Confusion about the proper time interval along an arbitrary worldline

Question

Let us consider the worldline $x^\mu(\lambda)$ of an observer moving in an arbitrary manner (i.e. possibly non-inertial). If we want to write down the proper time interval measured by that observer between two points $\lambda_1$ and $\lambda_2$ on its trajectory, we use the formula $$\Delta\tau=\int\limits_{\lambda_1}^{\lambda_2}d\lambda\sqrt{\eta_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}}$$ instead of $$\Delta\tau=\int\limits_{\lambda_1}^{\lambda_2}d\lambda\sqrt{g_{\mu\nu}(x)\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}}.$$

Aren't acceleration effects mimicked by the presence of a gravitational field ala the equivalence principle? If so, should we not use the second formula? I am sure that I hold a severe foundational misunderstanding here. I think I am equating "acceleration $\Longleftrightarrow$ presence of gravity" which is wrong?

Answer 1

Aren't acceleration effects mimicked by the presence of a gravitational field ala the principle of equivalence? If so, should we not use the second formula?

You can always use the second formula. The first formula is a special case of the second. Specifically, inertial coordinates in flat spacetime. The acceleration of the object is irrelevant.

Regarding the equivalence principle, for an accelerating worldline in flat spacetime you may choose to use an inertial coordinate system or a non-inertial coordinate system. If you use the inertial coordinate system then there is no "gravity field" (non-zero Christoffel symbols), and if you use a non-inertial coordinate system then there is. It is the acceleration of the coordinates that defines the presence or absence of the "gravity field" (non-zero Christoffel symbols). The acceleration of any given worldline is not directly relevant. It is only indirectly relevant to the extent that you choose to base your coordinate system around that worldline.