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.

I describe the Einstein’s gravitation of 3+1 dimensional spacetimes using the (2,2) formalism without assuming isometries. In this formalism, quasi-local energy, linear momentum, and angular momentum are identified from the four Einstein’s equations of the divergencetype, and are expressed geometrically in terms of the area of a two-surface and a pair of null vector fields on that surface. The associated quasi-local balance equations are spelled out, and the corresponding fluxes are found to assume the canonical form of energy-momentum flux as in standard field theories. The remaining non-divergence-type Einstein’s equations turn out to be the Hamilton’s equations of motion, which are derivable from the nonvanishing Hamiltonian by the variational principle. The Hamilton’s equations are the evolution equations along the out-going null geodesic whose affineparameter serves as the time function. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local quantities reduce to the Bondi energy, linear momentum, and angular momentum, and the corresponding fluxes become the Bondi fluxes. The quasi-local angular momentum turns out to be zero for any two-surface in the flat Minkowski spacetime. I also present a candidate for quasi-local rotational energy which agrees with the Carter’s constant in the asymptotic region of the Kerr spacetime. Finally, a simple relation between energy-flux and angular momentum-flux of a generic gravitational radiation is discussed, whose existence reflects the fact that energy-flux always accompanies angular momentum-flux unless the flux is an s-wave.

New Hamiltonian formalism and quasilocal conservation equations of general relativity

The main features of how to build a Born’s Reciprocal Gravitational theory in curved phase-spaces are developed. By recurring to the nonlinear connection formalism of Finsler geometry a generalized gravitational action in the 8D cotangent space (curved phase space) can be constructed involving sums of 5 distinct types of torsion squared terms and 2 distinct curvature scalars ${\mathcal{R}}, {\mathcal{S}}$ which are associated with the curvature in the horizontal and vertical spaces, respectively. A Kaluza-Klein-like approach to the construction of the curvature of the 8D cotangent space and based on the (torsionless) Levi-Civita connection is provided that yields the observed value of the cosmological constant and the Brans-Dicke-Jordan Gravity action in 4D as two special cases. It is found that the geometry of the momentum space can be linked to the observed value of the cosmological constant when the curvature in $\mathit{momentum}$ space is very large, namely the small size of P is of the order of $( 1/R_{\mathit{Hubble}})$. Finally we develop a Born’s reciprocal complex gravitational theory as a local gauge theory in 8D of the $\mathit{deformed}$ Quaplectic group that is given by the semi-direct product of U(1,3) with the $\mathit{deformed}$ (noncommutative) Weyl-Heisenberg group involving four $\mathit{noncommutative}$ coordinates and momenta. The metric is complex with symmetric real components and antisymmetric imaginary ones. An action in 8D involving 2 curvature scalars and torsion squared terms is presented.

Born’s Reciprocal Gravity in Curved Phase-Spaces and the Cosmological Constant

A unification model of 4D gravity and SU(3)×SU(2)×U(1) Yang-Mills theory is presented. It is obtained from a Kaluza-Klein compactification of 8D
 quaternionic gravity on an internal CP
 2=SU(3)/U(2) symmetric space. We proceed to explore the nonlinear connection $A^{a}_{\mu}( \textbf{x}, \textbf{y} ) $
 formalism used in Finsler geometry to show how ordinary gravity in D=4+2 dimensions has enough degrees of freedom to encode a 4D gravitational and SU(5) Yang-Mills theory. This occurs when the internal two-dim space is a sphere S
 2. This is an appealing result because SU(5) is one of the candidate GUT groups. We conclude by discussing how the nonlinear connection formalism of Finsler geometry provides an infinite hierarchical extension of the Standard Model within a six dimensional gravitational theory due to the embedding of SU(3)×SU(2)×U(1)⊂SU(5)⊂SU(∞).

J. H. Yoon

Papers

Quasi-local conservation equations in general relativity