7. C. Y. Lo, David P. Chan, Richard C. Y. Hui, A Counterexample for the Harmonic “Gauge Condition” in General Relativity

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Volume 15: Pages 77-86, 2002 (doi:http://dx.doi.org/10.4006/1.3025513)

A Counterexample for the Harmonic “Gauge Condition” in General Relativity

C. Y. Lo, David P. Chan, Richard C. Y. Hui

Applied and Pure Research Institute, 17 Newcastle Drive, Nashua, New Hampshire 03060 U.S.A.

It is pointed out that the applicability of the harmonic gauge has never been generally established if the resulting coordinate system is expected to be physically realizable. Mathematically, Hilbert's proof is incomplete, and also invalid. A gravitational plane wave is given to show that the harmonic “gauge” may not be applicable and that implications of the linearized gauge on plane waves can be invalid. Concurrently, it is shown that, for weak gravitational waves, the harmonic gauge is valid only if the Einstein tensor is of secondorder deviation from the flat metric. Since such an order is gauge invariant, a gravitational weak wave with an Einstein tensor of first order implies that the harmonic gauge can be a misnomer. Moreover, it is shown that the “gauge condition” may not be compatible with coordinate relativistic causality and the equivalence principle. Concurrently, it is also shown that the Ohanian and Ruffini approach in which, in a different way from that of Einstein, linear gravity is considered as derived from their gauge theory is not valid in physics.

Keywords: gauge condition, harmonic gauge, linear gravity, gravitational wave, weak gravity

Received: January 3, 2001; Published online: December 15, 2008