scholarly journals Kähler metrics via Lorentzian Geometry in dimension four

2019 ◽  
Vol 7 (1) ◽  
pp. 36-61 ◽  
Author(s):  
Amir Babak Aazami ◽  
Gideon Maschler
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Plane Waves ◽  
Killing Vector ◽  
Lorentzian Geometry ◽  
De Sitter ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Open Set ◽  
Sitter Spacetime

AbstractGiven a semi-Riemannian 4-manifold (M, g) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics gK is constructed, defined on an open set in M, which coincides with M in many typical examples. Under certain conditions g and gK share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK are computed under certain assumptions in terms of data associated to g. Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type D such as Kerr and NUT metrics, and metrics for which gK is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric.

scholarly journals The geometry of null-like disformal transformations

2019 ◽  
Vol 16 (11) ◽  
pp. 1950180 ◽  
Author(s):  
I. P. Lobo ◽  
G. G. Carvalho
Keyword(s):  
Vector Field ◽  
Conformal Transformation ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Spinor Structure ◽  
Schwarzschild Metric ◽  
Killing Vectors ◽  
Symmetry Properties ◽  
Geometric Objects

Motivated by the hindrance of defining metric tensors compatible with the underlying spinor structure, other than the ones obtained via a conformal transformation, we study how some geometric objects are affected by the action of a disformal transformation in the closest scenario possible: the disformal transformation in the direction of a null-like vector field. Subsequently, we analyze symmetry properties such as mutual geodesics and mutual Killing vectors, generalized Weyl transformations that leave the disformal relation invariant, and introduce the concept of disformal Killing vector fields. In most cases, we use the Schwarzschild metric, in the Kerr–Schild formulation, to verify our calculations and results. We also revisit the disformal operator using a Newman–Penrose basis to show that, in the null-like case, this operator is not diagonalizable.

Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

2015 ◽  
Vol 12 (10) ◽  
pp. 1550111 ◽  
Author(s):  
Mircea Crasmareanu ◽  
Camelia Frigioiu
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Conformal Killing ◽  
Space Forms ◽  
Potential Vector ◽  
Legendre Curves ◽  
Complex Space Forms

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

scholarly journals Charge conservation, entropy current and gravitation

2021 ◽  
Author(s):  
Sinya Aoki ◽  
Tetsuya Onogi ◽  
Shuichi Yokoyama
Keyword(s):  
Field Theory ◽  
Vector Field ◽  
Vector Fields ◽  
Plane Waves ◽  
Momentum Tensor ◽  
Entropy Density ◽  
Global Symmetry ◽  
Timelike Vector ◽  
First Law Of Thermodynamics ◽  
Entropy Current

We propose a new class of vector fields to construct a conserved charge in a general field theory whose energy–momentum tensor is covariantly conserved. We show that there always exists such a vector field in a given field theory even without global symmetry. We also argue that the conserved current constructed from the (asymptotically) timelike vector field can be identified with the entropy current of the system. As a piece of evidence we show that the conserved charge defined therefrom satisfies the first law of thermodynamics for an isotropic system with a suitable definition of temperature. We apply our formulation to several gravitational systems such as the expanding universe, Schwarzschild and Banãdos, Teitelboim and Zanelli (BTZ) black holes, and gravitational plane waves. We confirm the conservation of the proposed entropy density under any homogeneous and isotropic expansion of the universe, the precise reproduction of the Bekenstein–Hawking entropy incorporating the first law of thermodynamics, and the existence of gravitational plane wave carrying no charge, respectively. We also comment on the energy conservation during gravitational collapse in simple models.

scholarly journals Vector Field Energies and Critical Metrics on Kähler Manifolds

2001 ◽  
Vol 162 ◽  
pp. 41-63 ◽  
Author(s):  
Toshiki Mabuchi
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Bilinear Pairing ◽  
Fano Varieties ◽  
Holomorphic Vector Field ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kähler Einstein Metrics ◽  
Compact Kähler Manifold

Associated with a Hamiltonian holomorphic vector field on a compact Kähler manifold, a nice functional on a space of Kähler metrics will be constructed as an integration of the bilinear pairing in [FM] contracted with the Hamiltonian holomorphic vector field. As applications, we have functionals whose critical points are extremal Kähler metrics or “Kähler-Einstein metrics” in the sense of [M4], respectively. Finally, the same method as used by [G1] allows us to obtain, from the convexity of , the uniqueness of “Kähler-Einstein metrics” on nonsingular toric Fano varieties possibly with nonvanishing Futaki character.

scholarly journals Twisted Quadrics and Algebraic Submanifolds in $\mathbb {R}^{n}$

2020 ◽  
Vol 23 (4) ◽  
Author(s):  
Gaetano Fiore ◽  
Davide Franco ◽  
Thomas Weber
Keyword(s):  
Differential Geometry ◽  
Level Sets ◽  
Vector Fields ◽  
General Procedure ◽  
Killing Vector ◽  
Polynomial Equations ◽  
De Sitter ◽  
Killing Vector Fields ◽  
Gauss Theorem ◽  
Levi Civita Connection

AbstractWe propose a general procedure to construct noncommutative deformations of an algebraic submanifold M of $\mathbb {R}^{n}$ ℝ n , specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of$\mathbb {R}^{n}$ ℝ n , arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883–1911, 2006), whereby the commutative pointwise product is replaced by the ⋆-product determined by a Drinfel’d twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds Mc that are level sets of the fa(x), where fa(x) = 0 are the polynomial equations solved by the points of M, employing twists based on the Lie algebra Ξt of vector fields that are tangent to all the Mc. The twisted Cartan calculus is automatically equivariant under twisted Ξt. If we endow $\mathbb {R}^{n}$ ℝ n with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted M is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and ⋆-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in $\mathbb {R}^{3}$ ℝ 3 except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean $\mathbb {R}^{3}$ ℝ 3 and twisted hyperboloids embedded in twisted Minkowski $\mathbb {R}^{3}$ ℝ 3 [the latter are twisted (anti-)de Sitter spaces dS2, AdS2].

scholarly journals Killing Vector Fields in Generalized Conformalβ-Change of Finsler Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Mallikarjun Yallappa Kumbar ◽  
Narasimhamurthy Senajji Kampalappa ◽  
Thippeswamy Komalobiah Rajanna ◽  
Kavyashree Ambale Rajegowda
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Finsler Space ◽  
Killing Vector ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finsler Spaces ◽  
Killing Vector Fields ◽  
Necessary And Sufficient

We consider a Finsler space equipped with a Generalized Conformalβ-change of metric and study the Killing vector fields that correspond between the original Finsler space and the Finsler space equipped with Generalized Conformalβ-change of metric. We obtain necessary and sufficient condition for a vector field Killing in the original Finsler space to be Killing in the Finsler space equipped with Generalized Conformalβ-change of metric.

scholarly journals Gravitational thermodynamics of causal diamonds in (A)dS

2019 ◽  
Vol 7 (6) ◽  
Author(s):  
Theodore Jacobson ◽  
Manus Visser
Keyword(s):  
Killing Vector ◽  
Minkowski Spacetime ◽  
Conformal Killing ◽  
De Sitter Spacetime ◽  
Conformal Killing Vector ◽  
De Sitter ◽  
Thermodynamic Relation ◽  
Killing Field ◽  
Sitter Spacetime ◽  
Gravitational Thermodynamics

The static patch of de Sitter spacetime and the Rindler wedge of Minkowski spacetime are causal diamonds admitting a true Killing field, and they behave as thermodynamic equilibrium states under gravitational perturbations. We explore the extension of this gravitational thermodynamics to all causal diamonds in maximally symmetric spacetimes. Although such diamonds generally admit only a conformal Killing vector, that seems in all respects to be sufficient. We establish a Smarr formula for such diamonds and a ``first law" for variations to nearby solutions. The latter relates the variations of the bounding area, spatial volume of the maximal slice, cosmological constant, and matter Hamiltonian. The total Hamiltonian is the generator of evolution along the conformal Killing vector that preserves the diamond. To interpret the first law as a thermodynamic relation, it appears necessary to attribute a negative temperature to the diamond, as has been previously suggested for the special case of the static patch of de Sitter spacetime. With quantum corrections included, for small diamonds we recover the ``entanglement equilibrium'' result that the generalized entropy is stationary at the maximally symmetric vacuum at fixed volume, and we reformulate this as the stationarity of free conformal energy with the volume not fixed.

scholarly journals New application of the Killing vector field formalism: modified periodic potential and two-level profiles of the axionic dark matter distribution

2020 ◽  
Vol 80 (2) ◽  
Author(s):  
Alexander B. Balakin ◽  
Dmitry E. Groshev
Keyword(s):  
Dark Matter ◽  
Vector Field ◽  
Electric Fields ◽  
Vector Fields ◽  
Periodic Potential ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Covariant Model ◽  
Axion Field ◽  
Field Formalism

Abstract We consider the structure of halos of the axionic dark matter, which surround massive relativistic objects with static spherically symmetric gravitational field and monopole-type magneto-electric fields. We work with the model of pseudoscalar field with the extended periodic potential, which depends on additional arguments proportional to the moduli of the Killing vectors; in our approach they play the roles of model guiding functions. The covariant model of the axion field with this modified potential is equipped with the extended formalism of the Killing vector fields, which is established in analogy with the formalism of the Einstein–Aether theory, based on the introduction of a unit timelike dynamic vector field. We study the equilibrium state of the axion field, for which the extended potential and its derivative vanish, and illustrate the established formalism by the analysis of two-level axionic dark matter profiles, for which the stage delimiters relate to the critical values of the modulus of the timelike Killing vector field.

PRECOMPACTNESS OF THE CALABI ENERGY

1996 ◽  
Vol 07 (02) ◽  
pp. 245-254 ◽  
Author(s):  
SANTIAGO R. SIMANCA
Keyword(s):  
Continuous Function ◽  
Scalar Curvature ◽  
Complex Manifold ◽  
Vector Fields ◽  
Convergent Sequence ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Holomorphic Vector Fields ◽  
Extremal Kähler Metrics ◽  
Kahler Metric

For any complex manifold of Kähler type, the L2-norm of the scalar curvature of an extremal Kähler metric is a continuous function of the Kähler class. In particular, if a convergent sequence of Kähler classes are represented by extremal Kähler metrics, the corresponding sequence of L2-norms of the scalar curvatures is convergent. Similarly, the sequence of holomorphic vector fields associated with a sequence of extremal Kähler metrics with converging Kähler classes is convergent.

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