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For an observer moving along a time-like geodesic $x^{\mu}(\lambda)$ (parametrized by $\lambda$) the geodesic equations are satisfied $$ \ddot{x}^{\mu}(\lambda) + \Gamma^{\mu}_{\ \; \nu\rho} \; \dot{x}^{\nu}(\lambda) \dot{x}^{\rho}(\lambda) \ = \ 0 \ . $$ (As described on page 109 of Carroll's Spacetime and Geometry, I am aware that you can always re-parametrize $\lambda \to a \lambda + b$ such that the RHS more generally has the form $f(\lambda) \dot{x}^{\mu}(\lambda)$)

As I understand it, the proper-time experienced by the observer on the geodesic maximizes the functional $$ \tau \ = \ \int d\lambda\ \sqrt{ - g_{\mu\nu}(\lambda) \dot{x}^{\mu}(\lambda) \dot{x}^{\nu}(\lambda) } \ . $$

However, through use of some `external force' the observer can be put on a non-geodetic trajectory $x_{\mathrm{non}}^{\mu}(\lambda)$ where $\lambda$ parametrizes some path with $$ \ddot{x}_{\mathrm{non}}^{\mu}(\lambda) + \Gamma^{\mu}_{\ \; \nu\rho} \; \dot{x}_{\mathrm{non}}^{\nu}(\lambda) \dot{x}_{\mathrm{non}}^{\rho}(\lambda) \ \neq \ 0 \ . $$ (or more generally $\neq f(\lambda) \dot{x}^{\mu}(\lambda)$).

My question is how does one calculate the proper time experienced by the observer moving along $x_{\mathrm{non}}^{\mu}(\lambda)$? Is it still through an integration of $$ \int d\lambda\ \sqrt{ - g_{\mu\nu}(\lambda) \dot{x}_{\mathrm{non}}^{\mu}(\lambda) \dot{x}_{\mathrm{non}}^{\nu}(\lambda) } \ \ ? $$

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Yes, that is how you compute the proper time for any timelike path. See Wikipedia.

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