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

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

1993-01-15

arxiv.org/abs/gr-qc/9401012v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

39p, SNUTP 93-78, RevTex 3.0

Scientific paper

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 is equivalent to the Einstein-Hilbert Lagrangian, since it yields the same field equations as the E-H Lagrangian 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 diffeomorphism group of the 2-dimensional spacelike fibre space automatically appears. 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 flat space-times, the volume integrals of the proposed local gravitational energy density over suitable 3-dimensional hypersurfaces correctly reproduce the Bondi and the ADM surface integral, 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.

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