Jacek Tafel

Group Actions on Principal Bundles and Dimensional Reduction

Using the invariant geometrical interpretation of gauge and Higgs fields, a simple derivation is given of the dimensional reduction procedure. The underlying assumption with regard to the Riemannian structure, group orbits and invariant connection are clarified and the critical points of the Higgs potential are shown to have a natural geometrical interpretation.

Canonical connections on Riemannian symmetric spaces and solutions to the Einstein–Yang–Mills equations

It is shown that for any principal bundle over a Riemannian symmetric space G/G0 which admits G as automorphism group, the canonical G‐invariant connection satisfies the source free gauge field equations. Extending this to product manifolds V×G/G0 and assuming the metric and gauge fields decompose in a natural way, this result is still valid and the Einstein equations with gauge fields as source may also be satisfied. For G/G0, this is so automatically, but with a cosmological term present. For dimV=1 or 2, solutions are found, yielding metrics of the Robertson–Walker and Reissner–Nordstrom type.

On the Robinson Theorem and Shearfree Geodesic Null Congruences

Null electromagnetic fields and shearfree geodesic null congruences in curved and flat spacetimes are studied. We point out some mathematical problems connected with the validity of the Robinson theorem. The problem of finding nonanalytic twisting congruences in the Minkowski space is reduced to the construction of holomorphic functions with specific boundary conditions.

Einstein's equations and realizability of CR manifolds

It is shown that the vacuum Einstein equations imposed on a metric which admits shear-free null and geodesic congruence imply the realizability of an associated CR structure.

Can poles change color

The definition of the total nonabelian charge (‘‘color’’) in a classical Yang–Mills theory is shown to require a careful analysis of the boundary conditions at infinity imposed on the potentials and on gauge transformations. The color current of a nonabelian plane wave is found to be different from zero in the transverse gauge, though it vanishes in the null gauge. The color charge of a single pole, described by the Lienard–Wiechert potentials, is constant by virtue of the Yang–Mills equations. An approximate computation indicates that the total color charge of a system of particles may change in time, as a result of radiation. To make this result meaningful, it is necessary to find a method of fixing the allowed gauge transformations to those having a direction‐independent limit at infinity.

Bondi mass in terms of the Penrose conformal factor

Within the framework of the Penrose conformal approach to asymptotical flatness we find minimal conditions on the Ricci tensor of the physical metric which guarantee that the Bondi mass and momentum are well defined. The energy-momentum vector, the Bondi news functions and the energy loss formula are expressed in terms of the Penrose conformal factor. An approximate Bondi-Sachs form of the metric is constructed. The Robinson-Trautman metrics are considered as an example.

Algebraically special solutions of the Einstein equations with pure radiation fields

The Einstein equations Rmu v= Phi kmu kv, kmu being tangent to a twisting shear-free congruence of null geodesics, are formulated as equations in a three-dimensional Cauchy-Riemann space. If the NUT parameter M vanishes and the Cauchy-Riemann space is a hypersurface in C2 then the equations reduce to a single linear second-order equation. New gravitational solutions are found for the case of the Robinson congruence.

Symmetries of Cauchy-Riemann spaces

We simplify and generalize Cartans results on Cauchy-Riemann spaces admitting continuous groups of automorphisms. We describe all such spaces in terms of local coordinates.

A comparison of solution generating techniques for the self‐dual Yang–Mills equations. II

It is shown that Backlund transformations related to the Yang formulation of the self‐dual Yang–Mills equations reduce to Zakharov–Shabat transformations when the gauge group is GL(2,C).

Pure radiation field solutions of the Einstein equations

Solutions of the Einstein equations with pure radiation fields, are obtained in a class of algebraically special metrics related to the Cauchy-Riemann spaces admitting a group of symmetries of Bianchi type VIh.