Single kerr-schild metrics: A double view

Real-vacuum single Kerr-Schild (ISKS) metrics are discussed and new results proved. It is shown that if the Weyl tensor of such a metric has a twist-free expanding principal null direction, then it belongs to the Schwarzschild family of metrics — there are no Petrov type-II Robinson-Trautman metrics of Kerr-Schild type. If such a metric has twist then it belongs either to the Kerr family or else its Weyl tensor is of Petrov type II. The main part of the paper is concerned with complexified versions of Kerr-Schild metrics. The general real ISKS metric is written in double Kerr-Schild (IDKS) form. TheH andl potentials which generate IDKS metrics are determined for the general vacuum ISKS metric and given explicitly for the Schwarzschild and Kerr families of metrics.     

On the superposition of electromagnetic waves in general relativity

It is shown that, in a region of space-time containing two independent electromagnetic waves propagating in different directions, it is not possible for the two waves to follow simultaneously affinely parameterised shear-free and twist-free null geodesic congruences.     

Ghost neutrino fields in flat space-time

The Weyl neutrino equation is integrated in flat space-time assuming that the energy-momentum tensor of the neutrino field vanishes. It is shown that the flux vector of the neutrino field is tangent to a twist-free and shear-free congruence of null geodesics, which is a special Robinson congruence and constitutes a geometrical representation of a null twistor. It is also shown that, conversely, given such a congruence, a ghost neutrino field can be constructed.     

Conformally flat null electromagnetic field

A class of conformally flat solutions for null electromagnetic field is presented. Explicit forms of the field tensors are also given. The null field is characterized by shear-free, twist-free, expansion-free, and geodetic null congruences.     

A new class of exact solutions of the vacuum quadratic Poincaré gauge field theory

A class of algebraically special exact solutions of the vacuum quadratic Poincaré gauge field theory is presented. These solutions are of type III and type N and have a nonexpanding, shear-free and twist-free geodesic repeated principal null congruence. The metric is of Kundt's class, and the torsion components are solutions of certain differential equations. The solutions have been obtained using a generalised spin coefficient formalism.     

On solving Einstein's field equations in the Newman-Penrose formalism

Using the Newman-Penrose formalism and Penrose's conformai rescaling a method is presented for finding systematically solutions of (or, at least, reduced equations for) the general field equations. These solutions are necessarily (locally) asymptotically flat and are represented in a coordinate system based on a geodesic, twist-free, expanding null congruence. All redundant equations are disposed of and the freely specifiable data are clearly exhibited. Although the few equations that remain to be solved are, in general, intractable, well-known theorems guarantee the existence and uniqueness of solutions. The method applies to spaces and spaces as well as to real space-times.     

Singularities at real points of ℋ space

In the presence of gravitational radiation, there are ordinarily no shear-free slices of null infinity. A four-complex-dimensional set of shear-free slices of complexified null infinity do exist. They comprise the manifold space. In general, there are no preferred real subspaces of space associated with slices of real null infinity. However, for radiation fields possessing a twist-free axial symmetry, a two-parameter family of shear-free slices of real null infinity exist and therefore pick out a preferred two-dimensional real subspace of space. In this paper, we study the geometry of these 2-spaces for the particular case of quadrupole radiation fields for which determination of the shear-free slices reduces to the standard problem of determining orbits of a particle moving in a potential. Our principal interest is the investigation of possible singularities caused by sufficiently intense radiation fields. We find that such singularities do occur for radiation fields having the characteristic powerc ⁵/G.     

A class of algebraically general solutions of the Einstein-Maxwell equations for non-null electromagnetic fields. II

In a previous article the Einstein-Maxwell field equations for non-null electromagnetic fields were studied under the conditions that the null tetrad is parallel-propagated along both principal null congruences. A solution with twist and shear, but no expansion, was found and was conjectured to be the only expansion-free solution. Here it is shown that this conjecture is false; the general expansion-free solution is found to be a family of space-times depending on a single constant parameter which is the ratio of the (constant) twists of the two principal null congruences.We are grateful to R. G. McLenaghan and N. Tariq for kindly informing us that they have found essentially the same solution. Their article appeared recently inJournal of Mathematical Physics.     

Shear-free,twisting Einstein-Maxwell metrics in the Newman-Penrose formalism

The problem of finding algebraically special solutions of the vacuum Einstein-Maxwell equations is investigated using the spin coefficient formalism of Newman and Penrose. The general case, in which the degenerate null vectors are not hypersurface orthogonal, is reduced to a problem of solving five coupled differential equations that are no longer dependent on the affine parameter along the degenerate null directions. It is shown that the most general regular, shearfree, nonradiating solution of these equations is the Kerr-Newman metric.Based in part on a doctoral thesis submitted to the University of Pittsburgh (1970) while the author was NASA Predoctoral Trainee. Research also supported in part by the National Science Foundation under Grant GP-19378.     

An equation satisfied by the tangent to a shear-free,geodesic, null congruence

A tensorial equation satisfied by the tangent to a shear-free geodesic, null congruence is presented. If the congruence is neither twist-free nor expansion-free then the equation defines a second, unique, null direction previously obtained, using the spinor formalism, by Somers. Some further properties of the equation are discussed.     

WANDs of the black ring

Necessary conditions for various algebraic types of the Weyl tensor in higher dimensions are determined. These conditions are then used to find Weyl aligned null directions for the black ring solution. It is shown that the black ring solution is algebraically special, of type I_i, while locally on the horizon the type is II. One exceptional subclass – the Myers-Perry solution – is of type D.     

Self-dual Maxwell field on a null surface. II

The canonical formalism for the Maxwell field on a null surface has been revisited. A new pair of gauge-independent canonical variables is introduced. It is shown that these variables are derivable from a Hamillon-Jacobi functional. The construction of the appropriate C ^* algebra is carried out in preparation for quantization. The resulting quantum theory is similar to a previous result. It is then shown that one can construct the T-variables of Rovelli and Smolin on the null surface. The Poisson bracket algebra exhibits causal relations along the null rays, but is nonsingular if the loops are restricted to those whose projections along the null rays are not tangent and one-to-one. Finally, there is a brief discussion of the relevance of this work to general relativity.It is a pleasure to dedicate this paper to Fritz Rohrlich who has been a collegue at Syracuse University for the past 30 years. We both came to Syracuse at the same time. Indeed, Fritz called me to induce me to do so at a time when I was still considering the move. I have never regretted following his lead.     

Petrov classification of vacuum axisymmetric space-times

Using the null bivector approach, Petrov classification is studied for axisymmetric vacuum space-times with orthogonally transitive Killing vectors. It is shown that the equation on which the classification is based is biquadratic. This excludes that any such space-time can be type III. The only type-N manifolds are the radiative solutions considered by Hoffman. Van Stockum solutions are the most general type-II solutions. Degenerate Weyl solutions and Kinnersley solutions cover typeD. Stationary solutions with functional dependence of the potential are then examined. It is found that, except for special cases, Papapetrou and Lewis solutions are algebraically general.     

The generalized Taub-NUT congruence in Minkowski spaces

With any shear-free congruence of null geodesics in a Lorentzian geometry there is associated a Cauchy-Riemann three-space; and in certain spacetimes including the Ricci-flat spacetimes with expanding null shear-free (n.s.f.) congruences the deviation form of the congruence picks out an integrable distribution of complex two-spaces in the CR geometry. Conversely, given a CR geometry with an integrable distribution of two-spaces one can construct an associated family of spacetimes with a null, shear-free congruence. The interesting problem is the restrictionR _ab =0. We consider the case of n.s.f. congruences in Minkowski spacetime constructed from CR geometries of maximal symmetry. The special two-spaces are here taken to be those associated with either the Taub-NUT geometry or, as a limiting case, those associated with the Hauser twisting typeN solution. We obtain the most general solution for these cases.     

Core polarization effects and the random phase approximation solution

A metricg _ik=η _ik+Hξ _iξ_k+2Jξ _(iPk) is investigated. WhenJ=0 this reduces to the well-known Kerr metric. Conditions on the vectorp _i are obtained under which a geodetic, shear-free null congruenceξ _i in the Minkowskian space-time (with metricη _ik) will continue to remain geodetic and shear-free in the Riemannian space-time ofg _ik. A general solution of Einstein’s equationR _ik=σξ _iξ_k is obtained whenp _iξⁱ=0 andξ _i is twist-free.     

Boundary sources in the Doran-Lobo-Crawford spacetime

We take a null hypersurface (causal horizon) generated by a congruence of null geodesics as the boundary of the Doran-Lobo-Crawford spacetime to be the place where the Brown-York quasilocal energy is located. The components of the outer and inner stress tensors are computed and shown to depend on time and on the impact parameter b of the test-particle trajectory. The spacetime is a solution of Einstein’s equations with an anisotropic fluid as source. The surface energy density σ on the boundary is given by the same expression as that obtained previously for the energy stored on a Rindler horizon. For time intervals long compared to b (when the stretched horizon tends to the causal one), the components of the stress tensors become constant.      

一类半平面对称的电磁真空时空

本文证明了能够产生半平面对称度规的电磁场可以是平面对称的,也可以是半平面对称的。其中的平面对称电磁场一定是非类光电磁场,而半平面对称电磁场则可能是类光的,也可能是非类光的。求出由半平面对称类光(null)无源电磁场所产生的半平面对称时空通解。 关键词：     

On quadratic first integrals of the geodesic equations for type {22} spacetimes

It is shown that every type {22} vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries. In particular, the charged Kerr solution (with or without cosmological constant) is shown to admit a Killing tensor of valence 2. The Killing tensor, together with the metric and the two Killing vectors, provides a method of explicitly integrating the geodesics of the (charged) Kerr solution, thus shedding some light on a result due to Carter.     

Three-variable solution in the $$$$-dimensional null-surface formulation

The null-surface formulation of general relativity (NSF) describes gravity by using families of null surfaces instead of a spacetime metric. Despite the fact that the NSF is (to within a conformal factor) equivalent to general relativity, the equations of the NSF are exceptionally difficult to solve, even in 2+1 dimensions. The present paper gives the first exact \((2+1)\)-dimensional solution that depends nontrivially upon all three of the NSF’s intrinsic spacetime variables. The metric derived from this solution is shown to represent a spacetime whose source is a massless scalar field that satisfies the general relativistic wave equation and the Einstein equations with minimal coupling. The spacetime is identified as one of a family of \((2+1)\)-dimensional general relativistic spacetimes discovered by Cavaglià.