# Energy conservation on expanding universe [duplicate]

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?

• In GR in a curved spacetime energy is not conserved. Recall that energy conservation is, by Noether's theorem, a consequence of time translation invariance of Minkowski space. Just look at inflation to see an example where naive energy conservation is badly violated.
– user178876
Jun 6 '18 at 14:05
• Jun 6 '18 at 16:54

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 $\tau$.

• The problem is that this sort of energy cannot be added to the standard one associated to massive fields Huh? Not sure what you mean by this. IMO this answer misses the point, which is addressed in Lubos Motl's answer to the question that this one duplicates.
– user4552
Jun 7 '18 at 0:01

Let us take the simple case, we see the iron spectrum of a star shifted, and we can use energy conservation to assign a velocity to the star that would assure the Doppler shift of the spectrum.

The expansion of space was deduced because a redshift was measured that can only be interpreted as "every cluster of galaxies is moving away from every other cluster of galaxies ". This led to an image of an "explosion", i.e. the original Big Bang model. In an explosion energy conservation comes considering the whole system, original energy transferred to parts. The complication comes from General Relativity which does not have energy conservation as part of its structure. It is only in flat spaces where Lorenz transformations can be effectively applied , as in the case of a receding star and the spectrum of iron, that one can talk of conservation of energy.

So,imo, qualitatively energy in all its forms comes from the original Big Bang, but one cannot write down general conservation of energy equations, they have to be compatible with general relativity.

• Actually it is possible to speak of energy conservation in the presence of a timelike Killing vector, since the contraction of it with the stress energy tensor is a conserved current from Killing equation...In case of an expanding universe there is no timelike Killing vector, but there is a conformal timelike Killing vector. It gives a conservation law for systems with traceless stress energy tensor, like EM field. Jun 6 '18 at 15:08
• @ValterMoretti thanks for the correction . I do not know enough to develop it in my answer. If you will not write an answer maybe I will copy it with your name in the answer above, for completion. Jun 6 '18 at 15:12
• Ok, I will write my own answer. Jun 6 '18 at 15:14

while photons appear red-shifted for a remote observer who is receding away due to the expansion of the universe, they still retain the same wavelength and energy relative to the frame they originate from, thus no energy has been lost

Redshift happens when the wavelength of the photon is increased, or shifted to to the red end of the spectrum. Energy is never lost but transferred. So the answer to your question is NO - No energy is lost.

• "transferred" to where? Jun 6 '18 at 14:13
• Sorry i meant transformed not transferred, as in "law of conservation of energy" Jun 6 '18 at 14:25