Harmonic and Null Descriptions of Gravitational Radiation

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Abstract
We first consider an axially symmetric space-time containing gravitational waves in the usual Bondi null coordinate system. We then relate this Bondi frame to asymptotically harmonic coordinates by means of a characteristic initial-value problem. We use the resultant functional relationship between the null and asymptotically harmonic frames to transform Bondi's radiative metric into its equivalent asymptotically harmonic form. We then compare this with the radiative solution which Fock has calculated directly using harmonic coordinates. In agreement with Fock, we find that the harmonic form of the radiation metric contains logarithmic terms. Our treatment clearly shows that these logarithmic metric components do not result from the well-known logarithmic terms in the harmonic coordinate description of the light cones for Schwarzschild-like geometries. The principal result of this paper is that they arise from higher-order radiative corrections.

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