Gravitational waves are just space time variation which object follows. if this is true then object gains energy according to inertial frame because object starts to move under influence of gravitational waves.
You are absolutely right, everything is confusing. My opinion the whole GR and SR is not working with universe that we are exploring. Here is quantum mechanics gravity Article Quantum Mechanics Gravity, Mechanical Gravity of Newton is Myth
maybe it clear some fog out of your thought. best regard.
Yes gravitational waves do carry energy and momentum. No, the fact that they do doesn't imply that energy is quantized. At the scale, where the description in terms of waves is adequate, which is the case, the fluctuations due to gravitons, the quanta of the field, are negligible. Just as is the case for electromagnetism.
The fact that electrons bound in atoms can't absorb or emit arbitrary amounts of energy doesn't mean anything more than the fact that gravitational waves can't be detected by monitoring such processes.
Gravitational waves carry both energy and momentum and the values are equal (in consistent units) so they travel at the speed of light.
The first formal detection was an event that, during the last cycle or so, radiated more energy than all the stars in the universe, an astonishing gravitational luminosity.
The momentum carried by the waves is also expected to cause the resulting black hole to "recoil" after a merger, that's very important because it implies that BH formed in a dense cluster are likely to be ejected so not be able to merger a second time producing a hierarchy of masses.
The gravitational waves seen so far have frequencies up to a few tens of Hz for BBH mergers and a few kHz for the one BNS merger seen. That means the shortest wavelengths are around 100 km. You can't get two stars to orbit any faster than a pair of neutron stars.
The constant h is just a way of normalising measurement units, the equations apply to any massless effect.
The wave alters the metric of spacetime by a very small amount. The strain seen in the detections to date has been less than 1 part in 1023 so the diameter of the Earth which is around 104m would be altered by about 1-19m, or less than a millionth of the size of one atom accumulated over the width of the whole planet.
Space is never empty, it always carrying wave, if there is temperature. Space of universe also is intelligent and carrying character. But there is no movement of anything between galaxies, because there is no wave-temperature.
Please allow to refer to iSpace theory (as presented here on RG page for some time) able to have derived and hence predict *quantum of gravitational force* (and hence energy and mass relation as well) purely from integer geometric first principles, while based on a rarely used "changed distance definition" and unusual hybrid 10D (4D continuum space-time + 6D Calabi-Yau style discrete subspace) model. The papers and related very simple multiplicative only (!) equations really allow anyone immediately to follow taking 1-2h of time, that is. RG links below!