# Energy Conservation and an Expanding Universe

So, what’s the deal with this one? startswithabang.com reader Scott Stuart asks the following question:

I was reading “The First Three Minutes” last night and came across an
author states that as the universe expands, the number of photons
running around (in the CMB, for example) is unchanged, but their
wavelengths get stretched. The energy in a photon is, of course,
inversely proportional to its wavelength, so the energy content of a
photon decreases as its wavelength increases. That seems to mean that
the total energy content of the photons decreases due to the expansion
of space. Now the energy density clearly decreases as the volume
increases, but this argument says that the total energy decreases as
well. Does that mean that the expansion of space is not conserving
energy? Or is the energy “going” somewhere?

Remember the law of conservation of energy? It states that energy can neither be created nor destroyed, only transformed from one form into another. Now Scott asks how this works in an expanding Universe, because quite clearly the rules change!

His point is that if I have a bunch of photons in my Universe, and I stretch my Universe, the photons will change wavelength to accomodate the change in the size of the box. So if I double the size of the Universe, the energy in photons in the Universe halves.

What about matter? Both normal matter and dark matter don’t change their mass as the Universe expands, so that seems okay. But what about the energy in the gravitational field? After all, there is such a thing as binding energy, and as I increase the distance between objects, the gravitational binding energy (which is a form of negative energy) goes up (or closer to zero). Unfortunately, we don’t have an exact definition of gravitational field energy, so that gets sticky.

Now let’s throw dark energy in, and make the conundrum worse. All of the evidence for dark energy (currently) points towards it having a constant energy density. This means that as the Universe expands, and we wind up with more space, we are constantly creating more and more energy. So what’s the deal? Is energy conserved, or isn’t it?

If we define energy like we’re used to defining it, that is, locally, the answer is no. If we take the energy density of the Universe and multiply it by the “volume” of the Universe, we get a number for total energy that changes over time. Just after the big bang, most of the energy density was in radiation, which decreases faster than the volume increases due to the stretching talked about above. So at the start, it looks like the total energy of the Universe is dropping. Then the Universe becomes matter dominated, and then energy appears to be conserved, since the product of the matter’s energy density times the volume stays constant. But then the matter density drops below the dark energy density, and now, the Universe is dominated by dark energy. The product of the dark energy density (which is constant) times the volume (which is increasing) is increasing! So it looks like the total energy of the Universe first decreases, then becomes fairly constant, and then increases again.

How is this possible? The problem, as I’ve already alluded to, is that energy is only defined locally. That means that we have no idea how to define something like “the energy of spacetime” or “the energy of the Universe’s expansion.” Without that, all we can do is state the rules for how energies and energy densities change as the Universe expands and ages.

Maybe you don’t like my answer to this question. In that case, you can try Sean Carroll’s answer, or read Steve Carlip’s answer (the third one down). The big problem is that we don’t know how to define gravitational energy on cosmological scales. Clearly, there’s a lot of it! Maybe one interesting thing to do would be to define it in the one unique way that would conserve total energy, and to learn what that is? Then, perhaps, we can test it?

Thanks to Scott for a very tough, but very good question! You have one? Send it in!