general relativity - Are gravitational waves produced only by masses which experience a change in acceleration, and if so, is this connected to the spin 2 of the graviton?

I can't remember for sure, which is why I ask this question, but is it the case that gravitational waves are produced not by accelerated masses (like e.m. waves are produced by accelerated charges), but only by masses with a change in their acceleration?

And if so, is there a connection with the conjecture that the (hypothetical) graviton has spin 2?