectors

New Casimir wormholes in f(R, T) gravity admitting conformal killing vectors

Physically viable strange quark star models in modified teleparallel gravity

The aim of this paper is to develop the isotropic and anisotropic quark stars configurations in the context of [Formula: see text] gravity in the static spherically symmetric background. To explore the combined effects of torsion scalar [Formula: see text] and the trace of energy–momentum tensor (EMT) [Formula: see text] on relativistic astrophysics, we use diagonal as well as non-diagonal tetrad fields. By considering the conformal Killing vectors along with the MIT bag model, the interior solutions of the field equations corresponding to the linear [Formula: see text] model (in which [Formula: see text] are the constants and [Formula: see text] indicates the cosmological constant) are calculated. The feasibility of the obtained solutions is confirmed by implementing several physical tests. The model parameters are constrained subject to the existence and stability of the quark star models. We formulate the energy constraints, stability equations, mass function, compactness and redshift factor, and present the graphical analysis of all physical quantities. It is found that the derived solutions for both diagonal and non-diagonal tetrad exhibit well-behaved profiles in the framework of modified teleparallel gravity.

Shadows of Kerr–Vaidya-like black holes

In this work, we study the shadow boundary curves of rotating time-dependent black hole solutions which have well-defined Kerr and Vaidya limits. These solutions are constructed by applying the Newman–Janis algorithm to a spherically symmetric seed metric conformal to the Vaidya solution with a mass function that is linear in Eddington–Finkelstein coordinates. Equipped with a conformal Killing vector field, this class of solution exhibits separability of null geodesics, thus allowing one to develop an analytic formula for the boundary curve of its shadow. We find a simple power law describing the dependence of the mean radius and asymmetry factor of the shadow on the accretion rate. Applicability of our model to recent Event Horizon Telescope observations of M87 and Sgr A is also discussed.

Collapsing dynamics of relativistic fluid in modified gravity admitting a conformal Killing vector

The collapsing dynamics of relativistic fluid are explored in f(R) gravity in detailed systematic manner for the non-static spherically symmetric spacetime satisfying the equation of the conformal Killing vector. With quasi-homologous condition and diminishing complexity factor condition, exact solutions for dissipative as well as for non-dissipative system are found and the astrophysical applications of these exact solutions are discussed. Furthermore, it is demonstrated that f(R)=R, which is the extensive restriction of f(R) gravity, prior solutions of the collapsing fluid in general relativity, can be retrieved.

Generalized Vaidya spacetime: Horizons, conformal symmetries, surface gravity and diagonalization

In this paper, the different properties of generalized Vaidya spacetime are considered. We define the location of horizons. We show that the apparent horizon can contain the event horizon. The locations of all types of horizons are compared with the ones in the usual Vaidya spacetime. We investigate the time-like geodesics in this spacetime. New corrections to Schwarzschild and Vaidya cases appear and we give conditions when these corrections are not negligible. Also, we consider the conformal Killing vector and transform the metric to conformally static coordinates. We introduce a new constant of motion along null and time-like geodesics, which is generated by a homothetic Killing vector. The conformally static coordinates allow diagonalizing of the generalized Vaidya spacetime. The surface gravity has been calculated for the dust and stiff fluid cases.

Analytical solutions of spherical structures with relativistic corrections

This paper analyzes the characteristic of a non-static sphere along with anisotropic fluid distribution in the background of modified f(mathcal {G}) theory. Conformal Killing vector is a productive constraint for computing reliable results for modified field equations. The occurrence of conformal Killing vector indicates the existence of symmetries in spacetime and it permits us to choose the coordinates that reduce the number of independent variables. Subsequently, for different conformal Killing vector choices, we obtain several types of precise analytical solutions for both non-dissipative and dissipative systems. We compute the matching conditions in the context of f(mathcal {G}) gravity. In addition to this, we apply specific constraints to the matching conditions in an attempt to determine the significant results. Further, we proceed our investigation by utilizing quasi-homologous condition and vanishing complexity factor condition. Finally, we summarize all the important results which may help to understand the properties of astrophysical objects.

Compact Bardeen Stars in f(R,T)$f(\mathcal {R}, T)$ Theory with Conformal Motion

AbstractA set of exact spherically symmetric solutions for describing the interior of a relativistic star in the context of modified theory of gravity is presented. The existence of a Conformal Killing Vectors (CKV) and Bardeen's model to find the boundary conditions as an exterior space‐time geometry to achieve the metric potentials for current study is used. Moreover, matching conditions have been employed to obtain a set of free parameters for the physical analysis. Furthermore, some physically valid attributes like the behavior of energy density, pressure, equilibrium conditions, the adiabatic index, mass‐radius relation, surface red‐shifts, and compactness factor for the realistic stars such as , , , , and are investigated. It is observed that for the current investigation, the derived compact star solutions are physically acceptable and authentic in the presence of charge with conformal motion.

Vanishing Love numbers of black holes in general relativity: From spacetime conformal symmetry of a two-dimensional reduced geometry

We study the underlying structure of the vanishing of the Love numbers of both Schwarzschild and Kerr black holes in terms of spacetime conformal symmetry in a unified manner for the static spin-$s$ fields. The perturbations can be reduced with the harmonic decomposition to a set of infinite static scalar fields in a two-dimensional anti--de Sitter spacetime $({\text{AdS}}_{2})$. In the reduced system, each scalar field is paired with another, implying that all multipole modes of the perturbation can be regarded as symmetric partners, which can be understood from the property of the supersymmetry algebra. The generator of the supersymmetric structure is constructed from a closed conformal Killing vector field of ${\text{AdS}}_{2}$. The associated conserved quantity allows one to show no static response, i.e., vanishing of the Love and dissipation numbers. We also discuss the vanishing Love numbers of the Kerr black hole with the nonzero dissipation numbers for the nonaxisymmetric perturbations in terms of a radial constant found in a parallel manner as the axisymmetric field case even though the interpretation for the structure is controversial. The symmetric structure corresponds to the ``ladder'' symmetry in [Hui et al., J. Cosmol. Astropart. Phys. 01 (2022) 032] although the geometrical origin is different. Our ladder operator includes the generators of hidden symmetries in previous works.

Ladder symmetries of black holes and de Sitter space: love numbers and quasinormal modes

In this note, we present a synopsis of geometric symmetries for (spin 0) perturbations around (4D) black holes and de Sitter space. For black holes, we focus on static perturbations, for which the (exact) geometric symmetries have the group structure of SO(1,3). The generators consist of three spatial rotations, and three conformal Killing vectors obeying a special melodic condition. The static perturbation solutions form a unitary (principal series) representation of the group. The recently uncovered ladder symmetries follow from this representation structure; they explain the well-known vanishing of the black hole Love numbers. For dynamical perturbations around de Sitter space, the geometric symmetries are less surprising, following from the SO(1,4) isometry. As is known, the quasinormal solutions form a non-unitary representation of the isometry group. We provide explicit expressions for the ladder operators associated with this representation. In both cases, the ladder structures help connect the boundary condition at the horizon with that at infinity (black hole) or origin (de Sitter space), and they manifest as contiguous relations of the hypergeometric solutions.

New classes of wormhole model in [formula omitted] gravity by assuming conformal motion

The two models of wormhole in f(R,T) gravity using relationship between radial and transverse pressure is discussed in the article. To obtain the wormhole model, we used a separable functional form given by f(R,T)=f1(R)+f2(T). R is the Ricci scalar, and T is the trace of the energy momentum tensor, with f1(R) and f2(T) being arbitrary functions of R and T, respectively. We also explored the possibility of wormhole (WH) solutions by assuming that the spacetime admits conformal killing vectors. Using different values for the coupling parameter γ, we studied the viable solutions and graphically analyzed the various features of these models. It can also be noted that the energy conditions are violated for our anisotropic model and isotropic model violates null energy condition (NEC) for γ = 0,−2,−4 while holds NEC for γ = 2,4. Hence, the anisotropic WH throat is supported by exotic matter and the isotropic by exotic/non-exotic (i.e. γ = 0,−2,−4 & γ = 2,4) although not an interesting model as there is no opening throat.

Geometry of conformally symmetric generalized Vaidya spacetimes

In this paper, we consider conformally symmetric generalized Vaidya spacetimes with a composite null dust and null string matter distribution using the semi-tetrad covariant [Formula: see text] decomposition method. The important and novel result that emerges from our analysis is that all the geometric variables related to the time-like and the preferred space-like congruences are completely determined by the conformal vector and conformal factor. This result is unique to the specific matter distribution of the generalized Vaidya configuration. We further show that in the case of the pure null dust (or Vaidya) spacetime, a proper conformal Killing vector cannot be admitted.

A note on closed vector fields

<abstract><p>Special vector fields, such as conformal vector fields and Killing vector fields, are commonly used in studying the geometry of a Riemannian manifold. Though there are Riemannian manifolds, which do not admit certain conformal vector fields or certain Killing vector fields, respectively. Closed vector fields exist in abundance on each Riemannian manifold. In this paper, we used closed vector fields to study the geometry of the Riemannian manifold. In the first result, we showed that a compact Riemannian manifold $ (M^{n}, g) $ admits a closed vector field $\boldsymbol{\omega }$ with $ \mathrm{div} \boldsymbol{\omega }$ non-constant and an eigenvector of the rough Laplace operator, the integral of the Ricci curvature $ Ric(\boldsymbol{\omega }, \boldsymbol{\omega }) $ has a suitable lower bound that is necessarily isometric to $ S^{n}(c) $ and that the converse holds. In the other result, we found a characterization of an Euclidean space using a closed vector field $\boldsymbol{\omega }$ with non-constant length that annihilates the rough Laplace operator and squared length of its covariant derivative that has a suitable upper bound. Finally, we used the closed vector field provided by the gradient of the non-trivial solution of the Fischer-Marsden equation on a complete and simply connected Riemannian manifold $ (M, g) $ and showed that it is necessary and sufficient for $ (M, g) $ to be isometric to a sphere and that the squared length of the covariant derivative of this closed vector field has a suitable upper bound.</p></abstract>

Conformal Killing Vector Fields of Riemannian Manifolds

The main aim of this article to study about vector fields of manifold and how these vector fields will be Killing and Conformal Killing vector fields. Conformal transformation of Weyl rescaling which is conformally related to metrices from g to g, Levi-Civita connection Δ , Lie derivative, torsion with tensor concept of manifold N in a multi-linear map have been treated in this paper. Finally, we have been proved Example 3.02and established the theorem 6.02 on Conformal Killing vector fields. Dhaka Univ. J. Sci. 70(2): 18-22, 2022 (July)

Covariant classification of conformal Killing vectors of locally conformally flat n -manifolds with an application to Kerr–de Sitter spacetimes

We obtain a coordinate independent algorithm to determine the class of conformal Killing vectors of a locally conformally flat $n$-metric $\gamma$ of signature $(r,s)$ modulo conformal transformations of $\gamma$. This is done in terms of endomorphisms in the pseudo-orthogonal Lie algebra $\mathfrak{o}(r+1,s+1)$ up to conjugation of the its group $O(r+1,s+1)$. The explicit classification is worked out in full for the Riemannian $\gamma$ case ($r = 0, s = n$). As an application of this result, we prove that the set of five dimensional, $(\Lambda>0)$-vacuum, algebraically special metrics with non-degenerate optical matrix, previously studied by Bernardi de Freitas, Godazgar and Reall, is in one-to-one correspondence with the metrics in the Kerr-de Sitter-like class. This class exists in all dimensions and its defining properties involve only properties at $\mathscr{I}$. The equivalence between two seemingly unrelated classes of metrics points towards interesting connections between the algebraically special type of the bulk spacetime and the conformal geometry at null infinity