I'm a little confused about the proper way to find these null geodesics. From the line element,(adsbygoogle = window.adsbygoogle || []).push({});

$$c^2 d{\tau}^2=\left(1-\frac{r_s}{r}\right) c^2 dt^2-\left(1-\frac{r_s}{r}\right)^{-1}dr^2-r^2(d{\theta}^2+\sin^2\theta d\phi^2),$$

I think we can set ##d\tau##, ##d\theta## and ##d\phi## to ##0##, and find ##\frac{dr}{dt}##.

However, I wish I could find a reference for this, but I've also seen something like this done:

$$g_{\mu\nu}x^{\mu}x^{\nu}=0,$$

where ##g_{\mu\nu}## is the metric and I think ##x^{\mu}## is a position vector, maybe? Solving that equation should give the nulls. Setting the ##\theta## and ##\phi## components of ##x^{\mu}## to ##0## should give the radial nulls, then, I assume.

Are either of these correct? They seem to give two different answers, in this case. I'm ultimately trying to find a time coordinate delayed by the speed of the light, and the first method seems to give something in terms of a logarithm for a retarded time coordinate, while the second leaves me with me with something like a half integer power of ##r## over a square root, but I might just be blowing one or both. How do I find (specifically the radial, exterior, outgoing) null geodesics for Schwarzschild and for a general metric in general relativity?

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# A Radial, exterior, outgoing, null geodesics in Schwarzschild

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