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I break my question into three parts, with increasing "guessing status". Please tell me which of them (is any) is correct.

  • Suppose I am an observer in an inertial reference frame, away from all other masses influence. When relativistic velocities are implied, do Newton's laws of motion still hold if I use the concept of relativistic mass (that is, introducing the gamma factor within the momentum)?

  • Now suppose I am an observer in a non inertial reference frame (submit to a force, for example electromagnetic force), but still away from all other masses influence. When relativistic velocities are implied, do Newton's laws of motion still hold if I use the concept of relativistic mass AND introduce a fictitious force "in the classic way" (depending on my state of motion with respect to an inertial reference frame)?

  • Final step. Now suppose I am in a non inertial reference frame (submit to a force) AND under the influence of other masses. When relativistic velocities are implied, do Newton's laws of motion still hold if I use the concept of relativistic mass AND introduce a fictitious force in the classic way AND use Newton's law of gravitation?

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  • $\begingroup$ For your first point, are you asking does ${\bf F} = \gamma m_0 {\bf a}$? $\endgroup$ – jim Sep 15 '19 at 10:22
  • $\begingroup$ Yes indeed, without any reference to the "time component" of the Force Vector $\endgroup$ – Federico Toso Sep 15 '19 at 10:26
  • $\begingroup$ Note that "relativistic mass" is direction dependent. $\endgroup$ – Poutnik Sep 15 '19 at 10:33
  • $\begingroup$ @Poutnik yes I know. I should not be a problem since this is a vectorial equation (I just have to take care when I project it on the spatial axes) $\endgroup$ – Federico Toso Sep 15 '19 at 10:37
  • $\begingroup$ You last part is a really horrible mix of half-theories that just don't fit together. You would be far better served to just learn special and general relativity. $\endgroup$ – m4r35n357 Sep 15 '19 at 12:20

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