Ezra T. Newman

An Approach to gravitational radiation by a method of spin coefficients

A new approach to general relativity by means of a tetrad or spinor formalism is presented. The essential feature of this approach is the consistent use of certain complex linear combinations of Ricci rotation coefficients which give, in effect, the spinor affine connection. It is applied to two problems in radiationtheory; a concise proof of a theorem of Goldberg and Sachs and a description of the asymptotic behavior of the Riemann tensor and metric tensor, for outgoing gravitational radiation.

Metric of a Rotating, Charged Mass

A new solution of the Einstein‐Maxwell equations is presented. This solution has certain characteristics that correspond to a rotating ring of mass and charge.

Empty‐Space Generalization of the Schwarzschild Metric

A new class of empty‐space metrics is obtained, one member of this class being a natural generalization of the Schwarzschild metric. This latter metric contains one arbitrary parameter in addition to the mass. The entire class is the set of metrics which are algebraically specialized (contain multiple‐principle null vectors) such that the propagation vector is not proportional to a gradient. These metrics belong to the Petrov class type I degenerate.

Note on the Bondi-Metzner-Sachs Group

It is shown that, in space-times which are asymptotically flat, there are reasonable physical restrictionsthat allow one to impose coordinate conditions (in addition to the usual Bondi-type conditions)which restrict the allowed coordinate group to a subgroup of the Bondi-Metzner-Sachsgroup. This subgroup is isomorphic to the improper orthochronous inhomogeneous Lorentz group.

Spin‐s Spherical Harmonics and ð

Recent work on the Bondi‐Metzner‐Sachs group introduced a class of functions sYlm(θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the representation matrices of the rotation group R3 and the properties of ð are derived from its relationship to an angular‐momentum raising operator. The relationship of the sTlm(θ, φ) to the spherical harmonics of R4 is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the sTlm under this latter group is shown to realize a representation of the Lorentz group.

Behavior of Asymptotically Flat Empty Spaces

The asymptotic behavior of the Weyl tensor and metric tensor is investigated for probably all asymptotically flat solutions of the empty space Einstein field equations. The systematic investigation utilizes a set of first order differential equations which are equivalent to the empty space Einstein equations. These are solved asymptotically, subject to a condition imposed on a tetrad component of the Riemann tensor ψ0 which ensures the approach to flatness at spatial infinity of the space‐time. If ψ0 is assumed to be an analytic function of a suitably defined radial coordinate, uniqueness of the solutions can be proved. However, this paper makes considerable progress toward establishing a rigorous proof of uniqueness in the nonanalytic case. A brief discussion of the remaining coordinate freedom, with certain topological aspects, is also included.

New Conservation Laws for Zero Rest-Mass Fields in Asymptotically Flat Space-Time

Some recently discovered exact conservation laws for asymptotically flat gravitational fields are discussed in detail. The analogous conservation laws for zero rest-mass fields of arbitrary spin s= 0,½,1,...) in flat or asymptotically flat space-time are also considered and their connexion with a generalization of Kirchoff’s integral is pointed out. In flat space-time, an infinite hierarchy of such conservation laws exists for each spin value, but these have a somewhat trivial interpretation, describing the asymptotic incoming field (in fact giving the coefficients of a power series expansion of the incoming field). The Maxwell and linearized Einstein theories are analysed here particularly. In asymptotically flat space-time, only the first set of quantities of the hierarchy remain absolutely conserved. These are 4s + 2 real quantities, for spin s, giving a D(s, 0) representation of the Bondi-Metzner-Sachs group. But even for these quantities the simple interpretation in terms of incoming waves no longer holds good: it emerges from a study of the stationary gravitational fields that a contribution to the quantities involving the gravitational multipole structure of the field must also be present. Only the vacuum Einstein theory is analysed in this connexion here, the corresponding discussions of the Einstein-Maxwell theory (by Exton and the authors) and the Einstein-Maxwell-neutrino theory (by Exton) being given elsewhere. (A discussion of fields of higher spin in curved space-time along these lines would encounter the familiar difficulties first pointed out by Buchdahl.) One consequence of the discussion given here is that a stationary asymptotically flat gravitational field cannot become radiative and then stationary again after a finite time, except possibly if a certain (origin independent) quadratic combination of multipole moments returns to its original value. This indicates the existence of ‘tails’ to the outgoing waves (or back-scattered field),which destroys the stationary nature of the final field.

Structure of Gravitational Sources

The purpose of this paper is to propose a definition of multipole structure of gravitational sources in terms of the characteristic initial data for asymptotic solutions of the field equations. This definition is based upon a detailed study of the corresponding data for the linearized equations and upon the close analogy between the Maxwell and the linearized gravitational fields.

Spacetime perspective of Schwarzschild lensing

(November 29, 1999)We propose a definition of an exact lens equation without reference to a background spacetime,and construct the exact lens equation explicitly in the case of Schwarzschild spacetime. For theSchwarzschild case, we give exact expressions for the angular-diameter distance to the sources aswell as for the magnification factor and time of arrival of the images. We compare the exactlens equation with the standard lens equation, derived under the thin-lens-weak-field assumption(where the light rays are geodesics of the background with sharp bending in the lens plane, andthe gravitational field is weak), and verify the fact that the standard weak-field thin-lens equationis inadequate at small impact parameter. We show that the second-order correction to the weak-field thin-lens equation is inaccurate as well. Finally, we compare the exact lens equation with therecently proposed strong-field thin-lens equation, obtained under the assumption of straight pathsbut without the small angle approximation, i.e., with allowed large bending angles. We show thatthe strong-field thin-lens equation is remarkably accurate, even for lightrays that take several turnsaround the lens before reaching the observer.I. INTRODUCTION

The Theory of H-space

Abstract The theory of H -space, the four-dimensional manifold of those complex null hypersurfaces of an asymptotically flat space-time which are asymptotically shear-free, is reviewed. In addition to a discussion of the origins of the theory, we present two independent formalisms for the derivation of the basic properties of H -space: that it is endowed with a natural holomorphic complex Riemannian metric which satisfies the vacuum Einstein equations and whose Weyl tensor is self-dual. We show the connection of our work on H -space to that of Plebanski and to the theory of deformed twistor spaces, due to Penrose. Finally, there is a discussion of equations of motion in H -space.