Weyl Class Research Articles

Shells embedded in flat space

I give a solution of Einstein's equations representing a material surface obeying energy conditions and embedded in flat space. It is a member of Weyl's class of static, axially symmetric spacetimes, and can represent a plane distribution of counter-rotating photons or dust. A freely falling observer in the surrounding flat spacetime sees the surface as a moving hyperboloid.

Spinning test particles in Weyl spacetimes

The motion of spinning test particles along circular orbits in static vacuum spacetimes belonging to the Weyl class is discussed. Spin alignment and coupling with background parameters in the case of superimposed Weyl fields, corresponding to a single Schwarzschild black hole and single Chazy-Curzon particle as well as to two Schwarzschild black holes and two Chazy-Curzon particles, are studied in detail for standard choices of supplementary conditions. Applications to the gravitomagnetic "clock effect" are also discussed.

SUPERPOSITION OF WEYL SOLUTIONS: CIRCULAR ORBITS

Circular orbits are examined in static spacetimes belonging to the Weyl class of vacuum solutions which represent (nonlinear) superposition of the gravitational fields generated by certain collinear distributions of matter. In particular, solutions representing two and three Chazy–Curzon particles — all of them endowed with conical singularities — are considered. Conditions for geodesic motion in certain symmetry planes are discussed and results are summarized in a number of graphics too. All the discussion is developed in the framework of observer-dependent analysis of motion.

Matrix Finite-Zone Dirac-Type Equations

Weyl–Titchmarsh matrix functions play an essential role in the spectral theory of Dirac-type equations ( Oper. Theory: Adv. Appl. 107 (1999)). In this paper, we have constructed a class of Weyl–Titchmarsh matrix-functions generating potentials of finite-zone type. It has turned out that the corresponding potentials have derivatives of an arbitrary order. Using the above-mentioned results, we deduce the matrix analogue of the trace formula for finite-zone matrix potentials. In the last part of the paper, we consider separately the scalar case of Dirac-type equations. For this case, we have constructed finite-zone potentials in explicit forms and proved that these potentials are quasiperiodical. We note that for scalar Schrödinger equations the corresponding results are well known (see Invent. Math. 30 (1975), 217–274; “Soliton and the Inverse Scattering Transform,” SIAM, Philadelphia, 1981; Rev. Sci. Technol. 23 (1983), 20–50; “Theory of Solitons, The Method of Inverse Problem,” New York, 1984; “Inverse Sturm–Liouville Problems,” VSP, Zeist, 1987; “Inverse Spectral Theory,” Academic Press, New York, 1987).

Differentially Rotating Disks of Dust: Arbitrary Rotation Law

In this paper, solutions to the Ernst equation are investigated that depend on two real analytic functions defined on the interval [0,1]. These solutions are introduced by a suitable limiting process of Backlund transformations applied to seed solutions of the Weyl class. It turns out that this class of solutions contains the general relativistic gravitational field of an arbitrary differentially rotating disk of dust, for which a continuous transition to some Newtonian disk exists. It will be shown how for given boundary conditions (i. e. proper surface mass density or angular velocity of the disk) the gravitational field can be approximated in terms of the above solutions. Furthermore, particular examples will be discussed, including disks with a realistic profile for the angular velocity and more exotic disks possessing two spatially separated ergoregions.

Differentially Rotating Disks of Dust

We present a three-parameter family of solutions to the stationary axisymmetric Einstein equations that describe differentially rotating disks of dust. They have been constructed by generalizing the Neugebauer—Meinel solution of the problem of a rigidly rotating disk of dust. The solutions correspond to disks with angular velocities depending monotonically on the radial coordinate; both decreasing and increasing behaviour is exhibited. In general, the solutions are related mathematically to Jacobi's inversion problem and can be expressed in terms of Riemann theta functions. A particularly interesting two-parameter subfamily represents Bäcklund transformations to appropriate seed solutions of the Weyl class.

Geodesics in Lewis space–time

The geodesic equations are integrated for the Lewis metric and the effects of the different parameters appearing in the Weyl class on the motion of test particles are brought out. The appearance of a force parallel to the axial axis is without Newtonian analog and deserves particular attention.

Counter-rotating dust rings around a static black hole

Relativistic configurations describing counter-rotating dust discs with a central Schwarzschild black hole are analysed. It is shown that the corresponding solutions to the Einstein equations which belong to the Weyl class can be obtained from a boundary value problem for the Laplace equation where the boundary data are given at the two-dimensional dust ring. Using a result of Poole, one can prove existence and uniqueness of a solution to this problem. In adapted coordinates, the potential can be represented via eigenfunctions of the Lamé equation in the `flat ring' case which are not known explicitly. Following Kanwal, one can reformulate the boundary value problem in a system of Fredholm integral equations. As an example, a rigidly (counter-)rotating dust ring is considered. In the case of a dust ring stretching to infinity, it is possible to relate this problem via a Kelvin transformation to the case of a disc. An explicit solution is given for an isochrone dust ring.

The parameters of the Lewis metric for the Weyl class

We provide physical interpretation for the four parameters of the stationary Lewis metric restricted to the Weyl class. Matching this spacetime to a completely anisotropic, rigidly rotating, fluid cilinder, we obtain from the junction conditions that one of these parameters is proportional to the vorticity of the source. From the Newtonian approximation a second parameter is found to be proportional to the energy per unit of length. The remaining two parameters may be associated to a gravitational analog of the Aharanov-Bohm effect. We prove, using the Cartan scalars, that the Weyl class metric and static Levi-Civita metric are locally equivalent, i.e., indistinguishable in terms of its curvature.

On the parameters of the Lewis metric for the Lewis class

The physical and geometrical meaning of the four parameters of the Lewis metric for the Lewis class are investigated. Matching this spacetime to a completely anisotropic, rigidly rotating, fluid cylinder, we find from the junction conditions that the four parameters are related to the vorticity of the source. Furthermore, it is shown that one of the parameters must vanish if one wishes to reduce the Lewis class to a locally static spacetime. Using the Cartan scalars it is shown that the Lewis class does not include Minkowski as a special class globally, and that it is not locally equivalent to the Levi-Civita metric. It is also shown that, in contrast with the Weyl class, the parameter responsible for the vorticity appears explicitly in the expression for the Cartan scalars. Finally, to enhance our understanding of the Lewis class, we analyse the van Stockum metric.

Moment density of Zipoy's dipole solution

Zipoy used spheroidal coordinates to construct a family of static axisymmetric gravitational fields included in Weyl's class. We calculate the mass moment source for the dipole solution using techniques previously developed for magnetic and angular momentum densities and compare it with the Newtonian analogue studied by Bonnor and Sackfield.

Generalised soliton solutions of the Weyl class

The whole family of generalised soliton solutions of the Weyl class for Einstein's equations in vacuum is considered. A classification scheme for the distinct types of metrics belonging to this family is given and emphasis is placed on the asymptotically flat solutions. The physical interpretation of the occurring parameters is also briefly considered.

Stationary axisymmetric solutions of the Jordan-Brans-Dicke equations with electromagnetic sources

Any stationary axisymmetric solution of the coupled JBD-Maxwell field equations such thatF ab * taψb vanishes, can be determined from a composition of any stationary axisymmetric Einstein-Maxwell spacetime with the Weyl class of metrics.

A note on the Weyl solutions

The author gives the solutions for a metric of Weyl's class for sources possessing both inner and outer multipole moments.

Stationary axisymmetric solutions in the vacuum Jordan-Brans-Dicke theory

It is shown that any stationary axisymmetric solution to the vacuum field equations of the Jordan-Brans-Dicke (JBD) theory may be obtained from a “composition” of any stationary axisymmetric vacuum Einstein spacetime with the Weyl class of metrics. Thus, generating solution techniques can be used to obtain any stationary axisymmetric JBD vacuum solution. In this manner, C. B. G. McIntosh's results concerning this topic are improved upon.

Representations of E8 and other algebras

A simple procedure is given for determining whether or not an arbitrary weight of a simple Lie algebra is contained in an arbitrary irreducible representation. The procedure involves two steps, determining the Weyl class of the weight and determining the classes contained in the representation. The second step applies without alteration to all simple algebras, while the first step is given here only for E8. The weights of the shortest 31 Weyl classes of E8 are listed in a convenient, orthogonal basis.

An investigation of axially symmetric electrovac solutions

For the axially symmetric, electrostatic vacuum problem, there are three unknowns:g00,g11, and φ, the electrostatic field. Herlt, using his generation methods, has presented several new solutions by explicitly givingg00 andφ only. Theg11 is now determined in two cases and a detailed discussion of these solutions is given. The solutions do not, in general, possess the overlapping equipotential structure [g00=g00(φ)] which characterizes the Weyl class. One of the classes contains metrics which can be physically interpreted as representing exterior space-times of non-Weyl “two-body”-type charged “point” sources. With the specialization of the arbitrary parameters, the above solutions do reduce to known Weyl solutions. The Bonnor solution is a member of one of the above classes and consequently a possible physical reinterpretation of this solution is given. Kinnersley transformations applied to the above classes yield stationary space-times with line singularities and NUT-like asymptotic behavior.

On the stationary Einstein–Maxwell field equations

The stationary gravitational equations in the presence of the electromagnetic fields, outside charged gravitating sources, are investigated. (i) The action integral of Kramer–Neugebauer–Stephani (K.N.S.) is derived from the Hilbert action integral by using new variational techniques. (ii) It is shown that the classification scheme for the system of partial differential equations of general relativity depends on the coordinate system used. In particular, if orthogonal coordinates are chosen for the associated space then the system of Einstein–Maxwell equations is a hyperbolic one. (iii) The eigenvalues of the Ricci tensor of associated space are expressed in terms of the invariants of stationary electro-gravitational fields. It is proved that if these eigenvalues are equal then the fields must belong to the class of Peres–Israel–Wilson (PIW) solutions. (iv) The global integrability of some of the stationary Einstein–Maxwell equations and the consequent equilibrium conditions of the ’’bodies’’ are investigated. (v) Boundary value problems for some of the field equations are pursued. It is proved that ω≡ln‖g44‖ is neither subharmonic nor superharmonic and the boundary value problem for this function does not yield a unique solution in general. A nontrivial solution of the stationary equations with ω≡0 is given. A special boundary value problem is explicitly solved. (vi) The PIW solutions are generated from the charged Kerr–Tomimatsu–Sato–Yamazaki (KTSY) solutions. The complex axially symmetric harmonic functions of these PIW solutions can be obtained from the real axially symmetric harmonic functions of the static Weyl class of electrovac solutions by a complex scale transformation of the coordinates.

Static and stationary axially symmetric gravitational fields of bounded sources II. solutions obtainable from Weyl's class

This paper shows that a new class of axially symmetric static electrovacuum/magnetovacuum solutions is obtainable from Weyl's class of static vacuum solutions. The new class contains an infinite set of asymptotically flat solutions (in closed form) each of which involves an arbitrary set (d, αi) of parameters. These parameters have to be interpreted as functions of massm, chargee, and higher electric/magnetic multipole momentsα i of the particle. The cased = 0,α i =0 leads to the Darmois solution and the cased = 0,α i ≠0 leads to the results of [1]. The case d≠=0, e=αi=0 leads to the Schwarzschild solution, the cased≠ 0,α i =0,e≠ 0 leads to the Reissner-Nordstrom solution. To get more general examples is a lengthy but straightforward exercise.

New Class of Solutions of the Einstein-Maxwell Fields

Maxwell fields in static axially symmetric space-time are discussed. A new method for generating the Weyl class as well as a more general class of electromagnetic fields is outlined. It has been shown that by the same procedure a new class of Maxwell fields can be generated which are not of Weyl type. A particular solution of this class is derived which represents an asymptotically flat gravitational field of a body possessing an electric or magnetic dipole moment. At large distances and in the case of a vanishing dipole parameter, the gravitational field goes over to the Schwarzschild field.