How is it possible to define the energy for a massless particle in a Freedman-Robertson-Walker metric if the Lagrangian is not invariant under time-translation? I have postulated the simplest Lagrangian for a massless particle (a Lagrange multiplier times the constraint) and it is obviously not time-invariant. Not only: if I work since at the beginning using the proper time as a evolution parameter, the dynamics in the Lagrange parameter and the spatial components is physically inconsistent, giving meaningless results like a scale factor equal to the Lagrange multiplier. Is there something wrong in my assumption for the Lagrangian?
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