Suppose I am running near the speed of light, say 0.87c so that my gamma factor is exactly 2. What forces will I and the ground experience if I jump? Imagine it's a normal two-footed jump where I take off vertically but maintain all forward velocity. And how will gravity act on me?
My guess: I will exert some amount of force on the ground, say F Newtons. But from the ground's reference frame my acceleration is doubled so it experiences 2F Newtons of force. The reaction force is 2F Newtons also, which is 'translated' to F Newtons in my frame of reference, launching me upwards. I would experience half the acceleration from gravity, but I would also experience length contraction so I think I've jumped normal height but actually I've jumped double height. The impact with the ground would be F Newtons from my reference frame but 2F from the ground's.
Suppose now I'm vibrating on the spot at 0.87c, and I jump. Does anything change in this scenario?
Edit: Assume I'm on an infinite plane and acted upon by gravity, g=9.8m/s^2.