Weyl Class Research Articles

Stationary Axially Symmetric Generalizations of the Weyl Solutions in General Relativity

It is shown that a necessary condition that normal-hyperbolic solutions of the Einstein vacuum field equations for the metric tensor defined by the quadratic differential form $d{s}^{2}=fd{u}^{2}\ensuremath{-}2mdudv\ensuremath{-}ld{v}^{2}\ensuremath{-}{e}^{2\ensuremath{\gamma}}(d{x}^{2}+d{z}^{2})$ (where $f$, $l$, $m$, and $\ensuremath{\gamma}$ are functions of $x$ and $z$, and $fl+{m}^{2}={x}^{2}$) be of type III or $N$ is that ${x}^{\ensuremath{-}1}f$, ${x}^{\ensuremath{-}1}l$, and ${x}^{\ensuremath{-}1}m$ be functions of a single function $\ensuremath{\mu}$; it is further shown that no such nonflat solutions exist. Solutions having this functional dependence are found to belong to one of three classes: the Weyl class and two classes which may be obtained from it. One of these classes is characterized by Sachs-Penrose type-I stationary solutions having one real and two distinct complex-conjugate eigenvalues. The other class is characterized by Sachs-Penrose type-II stationary solutions admitting a single shear-, twist-, and expansion-free doubly degenerate geodesic ray which is also a null, hypersurface-orthogonal Killing vector. Further invariant properties of these classes are discussed, as well as the special case where $\ensuremath{\mu}$ depends only upon $x$.

Behaviour of Weyl–Levi Civita Coordinates for a Class of Solutions approximating the Schwarzschild Metric

THE search for energy sources for the ultra-high-energy phenomena in astrophysics discovered recently (quasars and pulsars, for example) has led to renewed interest in the study of general-relativistic gravitational collapse. (For a survey of relativistic astrophysics, see ref. 1.) Important unsolved problems include whether realistic models of collapsing bodies must pass through an event horizon, and, if so, whether they must continue collapsing until the inevitable development of a singularity. It has been conjectured2 that, once the simplifying assumption of spherical symmetry is dropped, it may be possible to avoid the formation of an event horizon (the generalization of the Schwarzschild radius) during collapse. In the hope that in the late stages of collapse the exterior field of the collapsing object might be well approximated by a static field, Israel2 has attempted to prove that “… there exists a general class of Weyl [static, axially-symmetric exterior gravitational] fields which deviate arbitrarily little from Schwarzschild's solution for r ≥ (1 + √(2)) m, which are analytic except for a particle-like singularity (well within the gravitational radius r = 2m) and which are free of event horizons”.

Certain Exact Solutions of the Equations of General Relativity with an Electrostatic Field

Certain exact solutions of the field equations of general relativity for empty space containing an electrostatic field are derived, and a physical interpretation is attempted. The canonical cylinder coordinates of Weyl are used, and all the solutions in which the electrostatic potential depends on only one of the two variables are obtained. Some of these are special cases of a class of solutions previously obtained by Weyl and they are shown to correspond either to a uniform electric field or to the field of a line-charge. Two of the solutions are not members of Weyl's class, and in these the electric field has no analogue in classical electrostatics and possesses the property of generating an orthogonal gravitational field.