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 $T^{ab}$: $$\nabla_a (K_bT^{ab}) =(\nabla_a K_b) T^{ab} + K_b \nabla_aT^{ab}= \frac{1}{2}(\nabla_a K_b) T^{ab} + \frac{1}{2}(\nabla_a K_b) T^{ba} +0$$ $$= \frac{1}{2}(\nabla_a K_b + \nabla_bK_a) T^{ab} = 0\:.$$ In case of an expanding universe there is no timelike Killing vector, but there is a conformal timelike Killing vector $K = \partial_\tau$ where $\tau$ conformal time. Conformal Killing equation reads $$\nabla_a K_b + \nabla_bK_a = \phi g_{ab}\:.$$ It gives a conservation law for systems with traceless stress energy tensor: $g_{ab}T^{ab}=0$, like the EM field with a procedure very close to that exploited above.
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 $\tau$.
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