Getting Nothing for Something (and Vice Versa)

Energy isn’t conserved. It can be — and is — created and destroyed. Your high school physics teacher lied to you. Or, more likely, your high school physics teacher was mistaken. And your college physics professor was probably mistaken too.

Just to be clear: this isn’t a trick or a technicality or anything like that,1 and I’m definitely not talking about E = mc2. (Matter’s just another form of energy.) Energy is actually not conserved in the universe as a whole, in a real and measurable way. In fact, we have measured this, albeit somewhat indirectly. And this doesn’t come out of string theory or some other theory that we’re not sure of yet; it’s a result of general relativity, which has been around for nearly 100 years.

Don’t worry, I can hear your skepticism from here. I’ll explain. (And yeah, if you’ve got a degree in physics, I know you’re screaming at me right now. Just be patient and I’ll get there.) I’m going to tell you a story about a remarkable woman named Emmy Noether, and how she taught us why energy is conserved — and, in turn, why we shouldn’t be surprised that it really isn’t.

Noether was one of the finest mathematical minds of her generation — but women in her time were generally considered to have inferior mental capacities. At the age of 18, in 1900, she became only one of two women enrolled at the University of Erlangen. At the age of 29, she completed her Ph.D. in mathematics, and took a job as a lecturer in mathematics at Erlangen for seven years; but being a woman, she was not paid for her work, and lived with her parents the whole time. Finally, in 1915, she received an offer for a paying job as a mathematics professor at the University of Göttingen, which she took — but the faculty in the other departments objected to a woman being a professor. Noether ended up working for nine years without pay at Göttingen, lecturing under the names of the male faculty, supported by her family once again. What makes this story even more outrageous is that, early in her time at Göttingen, she discovered the most elegant theorem in all of modern physics.

What Noether saw in 1915 was that the mathematical tools used in modern physics lead directly to a very simple and beautiful explanation for why things like energy and momentum are conserved. Specifically, Noether’s theorem shows that conservation laws come about naturally as a result of continuous symmetries in the laws of physics.

What does that mean? First, we need to know what we mean by the word symmetry. Something has a symmetry if there’s something you can do to it — turn it, flip it, whatever — that leaves it looking the same way it did before you did that thing. So, for example, this cube has rotational symmetry: if I turn it 90 degrees, it’ll look exactly the same as it did before.2

But what does “continuous symmetry” mean? Have a look at this sphere: it’s got rotational symmetry too, but not in the same way that the cube has it. You can rotate the sphere by any amount, and it’ll look the same as it did before, whereas the cube can only be rotated in multiples of 90 degrees. So the sphere’s got continuous rotational symmetry, but the cube’s only got discrete rotational symmetry.

So, finally, how can the laws of physics have continuous symmetries? Well, for example, the laws of physics look the same no matter where you are. Newton’s second law of motion, Maxwell’s laws, the Schrödinger equation — they all look exactly the same whether you’re in Detroit or Dar-es-Salaam. So that’s a continuous symmetry of the laws of physics, and by Noether’s theorem, that symmetry leads directly to the conservation of momentum. In other words: momentum is conserved because the laws of physics don’t change from place to place.

There are tons of other symmetries and conservation laws that are associated this way. The laws of physics have rotational symmetry too, like the sphere, and that leads directly to the conservation of angular momentum. There’s a symmetry called “electromagnetic gauge symmetry,” which I’m not ever going to touch with a ten-foot-pole on this blog, but that leads directly to the conservation of electric charge.

But what about energy?

You may have guessed it already: Noether’s theorem says energy is conserved because the laws of physics don’t change as we move through time. There’s no such thing as a free lunch, and that’s because the laws of physics are the same today as they were yesterday, and they’re going to be the same tomorrow.

Except they’re not. The laws of physics are not the same from day to day, not in the strict sense that Noether’s theorem requires, because in our universe, space is expanding. The expansion of the universe breaks the temporal symmetry,3 and thus Noether’s theorem states that energy doesn’t have to be conserved in our universe as a whole.4

And, in fact, we’ve seen this. Remember those cosmic microwave background photons that I talked about a while ago? When they were first emitted, 400,000 years after the Big Bang, they were in the near-infrared part of the spectrum, much more energetic than microwaves. But then they were redshifted by the expansion of the universe — their wavelength was stretched over a thousandfold, from about a micrometer to about a millimeter. The energy of a photon is inversely proportional to its wavelength, so the CMB photons today are over a thousandfold less energetic than they were 13.8 billion years ago. What happened to that energy?

It was destroyed. Gone, poof, no more. It didn’t “leak” out of the universe. It wasn’t transferred into the expansion of the universe in any meaningful sense. It just up and vanished. It ceased to be. It was no more. It was freaking gone. And those photons are still losing energy, even now, since the universe is still expanding. The same goes for any cosmically redshifted light: light from distant quasars, light from galaxies that are “only” hundreds of millions of light-years away, and everything in between. Those photons all had their wavelengths stretched — and thus had a fraction of their energy destroyed — by the expansion of the universe itself.5

There’s also some pretty solid evidence that energy is created in our universe too. Dark energy, the mysterious energy that accounts for over 70% of the stuff in the universe, was only discovered relatively recently, but we’re fairly sure it’s real — there are many different cosmic indications that it’s out there, and the Nobel committee tends to be fairly conservative about these things too. One of the most puzzling and strange things about dark energy is that its density in the universe appears to be roughly constant even as the universe expands. In other words, despite the fact that there’s more space as time goes on, the density of dark energy doesn’t go down — which means that more of it must come into existence as the universe expands, in order to keep up with the expansion. Specifically, dark energy must be created at the rate of roughly 4 billionths of a billionth of a billionth of a watt in each cubic meter, on average, across the universe. That doesn’t sound like much — and it’s not — but it does add up: every cubic light year generates 3 billion terawatts of dark energy, about 200 million times the rate at which the entire human race currently consumes energy.6 Better still, since dark energy is causing the expansion of the universe to accelerate, the rate at which dark energy is produced is also accelerating — in fact, it’s accelerating exponentially. So not only is energy not conserved, but energy is created at an exponential rate.7

So if energy is being created in most of the cosmos all of the time, then why does it seem like energy is conserved? The short answer is that number in the last paragraph is really tiny on the sorts of time and distance scales we normally deal with. The longer answer is that general relativity basically states that energy is almost conserved if you only look at a tiny area of spacetime, like we do.8 So we don’t notice the destruction and creation of energy around us in our everyday lives, because we’re too small and we don’t live long enough. Furthermore, on the small scales where we spend our time — on a ball of rock orbiting a star inside of a galaxy, all of which are billions and billions of times more dense than most places in the universe — space is not expanding, which is why I said “on average” in the last paragraph when I was talking about the creation of dark energy. But when we widen our view, and look at large swaths of the universe over billions of years, space is certainly expanding and energy is certainly not conserved. So despite appearances, we live in a universe that creates unending energy out of nothing at all.

  1. Specifically, it’s not the quantum mechanical energy-time uncertainty relation. I’m talking about permanent, deterministic increases and decreases in energy, not something temporary and probabilistic. []
  2. In this paragraph and the next, assume that one face of the cube stays flat on a table while Iʼm spinning it around its center (or about any axis that runs through the centers of opposite faces). []
  3. If you want to get real technical: in an expanding spacetime metric, there’s no timelike Killing vector, so there’s no associated conserved quantity by Noether’s theorem. Meaning energy isn’t conserved. []
  4. Still don’t believe me? Look at any textbook on general relativity. It’s on page 120 in Sean Carroll’s book, for example, and page 348 in the new edition of Schutz’s book. []
  5. The momentum of a photon is proportional to its energy — so how can momentum be conserved if energy isn’t? The answer is that while our usual expression for momentum is not conserved, canonical momentum is conserved. In other words, “momentum” is conserved, but I do not think “momentum” means what you think it means. (Thanks go to Bernard Schutz for clearing this up for me.) []
  6. Don’t get your hopes up about solving humanity’s energy needs with this. While energy is certainly being created, to the best of my knowledge it’s still a matter of some debate if any free energy (i.e. useful energy) is being created. Furthermore, the future evolution of dark energy is definitely still an open problem in cosmology, so this free lunch may be available For a Limited Time Only!™ Finally, even if free energy is being created indefinitely, the distances involved in “mining” useful amounts of this energy would be at least tens of millions of light-years, because you’d probably need to get outside of our galaxy cluster to really take advantage of this (hence the “on average” disclaimer above). So even in a best-case scenario, it would be at least tens of millions of years before such a system could be set up. []
  7. Before you ask: no, the creation of energy, in the form of dark energy, does not balance the destruction of energy from the cosmically-redshifted photons. Not only do the rates not match up, but there’s already way more dark energy than all the energy in all the photons in the universe — thousands of times more — and that number will only get bigger as time goes on. []
  8. Specifically, the equivalence principle says that any spacetime metric is asymptotically Minkowskian — flat — as you look at smaller scales. And GR also says that the only way to have a well-defined gravitational potential energy is if the metric is asymptotically flat as you go out to infinity. So if you only care about regions small enough that “infinity” is not large enough to actually see the curvature of our spacetime — say, the Solar System over the course of a few hundred millenia — then you can approximate a gravitational potential energy very well, and energy conservation comes roaring back, more or less. []

7 thoughts on “Getting Nothing for Something (and Vice Versa)

  1. Very interesting. I started reading this post wondering “what the blazes is he talking about,” but now it makes perfect sense, what with the expansion of the universe. It just goes to show how easy a variable can be to overlook sometimes (at least for the average person). However I do have one question. You mentioned the red-shifting of light being one means through which energy is destroyed. Does it then necessarily follow that energy is created when light is blue-shifted, or is that a bad assumption?

    1. Energy is destroyed when light is redshifted due to the expansion of space. This is called cosmological redshift. (There’s a nice animation of cosmological redshift here, and a good illustration of it here.) Cosmological redshift is different from Doppler redshift, when light gets redshifted because the source of the light is moving away from you. Doppler redshifts don’t cause energy to be destroyed, but cosmological redshifts do. So Doppler blueshifts don’t cause energy to be created — but if the universe were shrinking rather than expanding, then we’d have a cosmological blueshift, and energy would indeed be created in that case. So yes, energy is created in the right kind of blueshift, but we don’t really see that kind of blueshift in nature.

  2. Greatly enjoyed this post, though I am somewhat baffled that college physics profs in other fields are also still teaching that energy is conserved given what you’ve laid out here. I suspect this is a matter of it not directly applying to physicists looking only at the relatively short-term in our own solar system? Glad to see you blogging this, and hope that physicists from other fields are reading!

    Regarding footnote 5–would dark energy even be useable if we could access it? Or is that what you mean by “a matter of some debate if any free energy (i.e. useful energy) is being created”? Are there potentially different kinds of dark energy (i.e. useful and not useful), or is it just that we don’t know much about dark energy and its usefulness or lack thereof? Forgive me if this is a naive question–it’s been several years since I’ve taken a science course!

    1. Yeah, it’s kind of amazing that this fact hasn’t really become better-known among physicists outside of cosmology. I’m hardly the only one saying it….

      Yes, that’s exactly what I mean in footnote 5. Free energy = useable energy. The traditional example of energy that’s not free is the heat energy of a rock at the same temperature as its surroundings. Heat energy is just the energy of the random motions of the molecules in the rock, so there’s definitely energy there, but it’s not energy you could use to do anything. This is related to entropy, one of my favorite ideas in all of physics, and a likely subject for future posts here. (Entropy, in turn, is related to some of the ideas I talked about in this post from July.) So it’s not clear that the increase in energy in the universe due to the creation of dark energy actually leads to any energy we can use to do things. It’s less about the kind of dark energy that exists — there’s debate about what kind of thing dark energy is, but that’s not really relevant to this question — and more about how entropy works in the universe as a whole (which is an area of active research, though it’s not what I do). Honestly, the question of free energy in the universe is something that I’m still learning about; it’s possible that this question has been settled and I just don’t know it yet.

  3. Haha, yeah, Andrew linked me to that article earlier when I was like, wait, what? Is this common (astro)physicist knowledge?

    And thanks for the reply! I hadn’t read very much about dark energy before (I mean, I vaguely knew it existed, but that was about it!), and I think that clears it up a little for me.

    I’ll keep reading your blog! You explain things very well for people who only have the vaguest notion about this stuff!

  4. Fascinating, and kind of makes me smack myself saying “why didn’t I ever think about that?” At least the part about the cosmological redshift. The part about dark energy creation is completely nonintuitive though. Can you explain why dark energy creation is causing the universe to expand at an exponentially increasing rate?

    1. That’s an excellent question — but sadly, there isn’t an answer that’s both simple and satisfying. I could say dark energy is causing the expansion of the universe to accelerate because the density of dark energy doesn’t change appreciably as the universe expands, but that’s just another way of saying that dark energy is created as the universe expands. Ultimately, this is just something that falls out of general relativity. Specifically, it’s a direct consequence of the FRLW metric, which I talked about in this post. If you assume that the universe looks the same everywhere on large scales, then general relativity says that the universe must be expanding or contracting, and the rate at which that happens is determined by the overall energy density of the universe. (Again, matter’s just another kind of energy for our purposes, via E = mc2.) If the overall density of the universe is constant and positive, and if the overall geometry of the universe is flat (as we believe it is), then the universe must expand at an exponential rate. People will sometimes talk about a “negative pressure” associated with dark energy, and that’s true as far as it goes, but it doesn’t really help to understand why dark energy leads to an accelerating universe any more than what I’ve said, and maybe less. Sean Carroll talks about that (among other things) over at his excellent Dark Energy FAQ. So, long story short: if you want an answer better than “general relativity says that anything with a density constant everywhere in space and time causes a flat universe to expand at an exponentially increasing rate, because that’s how gravity works” you’ll have to actually dig into the guts of the Friedmann equations.

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