Timelike Killing Field Research Articles

States of scalar field on spacetimes with two isometries with timelike orbits

We define a pure state associated with every sufficiently regular state of linear quantum field theory. For a scalar field on a globally hyperbolic spacetime, the energy relative to a timelike vector field defines a quasifree state, whose associated pure state is a natural vacuum of the scalar field. If the energy state is defined relative to a globally timelike complete Killing field, its associated pure state is the well known `frequency splitting' stationary vacuum state. It is proved that such vacua corresponding to two commuting Killing fields are equal. It is conjectured that the vacua are equal even if the Killing fields do not commute. Implications for the particle interpretation of quantum field theory in curved spacetimes are discussed.

Hamiltonian structure of the Vlasov-Maxwell system in a curved background spacetime.

Working in the context of an ADM splitting into space plus time, this paper identifies a Hamiltonian formulation, i.e., cosymplectic structure, for the Vlasov-Maxwell system, as formulated in a fixed, albeit curved, background spacetime. The fundamental arena of physics is an infinite-dimensional phase space, coordinatized by the distribution function f, the spatial vector potential ${\mathit{A}}_{\mathit{i}}$, and the conjugate momentum ${\mathrm{\ensuremath{\Pi}}}^{\mathit{i}}$. This Hamiltonian formulation entails the identification of (i) Lie brackets 〈F,G〉, defined for functionals F[${\mathit{A}}_{\mathit{i}}$,${\mathrm{\ensuremath{\Pi}}}^{\mathit{i}}$,f] and G[${\mathit{A}}_{\mathit{i}}$,${\mathrm{\ensuremath{\Pi}}}^{\mathit{i}}$,f], and (ii) a Hamiltonian function H[${\mathit{A}}_{\mathit{i}}$,${\mathrm{\ensuremath{\Pi}}}^{\mathit{i}}$,f], so chosen that the equations of motion take the form ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$F=〈H,F〉, with ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$ a coordinate time derivative. This formulation is used to address the problem of stability of equilibria corresponding to time-independent electromagnetic fields in a spacetime admitting a timelike Killing field. An explicit expression is derived for the most general dynamically accessible perturbation \ensuremath{\delta}X\ensuremath{\equiv}(\ensuremath{\delta}${\mathit{A}}_{\mathit{i}}$,\ensuremath{\delta}${\mathrm{\ensuremath{\Pi}}}^{\mathit{i}}$,\ensuremath{\delta}f), and it is shown that all equilibria are energy extremals with respect to such \ensuremath{\delta}X, i.e., ${\mathrm{\ensuremath{\delta}}}^{(1)}$H[\ensuremath{\delta}X]\ensuremath{\equiv}0. The sign of the second variation ${\mathrm{\ensuremath{\delta}}}^{(2)}$H, also computed, is thus directly related to the problem of linear stability: If ${\mathrm{\ensuremath{\delta}}}^{(2)}$H[\ensuremath{\delta}X]>0 for all dynamically accessible perturbations, the equilibrium is guaranteed to be linearly stable. The existence of some perturbation \ensuremath{\delta}X for which ${\mathrm{\ensuremath{\delta}}}^{(2)}$H[\ensuremath{\delta}X]0 does not guarantee a linear instability. However, one does at least anticipate that equilibria admitting such perturbations will be nonlinearly unstable and/or unstable towards the effects of dissipation.

Quasilocal energy in static gravitational fields

The timelike Killing vector xi of any static spacetime defines a set of equipotentials on which mod xi mod =constant. The principle that flat spacetimes have zero energy allows a unique expression for the energy, E, within any equipotential. The energy, E, is the energy of the shell of stressed matter on the equipotential that gives the same external metric and a flat interior. The standard Arnowitt-Deser-Misner total energy is retrieved from this expression applied to a large equipotential. The authors compare E with the energy, EN, obtained from Noether's theorem when the actual spacetime is mapped onto a flat background. The mapping is one in which the timelike Killing fields are identified and the normal congruence of spacelike lines orthogonal to mod xi mod =constant is mapped differentiably on a normal congruence of spacelike lines in the flat background. For a shell it is shown that E=EN provided the mapping gives zero energy in the flat region. E is given explicitly for axially symmetric fields and the following examples are analysed: spherical symmetry, cylindrical symmetry, the metric of a rod and Lynden-Bell and Pineault's infinite disc of particles in oppositely rotating streams of constant speed. In each example the energy is positive whenever the field is generated by a physical source obeying the weak energy condition. It also increases as the equipotential is made larger.

Timelike isometries and killing fields

Timelike isometries and killing fields

Statistical mechanics of the gravitational field in a conformally static setting

This paper formulates a statistical description of a collection of N identical classical particles that interact relativistically via linear fields in a fixed background space-time that admits a conformal timelike Killing field. Attention focuses upon the special cases of a simple scalar interaction and a linearized gravitation interaction which should suffice to model many systems of astrophysical interest. The fundamental object of the theory is a complicated distribution function that depends upon appropriate variables for both the particles and the fields. By assuming that, in a first approximation, this distribution factorizes into an infinite product of reduced distribution functions, one recovers the type of mean-field theory developed by such authors as Ipser and Thorne. Alternatively, one may derive various exact and approximate relations which contain information about the interparticle correlations.

A stationary cylindrically symmetric electrovac space-time

A stationary cylindrically symmetric electrovac solution of the Einstein-Maxwell equations is derived in which the electromagnetic field is null. The resulting space-time contains no time-like hypersurface-orthogonal Killing fields so that it is non-static.

Complex Ernst-Harrison potential and nonrelativistic limits of stationary vacuum gravitational fields

Expressions are derived for determining the potentials of the nonrelativistic gravitational fields corresponding to stationary vacuum gravitational fields described by given complex Ernst-Harrison potentials E or ξ on a definite region of the three-space ¯V3 with given metric tensork (¯V3, is determined by the set of trajectories of a timelike Killing field). The dependence of the form of the nonrelativistic potentials on the parameters of the symmetry group SL(2, R) of the equations of the stationary vacuum gravitational fields with respect to transformations of the form of E is found.

A technique for analyzing the structure of isometries

A new technique is introduced to investigate the structure of isometry Lie algebras. Some general results are first proved by applying this technique to n-dimensional manifolds equipped with metrics of arbitrary signature. A restriction is then made to 3-manifolds representing the space of orbits of the timelike Killing field in stationary space–times. Under the assumption of asymptotic flatness at spatial infinity, a complete description of isometry Lie algebras of these 3-manifolds is obtained. As corollaries, several results about symmetries of stationary isolated systems in general relativity are proved.

Rotating steady state configurations in general relativity

Spacetimes with axial and timelike Killing fields ημ and ξμ respectively are studied. A natural basis is constructed for the steady state configurations of matter with 4-velocity uμ =αξμ+βημ. It is shown that the rate of shear tensor always has an eigenvector with zero eigenvalue lying in the 2-space of Killing orbits. A natural coordinate system is constructed on the 2-space of Killing orbits for rigidly rotating perfect matter. Gravitational isorotation is proved for the case of a perfect fluid with a one-parameter equation of state. Locally nonrotating frames are constructed for purely stationary spacetimes.