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I think your metric is wrong. The $dtd{\phi}$ component in particular. I just tried it with the metric from my notes and it's quite simple. Just find $\frac{d\vec{R}}{d\lambda}$ and use the metric to find its magnitude


For a ZAMO it is $$ \frac{{\rm d} t}{{\rm d} \tau} = \sqrt{g^{t t}}$$ for an object moving with local velocity $v$ relative to the ZAMO it is $$ \frac{{\rm d} t}{{\rm d} \bar\tau} = \frac{\sqrt{g^{t t}}}{\sqrt{1-v^2/c^2}}$$ and for an observer stationary with respect to the fixed stars it is $$ \frac{{\rm d} t}{{\rm d} \tilde\tau} = \frac{1}{\sqrt{g_{t ...


Yes, this is correct for an observer at rest in Boyer-Lindquist coordinates. The same reasoning applies to any metric. But it isn’t all that interesting because in general observers are more likely to be moving (e.g., orbiting).

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