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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?

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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.

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  • $\begingroup$ But depending on which formula we use, we'll get a different result. $\endgroup$ – mithusengupta123 Sep 29 '20 at 4:06
  • $\begingroup$ If spacetime is curved and you use a flat metric, you’ll get the wrong answer. What made you think you are supposed to use the first formula? $\endgroup$ – G. Smith Sep 29 '20 at 4:25
  • $\begingroup$ @G.Smith What I understand is that if spacetime is flat but the object is accelerating, then I can use the first formula. Is this part correct? Now, let me state the confusing bit. If an object is accelerating, isn't that acceleration effect replaced by/mimicked by a gravitational field? $\endgroup$ – mithusengupta123 Sep 29 '20 at 5:33
  • $\begingroup$ If spacetime is flat but the object is accelerating, then I can use the first formula Yes. If an object is accelerating, isn't that acceleration effect replaced by/mimicked by a gravitational field? No. When I accelerate in my car, I don’t curve spacetime. $\endgroup$ – G. Smith Sep 29 '20 at 5:37
  • $\begingroup$ @G.Smith "When I accelerate in my car, I don’t curve spacetime." This bugs me. If you are not allowed to look outside, how will you know whether you're really accelerating or there is a horizontal gravitational field that pushes me against my seat? I mean, I can vaguely guess that you must be right but I cannot fully convince myself. $\endgroup$ – mithusengupta123 Sep 29 '20 at 7:05

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