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.

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