scholarly journals Perturbations of the almost Killing equation and their implications

2021 ◽  
Vol 103 (8) ◽  
Author(s):  
Sumanta Chakraborty ◽  
Justin C. Feng
Keyword(s):  
Killing Equation

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

scholarly journals A cut-off tubular geometry of loop space

2017 ◽  
Vol 29 (03) ◽  
pp. 1750006
Author(s):  
Partha Mukhopadhyay
Keyword(s):  
Quantum Mechanics ◽  
Riemannian Manifold ◽  
Loop Space ◽  
Indirect Method ◽  
Infinite Sequence ◽  
Cartesian Product ◽  
Riemannian Metrics ◽  
Expansion Theorem ◽  
Killing Equation ◽  
The One

Motivated by the computation of loop space quantum mechanics as indicated in [14], here we seek a better understanding of the tubular geometry of loop space [Formula: see text] corresponding to a Riemannian manifold [Formula: see text] around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of [Formula: see text] around the diagonal submanifold, where [Formula: see text] is the Cartesian product of [Formula: see text] copies of [Formula: see text] with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to [Formula: see text] can be obtained by taking the limit [Formula: see text]. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [21] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-[Formula: see text] limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of [Formula: see text] which, for [Formula: see text], is the tangent bundle [Formula: see text].

scholarly journals INTEGRABLE CONFORMAL FIELD THEORY IN FOUR DIMENSIONS AND FOURTH-RANK GEOMETRY

1993 ◽  
Vol 02 (04) ◽  
pp. 413-429 ◽  
Author(s):  
VICTOR TAPIA
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Critical Dimension ◽  
Conformal Killing ◽  
Infinitely Many Solutions ◽  
Four Dimensions ◽  
Conformal Field ◽  
Killing Equation ◽  
Associated Operators ◽  
Line Elements

We consider the conformal properties of geometries described by higher-rank line elements. A crucial role is played by the conformal Killing equation (CKE). We introduce the concept of null-flat spaces in which the line element can be written as dsr=r!dζ1 … dζr. We then show that, for null-flat spaces, the critical dimension, for which the CKE has infinitely many solutions, is equal to the rank of the metric. Therefore, in order to construct an integrable conformal field theory in four dimensions we need to rely on fourth-rank geometry. We consider the simple model ℒ = ¼ Gμνλρ ∂μ ϕ ∂ν ϕ ∂λ ϕ ∂ρϕ and show that it is an integrable conformal model in four dimensions. Furthermore, the associated operators satisfy a Vir4 algebra.

scholarly journals On integrability of the Killing equation

2018 ◽  
Vol 35 (7) ◽  
pp. 075014 ◽  
Author(s):  
Tsuyoshi Houri ◽  
Kentaro Tomoda ◽  
Yukinori Yasui
Keyword(s):  
Killing Equation

scholarly journals Almost-Killing equation: Stability, hyperbolicity, and black hole Gauss law

2019 ◽  
Vol 100 (12) ◽  
Author(s):  
Justin C. Feng ◽  
Edgar Gasperín ◽  
Jarrod L. Williams
Keyword(s):  
Black Hole ◽  
Killing Equation

Dualities and geometrical invariants for static and spherically symmetric spacetimes

2016 ◽  
Vol 25 (09) ◽  
pp. 1641007
Author(s):  
Paola Terezinha Seidel ◽  
Luís Antonio Cabral
Keyword(s):  
General Class ◽  
Hamiltonian Formalism ◽  
Spherically Symmetric ◽  
Conserved Quantities ◽  
Singularity Structure ◽  
Killing Tensors ◽  
Symmetric Spacetimes ◽  
Killing Equation ◽  
Spherically Symmetric Spacetimes ◽  
Spacetime Metric

In this work, we consider spinless particles in curved spacetime and symmetries related to extended isometries. We search for solutions of a generalized Killing equation whose structure entails a general class of Killing tensors. The conserved quantities along particle’s geodesic are associated with a dual description of the spacetime metric. In the Hamiltonian formalism, some conserved quantities generate a dual description of the metric. The Killing tensors belonging to the conserved objects imply in a nontrivial class of dual metrics even for a Schwarzschild metric in the original spacetime. From these metrics, we construct geometrical invariants for classes of dual spacetimes to explore their singularity structure. A nontrivial singularity behavior is obtained in the dual sector.

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