Abstract:  In many textbooks, gravitational waves are described as perturbations of flat spacetime. Under the weak field approximation, gravitational interactions can be described in a similar form to the Maxwell equation, corresponding to the concepts of gravitational electricity and gravitational magnetism. The weak gravitational field naturally corresponds to the perturbation of flat spacetime. Due to the fact that gravitational waves, gravitational electricity, and gravitational magnetism can be described as perturbations of flat spacetime, it is interesting to ask the relationship between gravitational waves, gravitational electricity, and gravitational magnetism. The analysis in this article indicates that gravitational electricity and gravitational magnetism correspond to the behavior of the near region of the gravitational source, but cannot describe the wave behavior in the far region. That is to say, the gravitational electricity and gravitational magnetism commonly used in literature cannot describe the behavior of gravitational waves. 

Key words:  gravitational wave, gravitoelectric field, gravitomagnetic field, post-Newtonian gauge, transverse-traceless gauge