Why there exist no longitudinal waves in otherwise empty space?

By changing the pressure of a spherically symmetric mass we can vary the pull it has on a point test mass m, if m is located inside the spherical mass.

One may ask if such a radial, changing force can create longitudinal gravitational waves in space.

We do not think longitudinal waves are possible. The mechanism which seems to create transverse waves is that far parts of the field of some charge q should move faster than light to keep the static field in the usual form. There must be some "transverse" movement of q.

But if q moves directly toward us, we can keep its static field in the usual form without any faster-than-light movement.

The origin of transverse waves is in the lagging of the far field.

We have also remarked that the finite speed of light can simulate inertia in transverse movements of field lines. A wave requires some kind of inertia, or a delay, to exist.

It is harder to imagine how a longitudinal movement of a field line could simulate inertia.

If space is not empty, but contains plasma, then longitudinal waves can propagate. They get the required inertia from the plasma.