# 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. Shantanu says:

Sean any comments about 1002.3966? Would be interested to hear what you and others think
Thanks

2. Gavin Polhemus says:

KiwiDamien (25): P is the pressure at the boundary, which is always the same due to Newton’s Second Law. (In this case, the force per tiny area that the system exerts on its surroundings is equal and opposite to the force per tiny area that the surroundings exert on the the system.) In the case or a gas expanding into a vacuum, the pressure at the boundary is zero, so no work.

Although it works nicely for the expanding-into-a-vacuum example, it doesn’t make sense to say that P is the pressure of the environment in general, because the distinction between the system and the environment is a matter of choice.

Take, for example, a box of volume 2V divided into two equal parts by a removable partition. If one half is filled with gas at pressure P and the other half is vacuum, we can use your method. However, what if there is gas on both sides, which different pressures P1 > P2. Now we remove the partition. Gas 1 expands against an environmental pressure P2, so you would calculate that it is initially doing work, dW1 = -P2 dV. Meanwhile, gas 2 is being compressed by an environmental pressure P1, and therefore receiving work dW2 = -P1 dV. So work being done on gas 2 is greater than the work being done by gas 1! That is not right.

The correct way to deal with this situation is to look at the box as a whole The whole box doesn’t change volume at all. So no matter what sort of horrible, non-quasi-static event happens inside the box, the total work must be zero. In the case of a single gas that means its energy stays constant. In the case of the two different pressures, the gas will find a new pressure with the same total energy as the two gasses had before the partition was removed.

3. kiwidamien says:

Hi Gavin (27)

Thanks for the two box example — you are right and I had not carefully considered that before. I suppose the real moral of the story should be that I can only equate (nonparitial)W = -P dV for quasi-static processes.

I disagree that the pressure P at the boundary must be the same in both cases. If we stipulate everything is quasistatic then forces must indeed (quasi-)balance to maintain (quasi-)staticity, but in both the free expansion and mixed expansion cases horribly far from equilibruim there is a pressure differential at the boundary that drives the change. In the free expansion case there is a gas of pressure P released onto a P=0 vacuum, in the two gas case there is an interface with P1 meeting P2 at least initially.

So I think the objection that I have to using this explanation still stands — namely that I should not be considering my universe as embedded in a medium of pressure P that is equal to the current FRW pressure.

4. Gavin Polhemus says:

Kiwidamien: Probably we should only use E = -P dV on finite regions, not the whole universe. Then we are talking about the change in energy of that region, and the change in its volume. That removes any need to worry about something beyond the universe. The pressure inside the region is the FRW pressure, and outside is just more of the universe at the FRW pressure. Everybody’s happy.

5. “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.”

I think this is because most people don’t think about this, whereas dark energy (we really should get back to Sean’s “smooth tension” as a much better term) is something which is in the news now. Also, when photons lose energy in the expanding universe, many assume (incorrectly) that this is because they do work in the expansion. (The universe is not a steam engine!)

As always, my suggestion to anyone who is unclear on the basis ideas behind cosmology should consult Edward Harrison’s wonderful textbook. He spends a chapter on this, a chapter on why the cosmological redshift is not a Doppler shift, a chapter on horizons etc. One of my favourite books of all time, not just one of my favourite cosmology books.

6. lun says:

I thought that a Hamiltonian formulation of General relativity has been formulated, where energy integrated over a boundary surface is conserved. How does the cosmological constant look in that picture?

7. Iolaum says:

A Hamiltonian formulation of GR has been made but it involves constrained dynamics.

As long as you are upon the equations of motion the Hamiltonian equals zero because of the existence of constraints. (The Hamiltonian is weakly zero.)

8. Jason R says:

Sorry, but i’m not a physicist. Is this article on wikipedia regarding “Conservation of Energy” correct or incorrect, or partially correct?

http://en.wikipedia.org/wiki/Conservation_of_energy

9. Gavin Polhemus says:

Jason R (33), The short answer: Wikipedia is wrong. However, as Sean said, some people assign negative gravitational energy to the space, which balances the positive energy of everything else. In that case energy is conserved, we just don’t have a good local definition of energy any more, which makes me wonder what the point is.

However, the first law of thermodynamics, dE = dW + dQ, is still true and doesn’t require messing with gravitational energy.

10. Sean says:

Shantanu– I don’t have any especially deep thoughts about this paper. I agree that the idea of a cosmological constant is perfectly respectable, and is very likely to be the right explanation for why the universe is accelerating. I disagree that the coincidence problem and the small value of the vacuum energy aren’t important problems.

11. Aaron Sheldon says:

You need to be more careful with the partitioned box example (27), you must account for the change in entropy when removing the partition. The entropy is higher in the box without the partition, which allows for the extraction of heat from the reservoir, this is the principle that heat pumps work on. In the idealized binary partition case the entropy changes on the order of (N ln2).

12. P says:

Physics can only make solid statements about aspects of physical reality which are directly accessible to experiments. Cosmology deals with concepts and distances far removed from anything we can experimentally test which puts it on a very shaky ground.

The assumption that our simple theories which hold on our tiny little planet at this moment in time also hold in the whole expanse of the Universe at all times is well… a bit naive.

Yes, it was worth a try, but now that we know it forces us to accept dark matter, dark energy, naked singularities, accelerating expansion of space, inflation and other such exotic concepts completely inaccessible to experimental verification an intellectually honest person is forced to conclude that the assumption mentioned above is no longer justified and that despite many grandiose claims we still know very little about the Universe.

13. Aaron Sheldon says:

(37) To the extent that those distances have been tested, such as in the angular power spectrum of the microwave background, redshifts of supernovas and galaxies, the FLWR metric is in excellent agreement with observation. Which is remarkable given that the family of FLWR metrics are essentially parameterized by only a scalar pressure and scalar density, and the order of data collected is in the trillions of bytes. Other branches of science could only hope to be so lucky!

By comparison the Standard Model has 19 parameters that need to be fit, but many orders of magnitude more data collected.

Perhaps another way to popularize the problem is that stress-energy is a generalization of 4-momentum, which is a generalization of kinetic energy. Because the stress-energy is divergence-less, it has a type of conservation through orthogonally intersecting geodesics.

14. Joe says:

I’m now at the point of wondering why Noether’s theorem is so often associated with time-translation invariance, if GR proved that we don’t have time-translation invariance before Noether’s theorem was published.

Did it just take a long time to tease out this implication of GR? Or is time-translation invariance a good enough approximation that its treated as a no-harm, no-foul issue?

15. Doug Watts says:

Thanks Sean. I was totally lost in this post until you said this about Hubble red-shifting:

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.

We’ve “known” about photons’ loss of energy due to the expansion of space-time since Edwin Hubble, but it’s funny how when the concept is expressed in a different format we can miss the connection.

16. Doug Watts says:

For anyone:

when photons go through a gravitational lens, do they lose energy in addition to what they lose from a Hubble red-shift?

thx.

17. Cusp says:

> Cosmology deals with concepts and distances far removed from anything we can experimentally test which puts it on a very shaky ground.

Is this suggesting that cosmology is not an experimental science?

So the prediction, and then detection of the wobbles in the CMB was not experimental?

So the prediction that the time scale of distant supernovae would be time dilated was not a prediction?

So the prediction that the temperature of the CMB was hotter in the past was not experimentally verified?

Clearly many of those who criticize cosmology have no real clue about what happens on a day to day basis. We don’t just sit in the pub and make this stuff up! (well, except for the guys who publish in Phys Rev D).

18. Cusp says:

>when photons go through a gravitational lens, do they lose energy in addition to what they lose from a Hubble red-shift?

Nope – as generally the potential they fall into is as deep as the one they climb out of.

19. Sean says:

There is no net redshift if the lens they fall into is static (unevolving). If the lens itself is changing in time, the potential they fall into might not be the potential they climb out of, and there can be a redshift — that’s known as the “integrated Sachs-Wolfe effect.”

20. P says:

> To the extent that those distances have been tested, such as in the angular power spectrum of the microwave background, redshifts of supernovas and galaxies, the FLWR metric is in excellent agreement with observation.

Galaxy rotation curves are in stark disagreement with GR. In fact we cannot explain a whooping 95% of mass-energy content of the Universe and yet you claim there is “excellent agreement with observation” and that “other branches of science could only hope to be so lucky”?

>> Cosmology deals with concepts and distances far removed from anything we can experimentally test which puts it on a very shaky ground.

>Is this suggesting that cosmology is not an experimental science?

No, it’s saying that cosmology is on a very shaky ground – it’s based on limited experiments and breathtaking generalizations.

For example we cannot conclude that redshift of distant galaxies is due to expansion of space until we rule out the possibility that it’s an intrinsic property of all electromagnetic radiation. Unfortunately distances involved make such experiments technically impossible so the issue cannot be decided using scientific method.

21. Cusp says:

> For example we cannot conclude that redshift of distant galaxies is due to expansion of space until we rule out the possibility that it’s an intrinsic property of all electromagnetic radiation.

Every contender has been ruled out.

22. Aaron Sheldon says:

(45) Actually…the mathematics of GR itself is a proven truth, every smooth manifold has a well-defined stress energy as it is the only non-trivial contraction of the Bianchi identities, or conversely the Noether current on the local symmetries of the smooth manifold. As such it can never be proven or disproved, it is simply a mathematical truth. What is tested by observation is our understanding of the mechanisms that generate the stress-energy, or conversely the mechanisms that generate a metric.

So for dark matter in rotating galaxies, what we are told by experiment is that we do not know all the mechanisms that determine the stress-energy, but we can infer components of the stress-energy from the geodesics (rotations) within the galaxies.

As for the FLWR, it is quite literally the simplest possible stress-energy that conforms to the broad observations we have of the universe, such as angular isotropy. So the fact that it agrees with the data with so few free parameters is stunning.

A final example of a well confirmed metric is the Kerr metric which, as anyone who as used centimeter scale GPS can confirm, is in excellent agreement with observation of the curvature around a rotating body.

One can think of GR more as a really powerful tool, where the specific metrics or stress-energies are the theories being tested. The only way to dispense with GR is to dispense with the assumption that space-time can be represented by a smooth manifold, but testing such a hypothesis would require experiments that are beyond the scale of what can be accomplished in this century. So at the scale of energies and distances we have measured to, space-time is accurately described by a smooth manifold.

23. Brett says:

@Aaron Sheldon- Terrestrial gravitation measurements with e.g. GPS are not nearly sensitive enough to measure properties of spacetime that depend on the Earth’s rotation. They key such property is the dragging of inertial frames or the Lense-Thirring effect. Even a dedicated, space-based experiment to measure this phenomenon was not really successful; gravitomagnetic effects have not been unambiguously observed. (There are arguments that the same effect responsible for the Lense-Thirring precession has been observed already, in more mundane GR tests such as lunar laser ranging. However, whether or not this is actually the same turns out to be gauge dependent.)

24. Charon says:

Part of the problem is that even many physicists don’t know about this aspect of GR. Conservation of energy is hammered into people’s heads from first-year undergrad through grad school, and while probably almost every grad school requires quantum mechanics, many of them don’t even offer GR (and many of those that do make it optional).

Observational astronomers are intimately familiar with redshift, of course, but even they don’t think much about violation of conservation of energy, because when they’re doing flux calculations, they’re either doing them locally (where a static background is a very good approximation) or they’re using the luminosity distance, which by construction hides the non-conservation of energy.

25. P says:

> Every contender has been ruled out.

I am well aware of that but that is not a scientific argument and it’s not convincing in any case.
History provides countless examples of experiments discovering completely novel and unexpected phenomena which were not predicted by currently popular theories.

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. For all we know it may simply be a fact with no explanation at all, just like existence of electromagnetic radiation has no explanation at all.