New Hamiltonian formalism and quasilocal 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 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 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 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.

Hamiltonian boundary term and quasilocal energy flux

“…Nevertheless there still is considerable interest in this topic and some significant progress has been made recently; see in particular [13,14,15,16,17,18,19]. We include here some applications of our new natural energy flux expressions.…”

Quasilocal angular momentum of gravitational fields in (2+2) formalism

“…The equation (15) is the balance equation that relates the rate of change in L to the corresponding flux as the parameter u changes [2,6]. It should be emphasized that the quasilocal angular momentum can be also written in terms of geometric quantities as…”

“…In (2+2) formalism of Einstein's theory [2], the 4-dimensional spacetime E 4 is regarded as a fiber bundle that consists of a 2-dimensional base space M 1+1 of the Lorentzian signature and a 2-dimensional spacelike fiber N 2 at each point on M 1+1 . Then, the Einstein's theory can be interpreted as a gauge theory defined on M 1+1 with diff(N 2 ) as a gauge symmetry, the diffeomorphism group of N 2 [3].…”

On a class of 2-surface observables in general relativity

“…Similarly, the 'natural' boundary condition that the induced (n − 1)-metric q ab is fixed is preserved by the evolution equation (2.4.a) only if N a is vanishing on S or if (S, q ab ) admits N a as a Killing vector. It could be interesting to note that the quasi-local quantity L(N a ) of Yoon [17], obtained by following an (n − 1) + 2 analysis of the vacuum Einstein equations, as well as the '(generalized) angular momentum' of Brown and York [18], of Liu and Yau [19], and of Ashtekar and Krishnan [20] are just the observable O[N a ] provided N a on S is restricted to be tangent to S and δ e -divergence-free on S. Another (and quite obvious) observable is the surface integral of any integrable 'test' function f on S:…”