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Consider an arbitrary non-trivial metric $g_{ij}$ - like the Schwarzschild metric. Now, consider two observers $A$ and $B$, staying at fixed radii $R_A$ and $R_B$, respectively, with $R_A > R_B$. At the coordinate time $t_A^0$, observer $A$ starts to emit some light signal. Calculate the coordinate time $t_B^0$ at which the observer $B$ will start detecting this signal.

How does one solve a problem of this type? My first impulse would be to calculate the integral $$\int ds= \int \sqrt{g_{ij} \frac{\partial x^i}{\partial \tau} \frac{\partial x^j}{\partial \tau}} \mathrm{d}\tau.$$ However, that does not work for lightlike curves.

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