farolero said:
So there's a spaceship 5 light years away from Earth and they want to send a signal to Earth but a cloud of interestellar dust don't let them use light or radio signals so they decide to send a gravity signal.
So they produce a violent huge thermonuclear explosion that will send some particles CLOSE to lightspeed.
Then the mass of those particles increase hugely due to relativity and hence its gravitational field.
And this gravitational field would be detected as a perturbation of the force of gravity from earth.
Would this be correct?
Gravitational radiation does exist, but a "huge thermonuclear explosion" would almost surely be spherically symmetrical, expanding in all directions. And this implies that it wouldn't generate significant gravitational radiation - there is no such thing a spherically symmetric gravitational wave (this is a consequence of something called Birkhoff's theorem), so a spherically symmetric explosion wouldn't be a significant source of gravitational radiation.
See for instance the wiki article on Birhoff's theorem,
https://en.wikipedia.org/w/index.php?title=Birkhoff's_theorem_(relativity)&oldid=724240631
Let's ignore this problem for now, and ignore the fact that we don't have a good mechanism other than a binary inspiral for generating gravitational radiation, and just look at the figures for energy assuming we could arrange such an inspiral - or perhaps come up with some as-yet-unthought of similarly efficeint way of genreating gravitational radiation.
Looking at known sources of gravitational radiation, we turn to Ligo. The Ligo paper is online, at
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102. Ligo detected the signal due to a binary inspiral, something that is reasonably efficient at generating gravitational radiation. It involved two black holes, estimated at 36 and 29 solar masses, spiralling together (something that doesn't happen in Newtonian theory, but does happen in GR), to form a 62 (estimated) solar mass black hole. In this process it was estimated that three solar masses of energy were converted into gravitational radiation. So that's maybe 5 percent efficient. Which doesn't sound like much, but it's still better than fusion or fission as I recall - slightly better than fusion if my memory serves (but feel free to check).
This was about a billion light years (estimated) away (converting the megaparasecs to light years). Scaling this is tricky, because a smaller inspiral would generate a shorter pulse of gravitational radiation, which our current detectors are not designed to detect, and it's unclear to me what the noise floor of detectors that could detect it would be like. But let's take some seat-of-the pants guesses. First we'll assume that we can use the inverse square law to scale this to get a similar signal at 5 light years. The signal will be smaller in amplitude, but happen faster, so it'd have roughly the same power. (This is speaking rather loosely, by the way - power and energy are much trickier in GR than I'm letting on). Three solar masses divided by 200 million squared is on the order of 10^14 kg of energy release. If we assume a first power inverse scaling law, which would give the same amplitude of the signal (but higher power), of we'd need about 10^22 or 10^23 kg. This is the signal power, not the size of the masses you'd need to generate the signal via an inspiral - which would be 20 times larger.
It would be interesting to give a better answer for what would be required with known mechanisms - i.e. a binary inspiral. What would the size of the black holes required be? Would the holes be so small they'd evaporate before they could inspiral, or would it be feasible? But I'm not quite clear on the correct scaling rules for this, so rather than give a bad guess I'll leave it an open question.
The really short answer is that the energy and power requirements are literally astronomical.