Proper time, geodesic and external force

Deriving the geodesic equation

$$ (nabla_u u)^mu = 0 $$

one uses the variation of proper time $tau[C]$ along $C$

$$ delta int_C dtau = int_C dtau , g_{munu} , delta x^mu , (nabla_u u)^nu qquad (1) $$

with arbitrary infinitesimal variation $delta x$, 4-velocity $u = dot{x}$ along $C$ and parallel transport $nabla_u$.

On the other hand we know that the geodesic equation with finite external force $f$ is modified as

$$ (nabla_tilde{u} tilde{u})^mu = f^mu $$

where $tilde{u}$ is the 4-velocity along $tilde{C}$ which is not a geodesic due to non-vanishing $f$.

Fine.

Suppose we have explicit solutions $x$ and $tilde{x}$ for

$$ (nabla_u u)^mu = 0 $$

$$ (nabla_tilde{u} tilde{u})^mu = f^mu $$

Is there a way to define something like $delta x$, to find an expression like

$$ tau[tilde{C}] – tau[C] sim int_tilde{C} dtau , g_{munu} , delta x^mu , f^nu qquad (2) $$

in order to calculate the deviation of the proper time?

In contrast to (1) the general expression in (2) is neither infinitesimal nor arbitrary but would have to be derived exactly from the known solutions $x$ and $tilde{x}$.