New Hamiltonian formalism and quasilocal conservation equations of general relativity

I describe the Einstein’s gravitation of $3+1$ dimensional spacetimes using the (2,2) formalism without assuming isometries. In this formalism, quasilocal 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 quasilocal 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 quasilocal quantities reduce to the Bondi energy, linear momentum, and angular momentum, and the corresponding fluxes become the Bondi fluxes. The quasilocal angular momentum turns out to be zero for any two-surface in the flat Minkowski spacetime. I also present a candidate for quasilocal 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.

Figure 1

This figure shows the geometry of the (2,2) fibre bundle splitting of $3+1$ dimensional spacetime. The $1+1$ dimensional base manifold is spanned by $\{{\partial }_{\pm }\}$ and the two dimensional fibre space $N_2$ by $\{{\partial }_a\}$. The horizontal vector fields $\{{\widehat{\partial }}_{\pm }\}$ are orthogonal to $N_2$, and $A_{\pm }^a$ are the connections valued in the Lie algebra of the diffeomorphisms of $N_2$.

Figure 2

(a) This figure shows the spacetime geometry in the region where $2h>0$, and $v=$constant hypersurface is timelike. (b) This figure shows the spacetime geometry in the region where $2h<0$, and $v=$constant hypersurface is spacelike. In both cases $N_2$ is represented by a circle.