New Hamiltonian formalism and quasi-local conservation equations of general relativity

Abstract

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 divergence-type, 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 {\it non-vanishing} Hamiltonian by the variational principle. The Hamilton's equations are the evolution equations along the out-going null geodesic whose {\it affine} parameter 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 {\it 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.