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A Canonical Form of the Metric in General Relativity

If there is a null gradient field in a 1+3 dimensional Lorentzian manifold, we can establish a kind of light-cone coordinate system for the manifold from the null gradient field. In such coordinate system, the metric takes a wonderful canonical form, which is much helpful for resolving the Einstein's field equation. In this paper, we show how to construct the new coordinate system, and then explain their geometrical and physical meanings via examples. In the light-cone coordinate system, the complicated Einstein's field equation could be greatly simplified. This coordinate system might be also helpful to understand the propagation of the gravitational wave.