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Computing timelike geodesics in the Schwarzschild geometry is pretty straightforward using conserved quantities. You can treat the problem as a variational problem with an effective Lagrangian of

##\mathcal{L} = \frac{1}{2} (Q \frac{dt}{d\tau}^2 - \frac{1}{Q} \frac{dr}{d\tau}^2 - r^2 (\frac{d\theta}{d\tau}^2 + sin^2(\theta) \frac{d\phi}{d\tau}^2))##

where ##Q = 1 - \frac{2GM}{r}##

This "lagrangian" leads to the following conserved quantities:

In terms of these conserved quantities, the geodesics are completely determined by ##\frac{dr}{d\tau}##, which satisfies the one-D equation:

##\frac{1}{Q} (K^2 - \frac{dr}{dt}^2 ) - \frac{L^2}{r^2} = 1##

My question is: How are things changed if we are computing a null geodesic, instead of a timelike geodesic? The biggest change is that you can't use proper time as the parameter (since it's identically zero for null geodesics, by definition). If you replace ##\tau## by a different parameter, ##\lambda##, I'm assuming that it's still true that there is something like angular momentum that is conserved, but I'm not sure about the first equation, which is about the conservation of the ##t## component of momentum.

##\mathcal{L} = \frac{1}{2} (Q \frac{dt}{d\tau}^2 - \frac{1}{Q} \frac{dr}{d\tau}^2 - r^2 (\frac{d\theta}{d\tau}^2 + sin^2(\theta) \frac{d\phi}{d\tau}^2))##

where ##Q = 1 - \frac{2GM}{r}##

This "lagrangian" leads to the following conserved quantities:

- ##K = Q \frac{dt}{d\tau}##
- ##L = r^2 \frac{d\phi}{dt}## (You can choose ##\theta## and ##\phi## so that ##\theta = \frac{\pi}{2}##, so all the radial motion is due to changing of ##\phi##
- ##\mathcal{L}## itself, which is always equal to 1/2.

In terms of these conserved quantities, the geodesics are completely determined by ##\frac{dr}{d\tau}##, which satisfies the one-D equation:

##\frac{1}{Q} (K^2 - \frac{dr}{dt}^2 ) - \frac{L^2}{r^2} = 1##

My question is: How are things changed if we are computing a null geodesic, instead of a timelike geodesic? The biggest change is that you can't use proper time as the parameter (since it's identically zero for null geodesics, by definition). If you replace ##\tau## by a different parameter, ##\lambda##, I'm assuming that it's still true that there is something like angular momentum that is conserved, but I'm not sure about the first equation, which is about the conservation of the ##t## component of momentum.

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