“The geometric conditions imposed on our solutions, however, are too stringent to allow for completely realistic fields.”

If one considers GR as a valid theory and wants to confirm its results by experiment, then these results should obey general covariance, i.e. they should be expressed in quantities which do not change under arbitrary coordinate transformations.

Maybe, we were not precise when we wrote that Einstein’s equations do not have a wave solution. They might have a solution similar to what another nonlinear equation, e.g., Korteweg-de Vries’ equation, has – a solitary wave, but any solution of Einstein’s equations, which is to be related to observation, should be invariant under general coordinate transformations. For example, it is possible to consider a linear approximation of Einstein’s equation, or weak perturbation of a metric around a flat space-time, and find a plane wave solution, but there always exists a coordinate transformation (among all possible coordinate transformations) such that a plane wave solution disappears. In plane words, if we choose a coordinate system which oscillates together with a wave, then an observer in such a frame of reference will not “see” a gravitational wave. This is in contrast to the electro-magnetic theory. The solutions of Maxwell’s equations are invariant under Lorentz transformations, and consequently, there is no such a Lorentz transformation that can destroy a wave solution.

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