Since an accelerated frame is at rest in some inertial frame (MCRF) only at at a particular instance, I clearly have to use integration to calculate the total proper time elapsed for the accelerated frame. If I take an infinitesimally small spacetime interval along an arbitrary timelike path, then ##dτ^2=dt^2-dx^2-dy^2-dz^2=(1-\frac {dx^2} {dt^2} -\frac {dy^2} {dt^2} - \frac {dz^2} {dt^2})## ##dt^2##.(adsbygoogle = window.adsbygoogle || []).push({});

Now my main question is how to parametrize this is terms of another variable ##θ##. If I apply the chain rule on the equation above, then

##dτ = \sqrt{\frac {dt^2-dx^2-dy^2-dz^2}{dθ^2}}## ##dθ## (note: I'mnotusing polar coordinates here). I'm a little unsure here: can I write the numerator as ##-η_{μν} x_μ x_ν##?

If yes, then does the equation become $$τ = \int \sqrt{-η_{μν} \frac{dx_μ dx_ν}{dθ^2}} dθ$$ ?

EDIT: I'd accidentally written the numerator in the expression as $$x_μ x_ν$$ and forgotten to add the ##d## part.

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# Time elapsed along an accelerated world line

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