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Homework Statement
Homework Equations
There are 5 equations we can use.
We have the fact that Lagrangian is a constant for an affinely parameterised geodesic 0 in this case for a light ray : ##L=0##
And then the EulerLagrange equation for each of the 4 variables.
The Attempt at a Solution
The worked solution proceeds by letting ## \dot{y}=0## , where a dot denotes the derivative wrt the affine parameter ##s##.
It uses ##L=0## and then eliminates ##\dot{t}## and ##\dot{x}## via the constants of motion found from Noether's theorem/Euler Lagrange equations.
My question:
How do we know that we have the freedom to set one of the variables ## \dot{y}=0##. (I see by symmetry we could have equally chose ## \dot{x}=0##)..?
(I have worked through without setting ## \dot{y}=0## and then by the symmetry of x and y, using Noether's theorem/Euler  Lagrange we redefine another constant = C_1 + C_2 where C_1 and C_2 are the different constants associated with x/y respectively, however, from just looking at the question, before I start my working out, how do I know I have this freedom?) Thanks
Why do we not have the freedom to set both ## \dot{y}=0## and ## \dot{x}=0##?
(I see that we have 5 equations and 4 variables, does this have something to do with why have freedom to set one of ##\dot{x}## / ##\dot{y}## equal to zero?)
Many thanks in advance
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