Consider two masses $m_1$ and $m_2$ that are connected by a spring. Mass 1 follows the worldline $x_1(\tau)$ while mass 2 follows $x_2(\tau)$. Note that the argument $\tau$ is the proper time in the rest frame of mass 1 and in the rest frame of mass 2 respectively but we do not use a different symbol for both.
I want to calculate how the system reacts to an incident gravitational wave (GW).
Both worldlines fulfill the geodesic equation:
where $f^\mu_1$ and $f^\mu_2$ are forces that act on the different masses.
How do we calculate the four-forces? Do we need a specific frame for that?