# Have red shifted photons lost energy and where did it go?

I think the title says it. Did expansion of the universe steal the energy somehow?

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– Qmechanic♦ Feb 28 '12 at 23:05
– Michael Luciuk Feb 29 '12 at 13:00

## 2 Answers

Energy isn't a nice concept in GR, so all I'm giving is an intuitive way of looking at it.

For gravitationally redshifted stuff: A photon has energy, thus it gravitates (as energy can gravitate analogous to mass from $E=mc^2$), thus it has some (negative) gravitational potential energy when on the surface of a planet. If it's emitted, its GPE eventually becomes 0. So, this increase in GPE had to come from somewhere: the photon's redshift gave the energy. It's pretty much the same thing that happens when you throw a ball up. It loses kinetic energy (slows down).

The GPE in relativity is basically related to the energy stored in spacetime curvature; in a complicated way that I don't know.

For a normally redshifted photon from a moving body: Energy need not be conserved if you swith frames. Energy is different from each reference frame.

See the answers to the question provided by Qmechanic above as well. Over there, they're talking about the entire universe, though, which leads to additional issues.

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You know, it's like why's it expanding? No one say's oh yes, of course, why wouldn't it be. – sonardude Feb 29 '12 at 2:28
You can write GR in so many ways that there's no rightish way to look at it. – sonardude Feb 29 '12 at 2:32
@sonardude That's why every way of looking at it is wrongish :P. – Manishearth♦ Feb 29 '12 at 2:40

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This is not wrongish, but correct. Still, many people will tell you that it is wrong, even though it is correct, because they are uncomfortable with the energy in GR, because it is a pseudotensor, which is only globally interesting, and only uncontroversial for asymptotically flat spaces. But whatever. +1. – Ron Maimon Feb 29 '12 at 4:58
@RonMaimon Didn't know that, thanks . You've managed to to point out that I'm wrong when I think I'm right, and right when I think I'm wrong. Wierd. :P – Manishearth♦ Feb 29 '12 at 12:58
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The short answer is "yes". The energy lost from the photons is taken up by the energy in the gravitational field. Of course energy is a relative concept but if you take the simplest case of a spatially flat homogeneous cosmology with no cosmological constant then the equation for energy in an expanding volume $V(t) = a(t)^3$ is

$E = Mc^2 + \frac{P}{a} - \frac{3a}{\kappa} (\frac{da}{dt})^2 = 0$

$M$ is the fixed mass of cold matter in the volume, $\frac{P}{a}$ is the decreasing radiation energy in the volume with $P$ constant, and the third term is the gravitational energy in the volume which is negative. The rate of expansion $\frac{da}{dt}$ will evolve in such a way that the (negative) gravitational energy increases to keep the total constant and zero.

For a more general discussion of energy conservation in general relativity see my paper http://vixra.org/abs/1305.0034

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