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Why Newton's third law remain unchanged still now in relativity theory (as for example that is why we feel weight due to equal but opposite reaction of the Earth's surface)?

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Newton't third law is just the conservation of momentum, so the question is whether conservation of momentum applies in general relativity. This turns out to be a rather complicated issue, and for the details I suggest you look at General relativity and the conservation of momentum.

However you specifically ask why we feel weight due to equal but opposite reaction of the Earth's surface. For an observer spacetime locally appears to be flat i.e. locally special relativity applies rather than general relativity, and four-momentum is conserved in special relativity. Since you and the Earth are stationary with respect to each other we can ignore relativistic effects and Newton's third law applies. Things would be a bit different if you and the Earth were colliding at a significant fraction of the speed of light!

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  • $\begingroup$ But why does Einstein care much about "Momentum"instead of only energy density in his GR theory?please describe in detail. $\endgroup$ – Absaed May 15 '16 at 18:04
  • $\begingroup$ @Absaed: I don't understand what you are asking. As far as I know Einstein didn't place any special emphasis on momentum in GR. Are you referring to the fact that momentum flux appears in the stress-energy tensor? If so have a look at Intuitive understanding of the elements in the stress-energy tensor. $\endgroup$ – John Rennie May 15 '16 at 18:18

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