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Is energy conserved in an expanding universe?

  1. Aug 1, 2007 #1
    "General Relativity teaches us that Space is expanding." Ok, so as we discussed in a previous thread, a literal interpretation of this is nonsense. "Empty space" does not expand. Rather, objects move apart. Fine, but the universe as a whole is expanding, right? (Of course, we have little idea of the true size of our universe. But at the present moment in galactic history its volume is understood to be increasing, and not just in our observable Hubble volume. Also, the universe almost certainly is expanding at the same rate for many Hubble volumes outside our own.)

    But QM teaches us that empty space isn't truly empty. The vacuum energy is very small, but non-zero. So why doesn't the expansion of space mean increase the total amount of vacuum energy? Wouldn't this violate the law of conservation of energy? (Not that this would bother me all that much; the Big Bang itself seems to violate this law.) Or does my postulated increase in vacuum energy come at the expense of losing energy somewhere else?

    Any responses would be much appreciated! If this question was already addressed in some previous thread here on physicsforums, then by all means please let me know the appropriate link.


  2. jcsd
  3. Aug 1, 2007 #2
    I believe that I read somewhere (probably here of the FAQ for sci.physics.relativity) that it is unclear whether energy is conserved in the large scale of things. Perhaps someone will paste the URL to that web page.

    Good luck in your search. I'm also curious about this too. I've been ignoring it for a while due to other priorities but I think its time to get this down straight. Thanks for raising this interesting topic.

    Best wishes

  4. Aug 1, 2007 #3


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    I was not a part of that conversation, so maybe you could link me to it? At present, it is my belief that this is exactly what the current comological models propose. Objects are moving away from each other, but spacetime itself is litteraly expanding as well. This is how they account for objects at opposite ends of the known universe moving away from each other at speeds that appear to exceed c. The objects are moving at sub-c speeds relative to their local spacetime.
  5. Aug 1, 2007 #4


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    I thought it was pretty clear from the last thread that this was a drastic oversimplification. In particular, GR textbooks will teach no such thing. So why repeat it, especially if the only purpose of repeating it is just to shoot it down? It seems like there is a lot of straw (from this demolished strawman) strewn all over the floor here.

    As far as energy conservation goes:

    The sci.physics.faq on energy conservation in GR is at http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

    This is probably the link that Lurch was thinking of. I'll quote part of it to attempt to motivate people to read the entire link:

    So if you're interpreting energy conservation as a number, the "total amount of energy in the universe", that stays constant with time, GR does not conserve energy for the usual expanding cosmologies (FRW cosmologies).

    GR does have well-defined notions of energy for static and stationary space-times, and for asymptotically flat spacetimes. The general FRW expanding universe, however, is neither static, stationary, nor asymptotically flat.
  6. Aug 2, 2007 #5

    There are numerous well qualified writers that hold space is something that is expanding - Robertson is one that comes to mind. Unfortunately these boards are dominated by a different philosophy that is tantamount to Berkelys ideas of space as a sideless box.

    As to the issue posed by the thread - there are many cosmological models - in some of these energy is conserved - in others it is not - and in others the energy is increasing. We don't know because we do not have a model that explains beginnings and endings in a manner that can be tested
  7. Aug 2, 2007 #6


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    As far as I know, "expanding space", while indeed used by many authors, is mostly used in popularizations. Much like the "rubber sheet" analogy, the idea has definite limitations and can't really be taken too literally. That's why it's a part of popularizations, not really a part of the serious literature on GR.
  8. Aug 2, 2007 #7


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    Well, not actually, but it is a very interesting link and I thank you for incuding it. But I was referring to this...
    Robert100, to what "previous thread" are you referring? Like I said earlier, I'm pretty sure the expansion of space itself is the explanation cosmologists generally give for distant objects appearing to travel >c, but I feel like I'm arriving in the middle of a conversation. Maybe this point was dealt with in the other thread.
  9. Aug 2, 2007 #8

    According to these authors, (a) "empty space" certainly does not expand, and (b) the popular idea that space expands comes from loosely used, poorly defined terminology. Its been discussed here of late, and I haven't seen any serious disagreements with any of its main points.


    Expanding Space: the Root of all Evil?Matthew J. Francis, Luke A. Barnes, J. Berian James, Geraint F. Lewis
    (Submitted on 3 Jul 2007)

    Abstract: While it remains the staple of virtually all cosmological teaching, the concept of expanding space in explaining the increasing separation of galaxies has recently come under fire as a dangerous idea whose application leads to the development of confusion and the establishment of misconceptions. In this paper, we develop a notion of expanding space that is completely valid as a framework for the description of the evolution of the universe and whose application allows an intuitive understanding of the influence of universal expansion. We also demonstrate how arguments against the concept in general have failed thus far, as they imbue expanding space with physical properties not consistent with the expectations of general relativity.​

    All this is tangential to my main point. Since the volume of the universe increases, and since each volume constains a certain amount of vacuum energy, then what is happening to the total energy content of the universe? It would seem to be increasing. Contrary to what I was taught way back in college, I am learning here that GR is vague about conservation of energy, and that energy simply might not be conserved on a large scale. This, to me, seems bizarre.

  10. Aug 2, 2007 #9
    Sorry for being vague. Here are the recent discussions on this topic:

  11. Aug 2, 2007 #10


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    actually, i thought that the Big Bang and expansion of the universe did not violate conservation of energy as long as the rate of expansion was slowing down as it expanded (in a similar manner as the upward speed of the ball you throw up slows down as it is going up). but the problem is, sometime in the 90s they concluded that the rate of expansion of the universe is increasing. that disturbs the h*ll outa me.

    i queried about this at sci.physics.research once:


    ... and their reply (as i understand it) was that conservation of energy did not really apply with GR and cosmology. i don't get it, but i am woefully inadequate regarding GR, so that might explain it.
    Last edited: Aug 2, 2007
  12. Aug 2, 2007 #11


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    Thanks for the links. Haven't read them yet, but that abstract states that space does expand, that the authors have developed that notion in a way that is consistant with observation, and refuted the argumants against the idea. At least, that's how the abstract reads. But, I'm off now to read the article itself, and the previous threads.
    see you guys in a while!
  13. Aug 2, 2007 #12


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    OK, I read the article and yes, their position is that space does indeed expand. They even make refference to the superluminal recession rates of very distant objects as a phenominon that cannot be explained as simply "objects moving apart."
    Yeah, the acceleration of expansion was a real shocker for almost everyone, I think. But maybe I can clarify a little about the conservation question (both yours and the one in the thread title).
    Short answer; yes, energy is conserved in an expanding universe. Perhaps you've heard of the "heat death" predictions that came out right after it was discovered that the expansion is increasing. This prediction of a universe that gets colder and colder, until there isn't enough energy in any one place for anything to happen, is a direct result of conservation of energy within an expanding universe. It predicts that the amount of energy will not change (IOW; none will be created or destroyed), while the volume of the universe increases, so the same amount of energy is spread out over more volume, resulting in lower energy-density (IOW; colder temperature).
    As for conservation not applying to GR and Cosmology, they were probably referring to the frequent questions about the Oscilating Universe Model (Big Bang followed by Big Crunch, then Bang again, ad infinitum) being a violation of conservation. Any cosmologist you talk to has heard that one alot, or that the Big Bang all by itself constitutes "energy from nothing". The response to this is to point out that the law of conservation applies only within the universe. The Big Bang was an event that took place "before the universe began" (and even this is a term of convenience, since "before" cannnot have any meaning without time, which is a property of the physical universe), and the laws of this universe cannot be apllied to it.
    That make any sense at all?
  14. Aug 2, 2007 #13


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    I do not believe this is correct. GR does have several concepts of energy and its conservation, but they aren't as simple as assigning a number to the "total mass" (or total energy) of the universe which is constant.

    As mentioned in the FAQ, GR always has a local concept of the conservation of energy (in fact, this is built into the field equations). It also has some concept of global conservation, but only under special circumstances. The FRW metric is not one of those special circumstances which has a global conservation law.

    Do you have references for this statement about the conservation of energy in an expanding universe?

    Aside from the sci.physics.faq which I already posted a link to, MTW also disagrees with this

    [quote="Gravitation" pg 457]
    There is no such thing as the energy (or angular momentum, or charge) of a closed universe, according to general relativity, and this for a simple reason. To weigh something one needs a platform on which to stand to do the weighing ...

    To determine the electric charge of a body, one surrounds it by a large sphere, evaluates the electric field normal to the surface at each point on this sphere, integrates over the sphere, and applies the theorem of Gauss. But within any closed model universe with the topology of a 3-sphere, a Gaussian 2-sphere that is expanded widely enough from one point finds itself collapsing to nothingness at the antipodal point. Also collapsed to nothingness is the attempt to acquire useful information bout the "charge of the universe": the charge is trivially zero.

    This is also mentioned on pg 705

    And after giving the formula

    (volume) * (density) [itex]\approx[/itex] constant during a matter dominated era (where the pressure is zero) MTW adds:

    Last edited: Aug 2, 2007
  15. Aug 2, 2007 #14


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    Maybe it's because I'm not reading them in context, but this statement:
    and this one:
    seem to be contradictory. If there is no such thing as the mass-energy of the universe, how can it be said to be changing?

    At any rate, if the latter statement is true, "the amount of mass-energy in the universe changes from instant to instant," then doesn't that mean that there is no law of conservation? Only things that are in the universe can be said to exist, in any scientific sense. Therefore, if the amount of energy in the universe decreases, then energy that once existed has ceased to exist (been anihilated), and if the amount of energy in the universe increases, then energy which once did not exist has come into existance (been created). If energy can be both created and anihilated, what meaning can "conservation of energy" have?

    Now I'm really confused!
    Last edited: Aug 2, 2007
  16. Aug 2, 2007 #15


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    Think about a sphere of uniform density 1 meter in diameter. Perform a Lorentz boost. The sphere is now a squashed ellispsoid.

    It's volume goes down by a factor of 1/gamma due to the squishing effect.

    It's density (T00) goes up by a factor of gamma^2 due to the boost. Note that the SR/GR notion of "density" includes all forms of energy, specifically including kinetic energy, this may be different than what you're used to.

    Total energy contained in the sphere increases by a factor of gamma, as it should.

    So we can see that neither volume, nor density, nor their product (density * volume = energy) is a Lorentz invariant.

    There are similar problems with the definition of the "mass of the universe".

    To avoid getting hopelessly mired, let's try the following. Consider a series of observers, each moving with the "Hubble flow", i.e. for which the cosmic microwave background radiation appears isotropic.

    Using this special reference frame, we can define a "volume" and "density" for the universe. But unless we get this specific, we'll just get raised eyebrows if we start talking about "the" volume or "the" density of the universe. The problem is that volume and density are observer dependent quantities and we have to specify our observer.

    So, we get around this issue by specifying an observer. (Another option would be to refine our concepts to avoid coordinate dependence. This route is very fruitful, but a bit advanced, so I won't get into it).

    Our observer in this case moves with the hubble flow (uses synchronous coordinates) What's next?

    What's next is that

    volume * density (with the above definitions) is constant in a matter dominated universe, but is not constant in a radiation dominated universe (or in any universe with an appreciable amount of radiation).

    If you double the scale factor for a matter dominated universe, and increase the volume by a factor of 8, the density drops by a factor of 8, and density * volume stays constant.

    If you double the scale factor for a radiation dominated universe, the volume increases by a factor of 8, the density drops by a factor of 16. Volume * density does not stay constant. You can trace this to "redshifting" of the radiation. The product of volume * density drops.

    Density * volume^(4/3) stays constant for a radiation dominated universe. (See mtw pg 727).

    So if you have a mixture of radiation and matter in your universe, volume * density does not stay constant.
  17. Aug 2, 2007 #16


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    I decided that it would be worthwhile to get into this, but in a separate post.

    Basically, the easiest route to a global notion of energy in GR is via Noether's theorem. Noether's theorem associates energy conservation with a "time translation symmetry", and momentum conservation with a "space translation symmetry".

    Noether's theorem is based on and requires that the laws of physics be formulated in terms of an action principle. GR has such a formulation, so Noether's theorem applies to GR. In fact, it was the vexing problems of energy conservation in GR that led Hilbert to ask Emily Noether to work on the problem. For some historical background mixed with the mathematics, see for instance


    If your metric has a time translation symmetry, you're all set. This gives you one of the sorts of global energy that is defined in GR, the Komar Energy (aka Komar mass).

    If your metric doesn't have a time translation symmetry, you may still be able to get time translation symmetries "at infinity".

    This yields a couple of different notions of energy, the ADM energy (for spatial infinity), and the Bondi energy (for null infinity). Similarly, spatial translations at infinity (of whatever sort) yield a notion of conserved momentum for a system.

    Unfortunately, the standard FRW models don't have any time translation symmetry, nor do they have the asymptotically flat structure at infinity necessary to define an ADM or Bondi energy.

    In any event, when the necessary requirements are met (a time translation symmetry for Komar energy, an asymptotically flat spacetime for ADM or Bondi energy), one can get a specific energy and momentum just by writing down the metric, and any pair of metrics that are related by a simple coordinate variable substitution (more technically, by a diffeomorphism) will give the same energy and momentum. This defines the energy and momentum of a system in a manner that is independent of any particular "observer" or coordinate system, which is in the end much less confusing to talk about, though unfortunately it takes a while to master the necessary background.

    These coordinate independent approaches are the current standard way of approaching energy in GR, and they do not yield any notion of "total energy" for a FRW space-time.

    The approaches I outlined earlier are somewhat interesting, but because they rely on "special" coordinates they aren't very general or ultimately very convincing. So most people (including for example MTW) don't call the number that you get out of this process "the energy" or "the mass" of the universe, because they rely on singling out "special" coordinates for their defintion. However, they are useful enough that MTW wanted to talk about them anyway.
  18. Aug 2, 2007 #17
    A quick look in a few GR text confirms that to that extent.
    Perhaps he's seeking more opinions, i.e. for someone else to jump in? I personally don't see a problem of phrasing it that way. The expansion of the universe is defined by the increasing distances between galaxies. However if one were to ask for a rigorous definition of space then I believe the answer would be consistent with saying that space is expanding ... in my very humble opinion that is. :smile:

    Best regards

    Last edited: Aug 2, 2007
  19. Aug 2, 2007 #18


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    I'm not sure what he's after. Personally I think that "space is expanding" is an attempt to popularize GR that doesn't quite work if you take it too literally.
    Which actually appears to be more or less what the OP said:

    I would not word it quite as strongly as the OP did, though. The idea of expanding space has recieved enough airtime that it's not really "nonsense", even if it isn't all that good of an explanation when taken too seriously.

    I'm not even sure why this popularization is so popular. It looks like this didn't get hashed out well enough last time (I thought that it did, and didn't want to cover the same ground again and hijack the thread), but apparently people still want to argue about it.

    Let's take a look at a typical use of it (in a popular news report):


    So the idea behind the popularization seems to be to explain how recession velocities can be greater than 'c'. This explanation really isn't very good.

    In a popular news article, aimed at the lay audience, the basic idea appears to be to to impart some sort of general understanding, even if it's not totally accurate, rather than to get into all the fine technical details.

    It is probably IS simpler to use the above than getting into the muddle of the difference between velocity as a vector in a tangent space, and velocity as the generalized rate of change of any sort of distance with respect to any sort of time coordinate. The first sort of "velocity" is much more physical, will always be lower than 'c' at any point, and has the problem that you can't compare two velocities at different points in a curved space-time unless you define some mechanism to do so, such as both a means of parallel transport (defined by the metric, but probably not familiar to readers without a strong background in GR or differental geometry) AND a particular path along which the velocities are to be parallel transported.

    For an example of this sort of usage, See for instance Baez's remarks in http://math.ucr.edu/home/baez/einstein/node2.html )

    But Baez's defintion of velocity, even though it is much more physical, is not the sort that's used in Hubble's law.

    The definition of velocity used in Hubble's law is a "velocity" by the fact that it's the rate of change of some distance coordinate with respect to some time coordinate. Like all such "coordinate velocities", it is not very physical, does not have any physical limit on its value.

    Hubble's law is highly dependent on the usage of a particular set of coordinates, it can even be used to define those coordinates ala Ned Wright:

    OK, we now return you to the next part of the question, which is just about as confused as the first, the issue of energy and it's conservation in GR. Or not - it depends on what people (and the OP in particular) really want to talk about.
    Last edited: Aug 3, 2007
  20. Aug 4, 2007 #19
    Redirecting to the original question

    The many responses to my question have been fascinating, but problematic. No one addressed the actual question!

    This happens quite often. It concerns me that physicists often argue with each other because they don't listen to each other. They hear the bits that interest them, and miss the subtle points that are outside their area of interest, even though these subtle points are the heart of the issue.

    The many responses here are about general relativity and conservation of energy. But my actual question was about what relativity implied about a quantum mechanical phenomenon.

    Since the volume of the universe increases, and each volume contains a certain minimum amount of vacuum energy, then what is happening to the total energy content of the universe?

    This is about QM and vacuum energy. How can the total energy of the universe be constant, if the energy per cubic meter is constant, and the number of cubic meters is increasing? General relativity describes how the universe increases in volume, but seems to says nothing about QM.

    There may be a very simple answer. Or there may even be a problem with my question (I might be making some hidden assumption which renders my question invalid, or not precisely phrased enough to answer.) But nonetheless it appears as if my initial question hasn't been addressed at all. :yuck:

  21. Aug 5, 2007 #20


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    Bingo. There is a problem with your question which makes your question invalid.

    Given that GR offers us no definition for "the energy of the universe" (at least not a universe with a FRW metric), how can you possibly expect us to answer a question about what happens to said "energy of the universe" other than having us point out that it is not defined?

    You also might want to think about how productive it is to criticize physicists for "not understanding your questions" when presumably it is the people you think of as "physicists" who are answering them?

    I suspect that if you thought about it, you could find a way to eliminate a lot of extra "baggage" from your questions that would make them easier to ansewr, and less prone to digressions. This is something that you'll never accomplish (assuming you are interested in accomplishing it) if you place all the blame on "physicists" for their answers and don't stop to consider how your questions could have been improved to avoid digressions.
    Last edited: Aug 5, 2007
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