Due to the expansion of the universe, the photons emitted by the stars suffer redshift, Its mean that the energy is lowered a little bit. Does this mean that the energy is lost? Does the expansion of the universe violate some conservation principles according to Noether's theorem?
Answer
Actually it is possible to speak of energy conservation in curved spacetime in the presence of a timelike Killing vector K, since the contraction of it with the stress energy tensor is a conserved current from Killing equation and symmetry of Tab: ∇a(KbTab)=(∇aKb)Tab+Kb∇aTab=12(∇aKb)Tab+12(∇aKb)Tba+0
The problem is that this sort of energy cannot be added to the standard one associated to massive fields, so a common conservation law (EM field + matter) does not exist, though EM waves conserve their energy if referring to the conformal time τ.
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