There is a charge, i.e., mass-energy in gravitational waves: are there longitudinal waves, too?

In electromagnetics, a wave carries a zero charge. We believe that there exist no longitudinal electromagnetic waves.

But assume that the wave would carry an electric charge. Could there be longitudinal waves, and would they propagate at a speed less than light?

If we move to a frame where the wave is redshifted, then the charge would be less, and the waves would have a longer wavelength. This suggests that the waves would still propagate at the speed of light.

A rubber membrane model of gravity breaks Birkhoff's theorem. The metric of time and space can change around a spherically symmetric mass if we change the pressure. It could send spherically symmetric gravitational waves where the metric of time undulates. Are these waves longitudinal? It is monopole radiation.

The "displacement" in such a wave is not into any spatial direction. We could claim that these are transverse waves where the displacement is to the time direction. In a rubber membrane, if we remove a weight which was sitting on the membrane, then the depression in the membrane is removed. The depression described slowdown of time. The circular wave after that is transverse, and to the direction of a time speedup.

It is not clear what do "longitudinal" and "transverse" mean in this context. Maybe it is best just to describe the wave and refrain from categorizing it.

Anyway, changing pressure in a rubber membrane model may send monopole gravitational waves. Ordinary gravitational waves come from two circling dipoles which have a 180 degree phase shift. Ordinary waves are quadrupole waves.