Lie Algebra Of Killing Fields Research Articles

Symmetries of quaternionic Kähler manifolds with S1‐symmetry

We study symmetry properties of quaternionic K\"ahler manifolds obtained by the HK/QK correspondence. To any Lie algebra $\mathfrak{g}$ of infinitesimal automorphisms of the initial hyper-K\"ahler data we associate a central extension of $\mathfrak{g}$, acting by infinitesimal automorphisms of the resulting quaternionic K\"ahler manifold. More specifically, we study the metrics obtained by the one-loop deformation of the $c$-map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special K\"ahler manifold gives rise to a Lie algebra of Killing fields of the corresponding one-loop deformed $c$-map space. As an application, we show that this construction increases the cohomogeneity of the automorphism groups by at most one. In particular, if the initial manifold is homogeneous then the one-loop deformed metric is of cohomogeneity at most one. As an example, we consider the one-loop deformation of the symmetric quaternionic K\"ahler metric on $SU(n,2)/S(U(n)\times U(2))$, which we prove is of cohomogeneity exactly one. This family generalizes the so-called universal hypermultiplet ($n=1$), for which we determine the full isometry group.

The Rest Mass of an Asymptotically Anti-de Sitter Spacetime

We study the space of Killing fields on the four dimensional AdS spacetime $$AdS^{3,1}$$ . Two subsets $${\mathcal {S}}$$ and $${\mathcal {O}}$$ are identified: $${\mathcal {S}}$$ (the spinor Killing fields) is constructed from imaginary Killing spinors, and $${\mathcal {O}}$$ (the observer Killing fields) consists of all hypersurface orthogonal, future timelike unit Killing fields. When the cosmology constant vanishes, or in the Minkowski spacetime case, these two subsets have the same convex hull in the space of Killing fields. In presence of the cosmology constant, the convex hull of $$ {\mathcal {O}}$$ is properly contained in that of $${\mathcal {S}}$$ . This leads to two different notions of energy for an asymptotically AdS spacetime, the spinor energy and the observer energy. Chruściel et al. (J High Energy Phys 2006(11):084, 2006) proved the positivity of the spinor energy and derived important consequences among the related conserved quantities. We show that the positivity of the observer energy follows from the positivity of the spinor energy. A new notion called the “rest mass” of an asymptotically AdS spacetime is then defined by minimizing the observer energy and is shown to be evaluated in terms of the adjoint representation of the Lie algebra of Killing fields. It is proved that the rest mass has the desirable rigidity property that characterizes the AdS spacetime.

Geodesic orbit manifolds and Killing fields of constant length

The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian geodesic orbit manifold $(M,g)$ and the structure of its full isometry group. The Lie algebra of the full isometry group of $(M,g)$ is identified with the Lie algebra of Killing fields $\mathfrak{g}$ on $(M,g)$. We prove the following result: If $\mathfrak{a}$ is an abelian ideal of $\mathfrak{g}$, then every Killing field $X\in \mathfrak{a}$ has constant length. On the ground of this assertion we give a new proof of one result of C. Gordon: Every Riemannian geodesic orbit manifold of nonpositive Ricci curvature is a symmetric space.

Gravitational waves as exact solutions of Einstein field equations

Exact solutions of Einstein field equations invariant for a non-Abelian 2-dimensional Lie algebra of Killing fields are described. A sub-class of these gravitational fields have a wave-like character; it is shown that they have spin–1.

Nonlinear gravitational waves, their polarization, and realistic sources

We describe exact solutions of the Einstein field equations invariant under a non-Abelian two-dimensional Lie algebra of Killing fields. A subclass of these gravitational fields have a wavelike character. We show that they have spin 1. We also discuss some indirect observational effects. In particular, we show that vector cosmological perturbations can no longer be negligible in the presence of topological defects, which can be verified by examining the anisotropies of the cosmic microwave background.

Ricci flat metrics with bidimensional null orbits and non-integrable orthogonal distribution

We study Ricci flat 4-metrics of any signature under the assumption that they allow a Lie algebra of Killing fields with 2-dimensional orbits along which the metric degenerates and whose orthogonal distribution is not integrable. It turns out that locally there is a unique (up to a sign) metric which satisfies the conditions. This metric is of signature ( + + − − ) and, moreover, homogeneous possessing a 6-dimensional symmetry algebra.

SPIN-1 GRAVITATIONAL WAVES: THEORETICAL AND EXPERIMENTAL ASPECTS

Exact solutions of Einstein field equations invariant for a non-Abelian bidimensional Lie algebra of Killing fields are described. Physical properties of these gravitational fields are studied, their wave character is checked by making use of covariant criteria and the observable effects of such waves are outlined. The possibility of detection of these waves with modern detectors, spherical resonant antennas in particular, is sketched.

SPIN-1 GRAVITATIONAL WAVES

Gravitational fields invariant for a 2-dimensional Lie algebra of Killing fields [X,Y]=Y, with Y of light type, are analyzed. The conditions for them to represent gravitational waves are verified and the definition of energy and polarization is addressed; realistic generating sources are described.

An integrable model in general relativity

It is shown that gravitational fields invariant for a non Abelian 2-dimensional Lie algebra of Killing fields are parameterized either by solutions of a transcendental equation (the tortoise equation) or by solutions of a linear second order partial differential equation (the Laplace equation or the Darboux equation) on the plane. Those determined via Laplace or Darboux equations are exact nonlinear gravitational waves obeying to two nonlinear superposition laws.

Vacuum Einstein metrics with bidimensional Killing leaves. I. Local aspects

The solutions of vacuum Einstein's field equations, for the class of Riemannian metrics admitting a non-Abelian bidimensional Lie algebra of Killing fields, are explicitly described. They are parametrized either by solutions of a transcendental equation (the tortoise equation), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a 3-dimensional Lie algebra of Killing fields with bidimensional leaves.

Gravitational fields with a non-Abelian, bidimensional Lie algebra of symmetries

Vacuum gravitational fields invariant for a bidimensional non Abelian Lie algebra of Killing fields, are explicitly described. They are parameterized either by solutions of a transcendental equation (the tortoise equation) or by solutions of a linear second order differential equation on the plane. Gravitational fields determined via the tortoise equation, are invariant for a 3-dimensional Lie algebra of Killing fields with bidimensional leaves. Global gravitational fields out of local ones are also constructed.

The harmonic map and Killing fields for self-dual SU(3) Yang-Mills equations

Using symbolic computations, the unique metric in the space of fields, required to describe self-dual SU(3) Yang-Mills equations by a harmonic map, is determined. Moreover the complete Lie algebra of Killing fields for this metric is established.

Homogeneous Riemannian spaces and Lie algebras of Killing fields

The following new results are proved: (1) Every Riemannian space admitting a multiply transitive Lie algebra of Killing fields is locally isometric to a homogeneous Riemannian space. (2) For every closed connceted subgroupH 0 of the invariance group of a non degenerate quadratic form a homogeneous Riemannian space exists whose isotropy group containsH 0. (3) Necessary and sufficient conditions are derived for a Lie algebra $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{G} $$ to have a realization as multiply transitive Killing fields. These conditions are constructire in the sense that, for a given linear connected isotropy group, Lie algebras $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{G} $$ can be calculated algebraically, (4) It is shown how the Riemann tensor of a bomogencous space and its covariant derivatives can be expressed in terms of the metric at one point and the structure constants of the Lie algebra of Killing fields.