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I have simulated the geodesics of massive particles in the Kerr-Newman metric (though I don't think the metric matters) using Hamiltonian mechanics and I am now attempting to simulate null geodesics. However, I have a reached a problem in my simulation.

I wish to calculate $\frac{dt}{d\lambda}$ and thus the proper velocity from some initial coordinate velocity $v^i$ or $\frac{dx^i}{dt}$ measured by some observer at infinity. I know that null geodesics satisfy: $0 = g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}$ and that the length of $v^i$ is $c$ and therefore constant, but I don't know how to find $\frac{dt}{d\lambda}$ so that $\frac{dx^{\mu}}{d\lambda}$ has zero length. For the massive particles, I set $\lambda = \tau$ and used: $\frac{dt}{d\tau} = \frac{\sqrt{|g^{tt}|}}{\sqrt{1-\beta^2}}$ and therefore chain rule to obtain: $\frac{dx^{\mu}}{d\tau} = \frac{dx^{\mu}}{dt}\frac{dt}{d\tau}$ so its length would be $-c^2$, but for massless particles I don't know what the analogue is to use.

I understand that I could start directly using the proper velocity but I would prefer to calculate it starting from coordinate velocity. Any help would be appreciated!

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    $\begingroup$ You seem to have asked the same question twice. $\endgroup$
    – S.G
    Nov 25, 2023 at 22:47
  • $\begingroup$ I genuinely don't know how that happened lol. $\endgroup$ Nov 25, 2023 at 23:26

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