# Null-geodesic proper velocity from initial conditions

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!

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