Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

The present status of the quasi-local mass-energy-momentum and angular momentum constructions in general relativity is reviewed. First the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities are recalled. Then the various specific constructions and their properties (both successes and defects) are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned. This review is based on the talks given at the Erwin Schrodinger Institute, Vienna, in July 1997, at the Universitat Tubingen, in May 1998, and at the National Center for Theoretical Sciences in Hsinchu and at the National Central University, Chungli, Taiwan, in July 2000.

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Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article

The present status of the quasi-local mass, energy-momentum and angular-momentum constructions in general relativity is reviewed. First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities, are recalled. Then, the various specific constructions and their properties (both successes and deficiencies are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned.

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Quasi-Local Energy-Momentum and Angular Momentum in General Relativity.

The dynamical equations of an extended 1 + 3 orthonormal frame approach to the relativistic description of spacetime geometries are explicitly presented and discussed in detail In particular, the Bianchi identities for the Weyl curvature tensor occur in a fully expanded form, as they are given a central role in the extended formalism It is shown how one can naturally introduce local coordinates, both in the 1 + 3 threading and the ADM 3 + 1 slicing context By specializing the general 1 + 3 dynamical equations it is demonstrated how a number of problems of interest can be obtained In particular, the simplest choices of spatial frames for spatially homogeneous cosmological models, locally rotationally symmetric spacetime geometries, cosmological models with an Abelian isometry group and `silent' dust cosmological models are discussed

General relativistic orthonormal frame approach

The asymptotic structure of the gravitational field of isolated systems has been analyzed in great detail in the case when the cosmological constant ? is zero. The resulting framework lies at the foundation of research in diverse areas in gravitational science. Examples include: (i) positive energy theorems in geometric analysis; (ii) the coordinate invariant characterization of gravitational waves in full, nonlinear general relativity; (iii) computations of the energy?momentum emission in gravitational collapse and binary mergers in numerical relativity and relativistic astrophysics; and (iv) constructions of asymptotic Hilbert spaces to calculate S-matrices and analyze the issue of information loss in the quantum evaporation of black holes. However, by now observations have led to a strong consensus that ? is positive in our universe. In this paper we show that, unfortunately, the standard framework does not extend from the case to the case in a physically useful manner. In particular, we do not have positive energy theorems, nor an invariant notion of gravitational waves in the nonlinear regime, nor asymptotic Hilbert spaces in dynamical situations of semi-classical gravity. A suitable framework to address these conceptual issues of direct physical importance is developed in subsequent papers.

https://iopscience.iop.org/article/10.1088/0264-9381/32/2/025004/pdf

Asymptotics with a positive cosmological constant: I. Basic framework

The allowed asymptotic behavior of the Ricci tensor is determined for asymptotically flat space-times. With the aid of Penrose's conformai technique the asymptotic behavior of the components of the metric tensor, Weyl tensor, and spin coefficients in a suitable frame is calculated for such a space-time. For Einstein-Maxwell space-times these results reduce to those of Exton, Newman, Penrose, Unti, and Kozarzewski.

On asymptotically flat space-times

The complete class of conformally flat, pure radiation metrics is given, generalizing the metric given recently by Wils.

All conformally flat pure radiation metrics

This paper contains a classification of the Ricci tensorRαβ. The method of derivation is analogous to the spinor version of the Petrov classification of the Weyl tensor. It is shown how the various classes are related to the number and type of eigenvectors and eigenvalues ofRαβ. The classification is useful in the geometrization of various fields. The case of a real scalar field is treated in detail.

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Classification of the Ricci tensor

In order to achieve efficient calculations and easy interpretations of symmetries, a strategy for investigations in tetrad formalisms is outlined: work in an intrinsic tetrad using intrinsic coordi ...

Integration in the GHP formalism IV : A new Lie derivative operator leading to an efficient treatment of killing vectors

Held has proposed an integration procedure within the GHP formalism built around four real, functionally independent, zero-weighted scalars. He suggests that such a procedure would be particularly simple for the `optimal situation', when the formalism directly supplies the full quota of four scalars of this type; a spacetime without any Killing vectors would be such a situation. Wils has recently obtained a metric which he claims is the only conformally flat, pure radiation metric which is not a plane wave; this metric has been shown by Koutras to admit no Killing vectors, in general. Therefore, as a simple illustration of the GHP integration procedure, we obtain systematically the complete class of conformally flat, pure radiation metrics. Our result shows that the conformally flat, pure radiation metrics, which are not plane waves, are a larger class than Wils has obtained.

Garry Ludwig

Papers

On asymptotically flat space-times

All conformally flat pure radiation metrics

Classification of the Ricci tensor

Integration in the GHP formalism IV : A new Lie derivative operator leading to an efficient treatment of killing vectors

Integration in the GHP formalism III: Finding all conformally flat radiation metrics as an example of an `optimal situation'