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.

/pdf/quasi-local-energy-momentum-and-angular-momentum-in-general-wmf4x6598j.pdf

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

The definition of angular momentum recently given by Penrose (1982) is analysed on I. It is shown that this definition is essentially a supertranslation of previous definitions. The origin dependence of Penrose's angular momentum is shown to have the correct form. Based on Penrose's expression, the authors then define a momentum for all elements of the BMS Lie algebra, which is the first such expression with the property that its flux vanishes identically in Minkowski space.

https://iopscience.iop.org/article/10.1088/0264-9381/1/1/005/pdf

Angular momentum at null infinity

A priori, there is nothing very special about shear-free or asymptotically shear-free null geodesic congruences. Surprisingly, however, they turn out to possess a large number of fascinating geometric properties and to be closely related, in the context of general relativity, to a variety of physically significant effects. It is the purpose of this paper to try to fully develop these issues. This work starts with a detailed exposition of the theory of shear-free and asymptotically shear-free null geodesic congruences, i.e., congruences with shear that vanishes at future conformal null infinity. A major portion of the exposition lies in the analysis of the space of regular shear-free and asymptotically shear-free null geodesic congruences. This analysis leads to the space of complex analytic curves in an auxiliary four-complex dimensional space, 
${\mathcal H}$
-space. They in turn play a dominant role in the applications. The applications center around the problem of extracting interior physical properties of an asymptotically-flat spacetime directly from the asymptotic gravitational (and Maxwell) field itself, in analogy with the determination of total charge by an integral over the Maxwell field at infinity or the identification of the interior mass (and its loss) by (Bondi’s) integrals of the Weyl tensor, also at infinity. More specifically, we will see that the asymptotically shear-free congruences lead us to an asymptotic definition of the center-of-mass and its equations of motion. This includes a kinematic meaning, in terms of the center-of-mass motion, for the Bondi three-momentum. In addition, we obtain insights into intrinsic spin and, in general, angular momentum, including an angular-momentum-conservation law with well-defined flux terms. When a Maxwell field is present, the asymptotically shear-free congruences allow us to determine/define at infinity a center-of-charge world line and intrinsic magnetic dipole moment.

https://link.springer.com/content/pdf/10.12942/lrr-2012-1.pdf

Null Geodesic Congruences, Asymptotically-Flat Spacetimes and Their Physical Interpretation

Exploring the holographic principle in asymptotically flat spacetimes via the BMS group

Treating general relativity as a Yang-Mills theory based on Lorentz invariance of the second kind leads to the derivation of six spin currents, which are linear combinations of the Ricci rotation coefficients, together with a Gauss theorem for the integrated charges. For asymptotically flat Einstein-Maxwell manifolds these charges are calculated by performing two-surface integrals at future null infinity. The gauge dependence of these charges, which arises because the underlying group is non-Abelian, is removed by making a canonical alinement of the asymptotic frames of reference. The Lorentz generators then arise as a field in asymptotic spin space and defined on the set of all outgoing null cones, which is the carrier space of the B.M.S. group. In the absence of outgoing gravitational radiation the Lorentz generators transform under translations as a moment of momentum. However, the points about which moments are taken belong not to the original Riemannian manifold but rather to a four-parameter family of cones isomorphic to Minkowski space-time. From the angular momentum structure there arises a natural twistor structure. Points on the twistor and the centre of mass of the system are defined as world lines in the manifold of cones. The remainder of the paper is devoted to formulating physical laws in cone space. These include a Poynting's theorem for the radiation of spin. The importance of Noether's theorem in a quantum field theory cannot be overstressed. In the hope of obtaining a greater understanding of the structure of matter, the predominant aim of such a theory lies in the calculation of scattering amplitudes for particle states. Of course, this procedure implies the existence of some system of labelling by means of which the particle states may be classified.

Relativistic angular momentum for asymptotically flat Einstein-Maxwell manifolds

It is shown that the asymptotic twistor equation leads to the asymptotic alinement of frames of reference. In the absence of gravitational radiation, this alinement reduces to a parallel one.

The Alinement of Frames of Reference at Null Infinity for Asymptotically Flat Einstein-Maxwell Manifolds

An isolated system in general relativity makes a transition between stationary states. It is shown that the spin vectors of the system, long before and long after the emission of radiation, are supertranslation invariant and, hence, independent of the choice of Minkowski observation space.

B. D. Bramson

Papers

Relativistic angular momentum for asymptotically flat Einstein-Maxwell manifolds

The Alinement of Frames of Reference at Null Infinity for Asymptotically Flat Einstein-Maxwell Manifolds

The invariance of spin