On the transverse-traceless gauge condition when matter is presented

The transverse-traceless gauge condition is an important concept in the theory of gravitational waves. It is well known that a vacuum is one of the key conditions to guarantee the existence of the transverse-traceless gauge. Although it is thin, the interstellar medium is ubiquitous in the Universe. Therefore, it is important to understand the concept of gravitational waves when matter is presented. Bondi–Metzner–Sachs theory has solved the gauge problem related to gravitational waves. But it does not help with cases when the gravitational wave propagates in matter. This paper discusses possible extensions of the transverse-traceless gauge condition to Minkowski perturbation with matter presented.