# General 2D equations of motion in non-inertial relativistic frame of reference

Physics Asked by kbobrowski on August 26, 2020

Let’s assume that there is inertial frame of reference $$S$$, and the observer is at the origin of this frame at time (associated with this inertial frame) $$t = 0$$. The observer has own frame of reference $$S’$$ (with proper time $$tau$$), which coincides with inertial frame at $$t = tau = 0$$. From the perspective of this inertial frame, the observer starts moving with arbitrary, non-constant acceleration in $$xy$$ plane, this movement is given by functions $$x(t)$$, $$y(t)$$ in $$S$$. The moving observer observes the stationary point on $$xy$$ plane in $$S$$, which position is given by ($$x_p$$, $$y_p$$) in $$S$$.

How to obtain functions $$x_p'(tau)$$, $$y_p'(tau)$$ which describe apparent motion of this point from the observer perspective, in $$S’$$, given the movement of the observer ($$x(t)$$, $$y(t)$$) and position of this point ($$x_p$$, $$y_p$$) in $$S$$? I’m looking for some hints on numerical solution.