Inspired by the gravtiomagnetic analogy, I would expect that just as a charged tachyon would emit normal (electromagetic) Cerenkov radiation, any mass-carrying warp drive would emit gravitational Cerenkov radiation. The gravitomagnetic approximation may well break down near the mass, but "sufficiently far" from it, this would still be valid. Is that correct?
Specifically, let's suppose there is a moving closed surface S, such that on and outside S the gravitomagnetic equations are approximately valid (no assumptions about interior), such that it moves with a velocity greater than $c$, and such that it "carries mass", in the sense that the closed surface intergral of the gravitational field strength around S is negative (net inward gravitational field).
In general relativity, is this situation even possible? If so, would it emit gravitational radiation? If so, how fast would it lose energy (mass)?
I am motivated by the recent media hype around the Alcubierre metric. Nevertheless, it is a general question applying to any proposed "moving warp bubble" solution of general relativity. (As opposed to, say, a pair of "stargates", or a "warp corridor", or whatever — if a mass $M$ travels through a stargate, it might be that the gate through which it enters could get heavier by $M$, and the gate through which it leaves could get lighter by $M$. Then this particular question wouldn't arise.)
Best Answer
Let's take a look of the gravitational field of particles travelling at the speed of light in relativity. These are called gravitational shock-waves and they carry a space-time shock that travels with them at the exact same wave-front. However, as can be seen from the consideration of the work of Aichelburg and Sexl (1971), this shock is just the information about a point mass attracting you for a fleeting (shocking) moment. Specifically, it carries no energy transfer in the same sense as the Newtonian gravitational field of e.g. a meteor flying by also does not transfer any net energy.
It is attractive to think about the shock plane getting deformed into a Cherenkov-like shock cone as you breach the speed of light, but there is really no exact solution to support that. This is because relativity is built in a way so that if you put physical sources into a space-time, they must fulfill their dynamical equations, otherwise the space-time knows and rewards you with absolutely unpleasant singularities. And there is really no known physical dynamics that would get you to and beyond the speed of light.
So let's take a look at linearized relativity. In linearized relativity you can make the sources do whatever you want and you don't have to pay for it the same way as in full nonlinear relativity. Then, if you take a pointlike source that is moving superluminally (following a space-like worldline), you can always transform into a frame in which the source is just a static spatial line. Static spatial lines are static and they do not radiate. If you boost back to your original frame, you will just feel gravitomagnetic-type effects doing funny stuff to you without making any work.
Finally, let's take a look at the Alcubierre warp drive published in 1994. In Alcubierre's original proposal, the metric sees no naughty asymptotics. In fact, the Alcubierre metric goes to zero exponentially because Alcubierre wrote it down so that it does. So observers far away do not even know that a warp drive gravitationally is or was there. Alcubierre checked what makes the space-time behave like that only afterwards from the Einstein equations. If there is radiation, you can magick it away with fairy dust (the matter source of the Alcubierre metric), if you need radiation, you can magick it in. So I do not know what to take from that but no, there is no radiation in the proposed metric.
In summary. No, there is no gravitational Cherenkov radiation in relativity. Then again, who knows what can fairy dust do.