Canonical forms for axial symmetric space-times

The general canonical forms for axial symmetric space-times are investigated. Special forms such as static, stationary and cylindrical are considered. Our results are based only upon the symmetries assumed, not upon the field equations; thus they are applicable to vacuum, electromagnetic and matter field problems. In particular, we find a term that was missing in previous work. There can be in the general canonical form a metric coefficient between the axis of symmetry and the angle about this axis. This metric coefficient survives even for static or stationary space-times.     

Characterization of locally rotationally symmetric space-times

We give a simple characterization of locally rotationally symmetric space-times in terms of the existence of a canonical null tetrad or canonical orthonormal tetrad. The result is applied to space-times which satisfy the Einstein field equations with a perfect fluid or electromagnetic field as source.     

The Heterotic Supersymmetric Sigma Model in the Canonical Exterior Formalism

Starting from a classical 2D superconformal theory described by the Wess–Zumino–Witten action, the canonical exterior formalism on group manifold for the heterotic supersymmetric sigma model is constructed. The motion equations of the dynamical field and the constraints are found and analyzed from the geometric point of view. It can be seen how the use of the canonical exterior formalism is more adequate and simple because of its manifest covariance in all the steps. The relationship between the form brackets defined in the canonical exterior formalism and the Poisson-brackets is written. Later on, the Dirac-brackets are written by using the second class constraints provided by the canonical exterior formalism. As it can be seen the canonical exterior formalism allows to show how the canonical quantization of the heterotic supersymmetric sigma model is facilitated. Member of the Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina.     

The Hamiltonian constraint in the teleparallel equivalent of general relativity

In the teleparallel equivalent of general relativity the integral form of the Hamiltonian constraint contains explicitly theadm energy in the case of asymptotically flat space-times. We show that such expression of the constraint leads to a natural and straightforward construction of a Schrödinger equation for time-dependent physical states. The quantized Hamiltonian constraint is thus written as an energy eigenvalue equation. We further analyse the constraint equations in the case of a space-time endowed with a spherically symmetric geometry. We find the general functional form of the time-dependent solutions of the quantized Hamiltonian and vector constraints.     

Curvature collineations for type-N Robinson-Trautman space-times

The metric of type-N Robinson-Trautman space-times is generated by a real functionP satisfying certain field equations. Canonical forms forP are obtained under the assumption that at least one curvature collineation exists. In order to give an example of the improper subgroup structure of a group of curvature collineations all the curvature collineations are determined for the space-times corresponding to one of the two canonical forms.     

A class of empty spacetimes admitting a rational first integral of the geodesic equation

The empty space field equations are solved for one of the canonical forms obtained previously by Vaz and Collinson for the metrics of space-times admitting a surface generating Killing pair, one member of which is hypersurface orthogonal.     

A computer code for the study of spherically-symmetric cosmological space-times

We describe an Eulerian, explicit computer code for the study of spherically-symmetric cosmological space-times on a Robertson-Walker background. The Einstein equations and the relativistic-hydrodynamic equations are written down for such a space-time using the 3+1 formalism of Arnowitt, Deser, and Misner [1]. Time slices are selected by the constant-mean-curvature criterion. Numerical techniques utilized in the code, as well as several code tests, are discussed, with emphasis on the difficulties encountered and their possible causes and cures.     

Cauchy's problem and Huygens' principle for the linearized Einstein field equations

The linearized Einstein field equations with Einstein space-times as background are studied by use of the harmonic gauge. By means of Riesz' integration method a representation theorem for the solution of Cauchy's problem, using the constraints of the Cauchy data and the calculus of symmetric differential forms, is proved. We introduce some linear differential operators, which map the set of symmetric differential forms into the subset with vanishing divergence and trace and use these operators to derive necessary conditions for the validity of Huygens' principle from which it follows that the linearized field equations satisfy Huygens' principle only in flat space-times.Dedicated to Gudrun Schmidt on the occasion of her 50th birthday.     

Higher-spin fields in a curved space-time

Algebraic constraints are derived for higher-spin fields in a curved space-time manifold. Comparison is made with previously obtained results. A particular solution of the zero-restmass field equations is given for the plane wave Einstein-Maxwell space-times.     

Killing pairs and the empty space field equations

The empty space field equations are investigated for each of the canonical forms obtained previously for the metrics of space-times admitting a surface generating Killing pair, one member of which is hypersurface orthogonal. It is found that the rational first integral of the geodesic equation, corresponding to the Killing pair, is always necessarily the ration of two linear first integrals.     

Simplification of the constraints in the dirac formalism for general relativity

In any classical theory in canonical form, the Poisson bracket relations between the constraints are preserved under canonical transformations. We show that in the Dirac formalism for general relativity this condition places certain limits on the degree to which one can simplify the form of the constraints. It implies, for instance, that the constraints cannot all be written as canonical momenta. Furthermore, it is not even possible to reduce them all to purely algebraic functions of the momenta by means of a canonical tansformation which preserves the original configuration space subspace of phase space.     

Einstein's equations near spatial infinity

A new class of space-times is introduced which, in a neighbourhood of spatial infinity, allows an expansion in negative powers of a radial coordinate. Einstein's vacuum equations give rise to a hierarchy of linear equations for the coefficients in this expansion. It is demonstrated that this hierarchy can be completely solved provided the initial data satisfy certain constraints.Work supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project no. 4069     

Simplification of the constraints in the dirac formalism for general relativity

In any classical theory in canonical form, the Poisson bracket relations between the constraints are preserved under canonical transformations. We show that in the Dirac formalism for general relativity this condition places certain limits on the degree to which one can simplify the form of the constraints. It implies, for instance, that the constraints cannot all be written as canonical momenta. Furthermore, it is not even possible to reduce them all to purely algebraic functions of the momenta by means of a canonical tansformation which preserves the original configuration space subspace of phase space.     

Canonical quantization of general relativity in discrete space-times

It has long been recognized that lattice gauge theory formulations, when applied to general relativity, conflict with the invariance of the theory under diffeomorphisms. We analyze discrete lattice general relativity and develop a canonical formalism that allows one to treat constrained theories in Lorentzian signature space-times. The presence of the lattice introduces a "dynamical gauge" fixing that makes the quantization of the theories conceptually clear, albeit computationally involved. The problem of a consistent algebra of constraints is automatically solved in our approach. The approach works successfully in other field theories as well, including topological theories. A simple cosmological application exhibits quantum elimination of the singularity at the big bang.     

Einstein-Weyl space-times with geodesic and shear-free neutrino rays: Asymptotic behaviour

We consider a neutrino field with goodesic and shear-free rays, in interaction with a gravitational field according to the Einstein-Weyl field equations. Furthermore we suppose that there exists a Killing vector r^μ whose magnitude is almost everywhere bounded at the future and past endpoints of the neutrino rays. The implications of the asymptotic behavior of r^μ on the structure of space-time are investigated and a useful set of reduced equations is obtained. It is found that under these hypothesis the space-time cannot be asymptotically flat if the neutrino field is nonvanishing. All the Demianski-Kerr-NUT-like space-times as well as the space-times which admit a covariantly constant null vector are explicitly obtained.     

Grand Canonical Ensembles in General Relativity

We develop a formalism for general relativistic, grand canonical ensembles in space-times with timelike Killing fields. Using that, we derive ideal gas laws, and show how they depend on the geometry of the particular space-times. A systematic method for calculating Newtonian limits is given for a class of these space-times, which is illustrated for Kerr space-time. In addition, we prove uniqueness of the infinite volume Gibbs measure, and absence of phase transitions for a class of interaction potentials in anti-de Sitter space.     

Construction of Hadamard States by Pseudo-Differential Calculus

We give a new construction based on pseudo-differential calculus of quasi-free Hadamard states for Klein–Gordon equations on a class of space-times whose metric is well-behaved at spatial infinity. In particular on this class of space-times, we construct all pure Hadamard states whose two-point function (expressed in terms of Cauchy data on a Cauchy surface) is a matrix of pseudo-differential operators. We also study their covariance under symplectic transformations. As an aside, we give a new construction of Hadamard states on arbitrary globally hyperbolic space-times which is an alternative to the classical construction by Fulling, Narcowich and Wald.     

On the algebraic structure of second order symmetric tensors in 5-dimensional space-times

A new approach to the algebraic classification of second order symmetric tensors in 5-dimensional space-times is presented. The possible Segre types for a symmetric two-tensor are found. A set of canonical forms for each Segre type is obtained. A theorem which collects together some basic results on the algebraic structure of the Ricci tensor in 5-dimensional space-times is also stated.     

Plane rotations and Hamilton—Dirac mechanics

Canonical formalism for SO(2) is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretations are given. The Euler-Lagrange and Hamiltonian canonical equations coincide with the Lie equations. It is shown that the constraints satisfy CCR. Consistency of the constraints is checked. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Stationary vacuum fields with a conformally flat three-space. II. Proof of axial symmetry

The conjecture is proved that stationary vacuum space-times having a conformally flat three-space are axially symmetric. The proof uses the Ernst potential and the complex conjugate potential as independent coordinates. Two field equations: a combination of the Einstein equations and an integrability condition are algebraic in one of the field variables. Their coefficients, computed by employing a REDUCE program, separately vanish unless axial symmetry holds. Solution of the coefficient equations yields the proof of axial symmetry. Certain special classes of metrics must be excluded from the discussion. The axial symmetry of these exceptional classes has been proved in I.