Energy Is Not Conserved

I’ve been meaning to link to this post at the arXiv blog, which is a great source of quirky and interesting new papers. In this case they are pointing to a speculative but interesting paper by Martin Perl and Holger Mueller, which suggests an experimental search for gradients in dark energy by way of atom interferometry.

But I’m unable to get past this part of the blog post:

The notion of dark energy is peculiar, even by cosmological standards.

Cosmologists have foisted the idea upon us to explain the apparent accelerating expansion of the Universe. They say that this acceleration is caused by energy that fills space at a density of 10-10 joules per cubic metre.

What’s strange about this idea is that as space expands, so too does the amount of energy. If you’ve spotted the flaw in this argument, you’re not alone. Forgetting the law of conservation of energy is no small oversight.

I like to think that, if I were not a professional cosmologist, I would still find it hard to believe that hundreds of cosmologists around the world have latched on to an idea that violates a bedrock principle of physics, simply because they “forgot” it. If the idea of dark energy were in conflict with some other much more fundamental principle, I suspect the theory would be a lot less popular.

But many people have just this reaction. It’s clear that cosmologists have not done a very good job of spreading the word about something that’s been well-understood since at least the 1920’s: energy is not conserved in general relativity. (With caveats to be explained below.)

The point is pretty simple: back when you thought energy was conserved, there was a reason why you thought that, namely time-translation invariance. A fancy way of saying “the background on which particles and forces evolve, as well as the dynamical rules governing their motions, are fixed, not changing with time.” But in general relativity that’s simply no longer true. Einstein tells us that space and time are dynamical, and in particular that they can evolve with time. When the space through which particles move is changing, the total energy of those particles is not conserved.

It’s not that all hell has broken loose; it’s just that we’re considering a more general context than was necessary under Newtonian rules. There is still a single important equation, which is indeed often called “energy-momentum conservation.” It looks like this:

\nabla_\mu T^{\mu\nu} = 0,.

The details aren’t important, but the meaning of this equation is straightforward enough: energy and momentum evolve in a precisely specified way in response to the behavior of spacetime around them. If that spacetime is standing completely still, the total energy is constant; if it’s evolving, the energy changes in a completely unambiguous way.

In the case of dark energy, that evolution is pretty simple: the density of vacuum energy in empty space is absolute constant, even as the volume of a region of space (comoving along with galaxies and other particles) grows as the universe expands. So the total energy, density times volume, goes up.

This bothers some people, but it’s nothing newfangled that has been pushed in our face by the idea of dark energy. It’s just as true for “radiation” — particles like photons that move at or near the speed of light. The thing about photons is that they redshift, losing energy as space expands. If we keep track of a certain fixed number of photons, the number stays constant while the energy per photon decreases, so the total energy decreases. A decrease in energy is just as much a “violation of energy conservation” as an increase in energy, but it doesn’t seem to bother people as much. At the end of the day it doesn’t matter how bothersome it is, of course — it’s a crystal-clear prediction of general relativity.

And one that has been experimentally verified! The success of Big Bang Nucleosynthesis depends on the fact that we understand how fast the universe was expanding in the first three minutes, which in turn depends on how fast the energy density is changing. And that energy density is almost all radiation, so the fact that energy is not conserved in an expanding universe is absolutely central to getting the predictions of primordial nucleosynthesis correct. (Some of us have even explored the very tight constraints on other possibilities.)

Having said all that, it would be irresponsible of me not to mention that plenty of experts in cosmology or GR would not put it in these terms. We all agree on the science; there are just divergent views on what words to attach to the science. In particular, a lot of folks would want to say “energy is conserved in general relativity, it’s just that you have to include the energy of the gravitational field along with the energy of matter and radiation and so on.” Which seems pretty sensible at face value.

There’s nothing incorrect about that way of thinking about it; it’s a choice that one can make or not, as long as you’re clear on what your definitions are. I personally think it’s better to forget about the so-called “energy of the gravitational field” and just admit that energy is not conserved, for two reasons.

First, unlike with ordinary matter fields, there is no such thing as the density of gravitational energy. The thing you would like to define as the energy associated with the curvature of spacetime is not uniquely defined at every point in space. So the best you can rigorously do is define the energy of the whole universe all at once, rather than talking about the energy of each separate piece. (You can sometimes talk approximately about the energy of different pieces, by imagining that they are isolated from the rest of the universe.) Even if you can define such a quantity, it’s much less useful than the notion of energy we have for matter fields.

The second reason is that the entire point of this exercise is to explain what’s going on in GR to people who aren’t familiar with the mathematical details of the theory. All of the experts agree on what’s happening; this is an issue of translation, not of physics. And in my experience, saying “there’s energy in the gravitational field, but it’s negative, so it exactly cancels the energy you think is being gained in the matter fields” does not actually increase anyone’s understanding — it just quiets them down. Whereas if you say “in general relativity spacetime can give energy to matter, or absorb it from matter, so that the total energy simply isn’t conserved,” they might be surprised but I think most people do actually gain some understanding thereby.

Energy isn’t conserved; it changes because spacetime does. See, that wasn’t so hard, was it?

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56 Responses to Energy Is Not Conserved

  1. Aaron Sheldon says:

    (48) I was actually referring to the use of GR corrections in the time coordination algorithms both in the ground devices and on the satellites, without which GPS wouldn’t be nearly as accurate.

  2. Cusp says:

    > The possibility that all electromagnetic radiation experiences intrinsic redshift on cosmic scales cannot be excluded simply because we have no idea why it should be so or how to explain it.

    There is more to expansion than just the redshift – bit only would photons have to magically lose energy, they would have to space themselves out so the surface brightness drops as (1+z)^4 – the Tolman surface brightness test.

    Yes, there *may* some magic that can do this, but currently there is no mechanism for it and not even remotely workable theory (other than random handwaving) for it. Hence, cosmology in terms of GR is it.

    I am not suggesting that we not look for it, but damning cosmology because of some unknown physics is silly – you may as well damn all of science.

  3. P says:

    >There is more to expansion than just the redshift – bit only would photons have to magically lose energy, they would have to space themselves out so the surface brightness drops as (1+z)^4 – the Tolman surface brightness test.

    Tolman test is a fiction. Check out the linked article for example, the measured value of the exponent is between 2.6 and 3.4 but the authors conclude that the difference from 4 is due to galactic evolution and locate some model which gives such an evolution. Seriously, if we have to assume how a particular galaxy evolved billions of years ago then the test is worthless. Tolman original idea was that the measurement should be 4 to begin with so the above value can be seen as a refutation of simple expansion model.

    But there are other problems with the test, to do it we have to know gravitational potential difference and relative speed to account for gravitational redshift and Doppler effect and there is no way to know them meaning no way to do the test.

    > I am not suggesting that we not look for it, but damning cosmology because of some unknown physics is silly – you may as well damn all of science.

    Certainly not all of science, only those disciplines where many experiments needed to verify theories cannot be performed for technical reasons. Most of natural sciences don’t have such problems.

    And I am not trying to “damn” cosmology, I am trying to point out that many of it’s predictions are nowhere near the level of reliability that many people expect from natural sciences. For example it is far to early to conclude with certainty that our Universe started this way or that way – for that we have to first understand more then just 5% of its mass-energy content! We also need a proper theory of matter, meaning successfully marrying QM and GR, we need to be able to correctly predict vacuum energy, we need to know where the parameters of SM come from and so on.

  4. A universe with a fixed finite mass at its origins, (necessary for the universe to develop roughly according to the predictions of the standard model) strongly implies a strict energy conservation principle. This is especially true in a self-contained, everywhere universe with a marginally closed geometry.

    The fact that open geometry GR does not have a conservation of energy principle casts doubt, I would think as to the veracity of the concept. Einstein’s grand proportion itself implies a cosmological relationship between all (or any part) of what we identify as “matter” and “energy”.

    “Space” is created by the way energy densities are observed…their relationships relative to the observer within the manifold.

  5. Reptile says:

    The concept of the graviton, and relationship to GR, has always bothered me. I always took it that gravitational waves are distortions of space time, nothing to do with the wave/particle duality of, say, a photon. Thus the exquisite use of interferometry to detect them over long baselines.

    If the graviton has any relevance/existence, how would it play out with gravitational effects (much less energy conservation) arising from an expanding universe.

    Too much to hold in my mind. But then the idea that there is may be no exchange particle creating gravitational effects has always boggled my mind also.

  6. Just a quick stab at it…there may be something amorphous and simultaneous about the entire universe and all the information/complexity it contains. This probable fact makes the study of the nature of the singular condition a key to our understanding the nature of existence.

    As Einstein and Hawking point out, all the above means that time (space-time)is an illusion.

    The very existence of time (and space) comes about by the process of observation…electromagnetic observation. The electromagnetic wave/particle duality implies we are on a Holo-Deck. We ARE real, but cosmologically, all this stuff about the expansion of the universe- all motion and change- is frame of reference “bull shit”. We observe “expansion”, spin (implying motion) and particles, but essentially (cosmologically), the universe is none of these things.

    conumdroms like “wave-particle duality” and the fact that we measure gravity as propagating at the speed of light when we know full well such a thing is impossible are natures warnings to us….they are the door to the outside of the Holo-Deck…except that in the case of the universe, existence, motion and change are what we observe and measure in a very certain way- there is really NO “outside” or “beyond”. We and everything else we observe exist -permanently- ONLY inside.