The LHC website states that LHC can accelerate protons to 0.999999991 c.

Let's say that this acceleration costs the energy of one power plant for 1 hour (it's an hypothesis, i don't know the real energy cost).

Before acceleration, the proton was in the same inertial reference frame that the rest of the univers (if we consider univers to be static); but after the acceleration the univers and the proton are in two distincts inertials references frames with relative speed of 0.999999991 c. As in SR there is no prefered referential frame, can we conclude that, from the point of view of the proton, that accelerating the universe to 0.999999991 c costed the energy of one power plant ?

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    $\begingroup$ 1. The proton’s rest frame is highly non-inertial because it is undergoing circular motion at extremely high speed. 2. When the proton accelerates, it can still be described in the laboratory frame. You do not have to consider it to be in a different frame. $\endgroup$ – G. Smith Jun 3 '19 at 16:22
  • $\begingroup$ When they turn on the LHC, do you feel any acceleration of your body and the Earth to $0.999999991c$? $\endgroup$ – G. Smith Jun 3 '19 at 16:55
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    $\begingroup$ To run the cooling plants at high power and the magnets and to bring the whole beam up to speed is a significant enough power load that the utility needs to know in advance to bring capacity on line. But the kinetic energy of a single proton is still much less than a joule. It's just that there are a lot of them. $\endgroup$ – dmckee --- ex-moderator kitten Jun 3 '19 at 19:03

The proton accelerates to (say) $0.999999991c$ (relative to the lab) and then stays at that new velocity. In the proton's new frame, the lab is traveling at speed $0.999999991c$ and always has been. Since the lab hasn't accelerated, no force was ever applied to it, and no energy was ever needed.

(Likewise, at any point during the acceleration, the proton has an instantaneous frame in which the lab has not accelerated. This must be true, because the lab --- which I take to be inertial --- has not accelerated in its own frame, and all inertial frames must agree on the acceleration of the lab.)


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