
#1
Apr1411, 07:40 PM

P: 315

Okay, my title may have been a bit overdramatic ... but my question is still about something a bit troubling I read in Sean Carroll's GR text.
Basically, he claims that the total energy in an expanding Universe is not typically conserved. See pages 137138 and 344. This is because expansion means the metric is changing in time, and therefore there is no isometry in this direction. This, in turn, means there are no timelike Killing vectors. Killing vectors generate the symmetries in a curved spacetime. No timelike Killing vectors means no symmetry in time and, as we all remember, conservation of energy is related to symmetry in time. He goes on elsewhere, IIRC, to claim that energy is conserved locally. It's just that globally the total energy is no longer conserved (in an expanding curved spacetime) So, is this viewpoint generally regarded as true? If so, what are the ramifications? 



#2
Apr1411, 08:19 PM

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See also here for some more details:
Is Energy Conserved in General Relativity? (from the Usenet Physics FAQ) 



#3
Apr1511, 09:42 PM

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PF Gold
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FAQ: How does conservation of energy work in general relativity, and how does this apply to cosmology? What is the total massenergy of the universe?
Conservation of energy doesn't apply to cosmology. General relativity doesn't have a conserved scalar massenergy that can be defined in all spacetimes.[MTW] There is no standard way to define the total energy of the universe (regardless of whether the universe is spatially finite or infinite). There is not even any standard way to define the total massenergy of the *observable* universe. There is no standard way to say whether or not massenergy is conserved during cosmological expansion. Note the repeated use of the word "standard" above. To amplify further on this point, there is a variety of possible ways to define massenergy in general relativity. Some of these (Komar mass, ADM mass [Wald, p. 293], Bondi mass [Wald, p. 291]) are valid tensors, while others are things known as "pseudotensors" [Berman 1981]. Pseudotensors have various undesirable properties, such as coordinatedependence.[Weiss] The tensorial definitions only apply to spacetimes that have certain special properties, such as asymptotic flatness or stationarity, and cosmological spacetimes don't have those properties. For certain pseudotensor definitions of massenergy, the total energy of a closed universe can be calculated, and is zero.[Berman 2009] This does not mean that "the" energy of the universe is zero, especially since our universe is not closed. One can also estimate certain quantities such as the sum of the rest masses of all the hydrogen atoms in the observable universe, which is something like 10^54 kg. Such an estimate is not the same thing as the total massenergy of the observable universe (which can't even be defined). It is not the massenergy measured by any observer in any particular state of motion, and it is not conserved. MTW: Misner, Thorne, and Wheeler, Gravitation, 1973. See p. 457. Berman 1981: M. Berman, unpublished M.Sc. thesis, 1981. Berman 2009: M. Berman, Int J Theor Phys, http://www.springerlink.com/content/357757q4g88144p0/ Weiss and Baez, "Is Energy Conserved in General Relativity?," http://math.ucr.edu/home/baez/physic...energy_gr.html Wald, General Relativity, 1984 



#4
Apr1611, 12:50 AM

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Conservation of Energy Down The Drain?http://www.physics.ucla.edu/~cwp/art...g/noether.html If an exact timetranslation symmetry exists, it yields a conserved Komar mass via Noether's theorem. It's possible, however, to have some useful notions of conserved energy even in the lack of such an exact symmetry. One may still have an asymptotic timetranslation symmetry which is sufficient to give a conserved Bondi or ADM energy  notions which are very useful, though they require asymptotic flatness to be applied, something that our universe seems to lack. 



#5
Apr1811, 06:36 PM

P: 315

Thanks for the info guys. I'll have to read through all the links provided.
one additional question related to this: If the vacuum energy throughout space is constant and space is expanding, wouldn't that indicate the total energy of the Universe is increasing (with time)? 



#6
Apr1811, 07:07 PM

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#7
Apr1811, 07:28 PM

P: 315

I'm not sure the simple answer is a plain "no", is it?
I understand energy is not defined globally per my OP and the lack of timelike killing vectors. But, it can be defined locally, at least approximately. If you consider my question again locally, and then extrapolate to cosmological scales, it seems like an energy increase is implied, at the very least, doesn't it? Besides, if energy truly isn't conserved on a global scale, it seems to me, it could increase, or decrease, or do whatever it wants, as far as we know and can predict. Unless, of course, the lack of it being defined is an artifact of GR, and/or a limitation of the theory. This is why I originally asked what are the ramifications of it not being conserved under GR? 



#8
Apr1811, 08:07 PM

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Ben 



#9
Apr1811, 08:36 PM

P: 315

If we are unable to define something I don't think we can rightly make any absolute statements about that something. And, if you are, we cannot completely rule out was is being implied by the extrapolation, if it indeed falls outside the domain of validity of the theory. It seems sort of hard to escape the conclusion that energy would increase globally when you're adding up all the local increases. Or, at the very least, just claiming it isn't defined seems like an answer that leaves something desired. So: (1) Is energy truly not defined on a global scale? (2) or, are we unable to define it, given our current understanding? 



#10
Apr1811, 08:43 PM

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#11
Apr1811, 09:52 PM

P: 315

Hmmm, energy consideration using 4Vectors is more like SR, and it is not the total energy we are talking about. Under GR, energy is usually considered in a more sophisticated way within the energymomentum tensor, which is divergence free and can be interpreted as a "local" energy conservation. Also, as part of a stationary spacetime, the Komar Integral can be interpreted as the total energy. But, with an expanding space, total energy is undefined under GR. All this is besides the point I was trying to make. So, leaving all that aside .....
I repeat: (1) Is total energy truly not defined on a global scale? (2) or, are we unable to define it, given our current understanding? If you answer #2, the statement in your last post doesn't mean much for the question at hand. It could be trying to address a problem in the language of a theory that may be ill equipped for the task at hand. That's the main gist of the Point I wanted to make for #2. If you answer #1, well, then all sorts of fun, interesting questions can get raised, which would be a great topic in its own right. (although, one likely to get banned on this forum) Anyhow, this is why the main point of my OP was ...... "What are the ramifications of total energy NOT being conserved under GR?" 



#12
Apr1811, 10:57 PM

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#13
Apr1811, 11:17 PM

P: 315

I said the SR 4Vector component is not related to the total energy of the Universe that we were talking about, which you also stated when you claimed the total energy is undefined. I personally suspect total energy will be definable at some point, and if it isn't conserved, we'll be able to say why. Perhaps, quantum gravity will help out here, whenever it gets figured out. 



#14
Apr1811, 11:21 PM

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#15
Apr1911, 01:32 AM

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Divide spacetime up into a bunch of small pieces. In each small piece: Take the energy density rho in some local frame multiply it by the volume element in said local frame Sum them all together. One of the problems with this idea is that it isn't in general independent of how one cuts up spacetime into pieces. But, we can take a specific case, a static geometry, where there is an "obvious" way to cut up spacetime into pieces  and see that said number we compute by this method STILL fails to give us the total energy. This gives us some insight into why the number one computes in this manner isn't energy. What you'd compute by this technique in a static spacetime is what MTW calls the "energy before assembly". If you have this text, see chapter 23 on "spherical stars". The actual energy, which MTW also computes, is lower than this number  we can say "actual energy" because the geometry is static so we have a defined energy to compare your number to. MTW calls the difference "gravitational binding energy", and demonstrates how the difference approaches the expected Newtonian value in the Newtonian limit So, since the actual energy is lower than your number, showing that your number increases doesn't demonstrate anything, since we have reason be believe by example that the actual energy is lower than your number because of the "gravitational binding energy" in a simple situation where we can define both your number and energy. Preferred frame theories are possible, but are out of vogue  and are not GR. For instance, Self Creation Cosmology would be an example of a nonGR theory that would have a conserved energy, getting around Noether's theorem by having a scalar field with an associated Jordan Frame, a bit like BranseDicke theory. Unfortunately, both BranseDicke theories and SelfCreation Cosmology (as originally proposed) appear to be inconsistent with experiment  the later predicted different results for Gravity probe B. So, unless my understanding is wrong, GR won't have a conserved energy because it doesn't have a preferred frame. Examples of theories that have a conserved energy also have preferred frames. 



#16
Apr1911, 07:56 AM

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I read the WP article on SCC. I don't see anything in it about global conservation of energy, only local conservation. Of course, reading a WP article doesn't make me an expert, but are you sure that it's the case that there's a conserved global energy in all spacetimes? For example, SCC is supposed to be equivalent to GR in the case of vacuum solutions, but GR doesn't have a conserved global energy that applies to all vacuum solutions. 



#17
Apr1911, 10:51 AM

P: 315

Take for example singularities. GR, more or less, predicts them, but then offers no explanation for them. This is mainly because we're now outside the scope of the theory. Well, GR also "predicts" that the total energy of the Universe is undefined. Is it really? Or, are we outside the domain of the theory again? Is it just a mathematical artifact? Many seem to think that spacetime is NOT really curved, and is just a convenient analogy to describe a mathematical model that happens to work, as far as giving predictions. Well, could an undefined total energy be an unfortunate consequence of using a model that doesn't truly physically represent reality? Or, could it just be the theory is incomplete and quantum gravity will fill in some holes? Or, has energy been conclusively proven to be undefined in GR? I think the answer to the last question is a "no". But, the jury seems to be out on the other two. I personally have a hard time imagining that total energy is truly undefined. Everything in the Universe is essentially energy. We are not special in our little corner of the Universe. If we would argue that energy is defined locally and energy "conservation" is valid locally, well, then, so would everybody else in their little corner of the Universe, no matter where they are. How then can it be undefined globally in an absolute sense? Anyhow, don't take my last paragraph as a scientific theory and analyze it. It's just meant as a simple analogy. ;) 



#18
Apr1911, 11:01 AM

P: 315

sorry for the confusion Thanks, this getting along the lines of what I was looking for. Yes, preferred frames definitely don't seem like the right way to go. Could it be possible to not necessarily use a perferred frame, but rather use an entirely coordinateindependent way to define total energy, as JesseM also stated above? I guess we don't know the answer to that though. Have there been any attempts to do so, though? Coordinate systems and reference frames are manmade mental constructs. Perhaps they are getting in the way here? 


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