I have this metric: ds2=−dt2+etdx2
and I want to find the equation of motion (of x). for that i thought I have two options:
using E.L. with the Lagrangian: L=−˙t2+et˙x2.
using the fact that for a photon ds2=0 to get: 0=−dt2+etdx2 and then: dt=±et/2dx.
The problem is that (1) gives me x=ae−t+b and (2) gives me x=ae−t/2+b.
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