A Proposal of Positive-Definite Local Gravitational Energy Density in General Relativity

We propose a 4-dimensional Kaluza-Klein approach to general relativity in the (2,2)-splitting of space-time using the double null gauge. The associated Lagrangian density, implemented with the auxiliary equations associated with the double null gauge, is equivalent to the Einstein-Hilbert Lagrangian density, since it yields the same field equations as the E-H Lagrangian density does. It is describable as a (1+1)-dimensional Yang-Mills type gauge theory coupled to (1+1)-dimensional matter fields, where the minimal coupling associated with the infinite dimensional diffeomorphism group of the 2-dimensional spacelike fibre space automatically appears. The physical degrees of freedom of gravitational field show up as a (1+1)-dimensional non-linear sigma model in our Lagrangian density. Written in the first-order formalism, our Lagrangian density directly yields a non-zero local Hamiltonian density, where the associated time function is the retarded time. From this Hamiltonian density, we obtain a positive-definite local gravitational energy density. In the asymptotically e-mail address: [email protected] 1 flat space-times, the volume integrals of the proposed local gravitational energy density over suitable 3-dimensional hypersurfaces correctly reproduce the Bondi mass and the ADM mass expressed as surface integrals at null and spatial infinity, respectively, supporting our proposal. We also obtain the Bondi mass-loss formula as a negative-definite flux integral of a bilinear in the gravitational currents at null infinity. PACS numbers: 04.20.-q, 04.20.Cv, 04.20.Fy, 04.30.+x Typeset using REVTEX 2