## [SOLVED] Energy conservation in an expanding universe

I've been looking through several old threads that address the issue of
cosmic red shift, dark energy and conservation of energy in an
accelerating, expanding universe. Since I'm a total outsider to the
field of cosmology I have no way to determine the relative merits of the
(seemingly) opposing assertions made except intuitions based on style or
process (which probably isn't a very good method of determining truth).

would help me immensely if a knowledgeable professional can clarify
which of the following issues are generally considered resolved and
which are still open for vigorous debate, or whether I've worded the
question so poorly it can't be answered. So, I'm not asking anyone for
their own conclusion on the issue so much as for their subjective
perceptions of consensus, or not, in the professional physics community.

1) Is it generally accepted that GR guarantees conservation of energy
(or read "energy + momentum" wherever I write only "energy") locally,
but not globally (in the universe as a whole) in an expanding universe?

2) Whether energy is conserved universally, does it depend on how one
defines "energy" and "conserved"? Are there generally accepted
definitions of those terms such that *most* physicists would answer the
question of conservation in a particular way?

3) Does it also depend on which model of space-time geometry is
*postulated* (not sure if that's the right word)? Is there general
agreement on which model of space-time geometry is the closest
approximation to the universe we observe?

4) I've read that energy is strictly conserved in SR due to an
underlying assumption of flat space-time but not in GR due to an
underlying assumption of curved space-time. I've also read that
numerous observational methods show that the universe we observe is, or
is very close to, spatially flat. Does that mean that even if GR does
not conserve energy in principle, the real universe should still come
very close to conserving energy?

5) In the case that it is accepted that energy is *not* conserved in GR,
does it necessarily follow that either GR or the Conservation Law must
be abandoned or is it still plausible that reasonable modifications of
definitions or other underlying assumptions will result in an accurate
cosmological model that preserves both? If either GR or Conservation
must be tossed out or significantly modified in order to build a
cosmological model that accurately reflects current data, is there a
widely accepted and objective reason for favoring one theory over the
other? (For example my personal bias is toward Conservation over GR
since I was a chemical engineering student who studied thermodynamics
many years ago. If push came to shove, I'd rather keep good old,
familiar Conservation and toss out GR which I never studied in detail
and don't understand intuitively. But that's just a personal,
subjective bias with no objective substance. Is it possible that
cosmologists are more likely to have a similar personal bias toward GR
over Conservation?)

6) In the case that energy *is* conserved by GR in an expanding
universe, I've seen 2 explanations: a) the "lost" radiant energy of
red-shifted photons (due to expansion of space-time) is not actually
lost but converted to gravitational potential energy (which also changes
due to the expansion of space-time), and b) as total dark energy
increases (dark energy *density* staying constant) it is offset by the
negative work it does on surrounding volumes. Are either of these
explanations generally accepted?

7) In the case that both of the above explanations are accepted as valid
interpretations, are the 2 explanations considered equivalent? That is,
describing the same physical phenomenon but in different terms? (To be
more explicit: I've seen suggestions that dark energy *is* the energy
"lost" by the cosmic red shift, but I've also seen that idea summarily
dismissed as ridiculous. Yet if increasing total dark energy is the
*cause* of the cosmic expansion and the cosmic red shift is the
*consequence* of that expansion, why isn't it a valid interpretation to
consider that as a change from one form of energy to another with total
energy conserved? I have a vague sense there is general consensus that
cosmic red shift is not the source of dark energy but I don't understand
the subtle interpretations of the relationship.)

8) Have the explanations offered in #6 been experimentally tested? It
seems that we have good quantification of energy "lost" by photons due
to cosmic expansion (at least for individual photons, but I'm not sure
if that necessarily leads to accurate estimation of the average
*density* of the "lost" energy) and there are also reasonable estimates
of the density of vacuum energy or dark energy based on observation.
Are there also accurate estimates of the change in gravitational
potential energy due to the expansion or of the amount of negative work
that dark energy does on its surrounding space to expand it? If there
are no good estimates of those quantities can they be calculated from
theory? Is there consensus on how to test these concepts
experimentally?

9) Is Edward Harrison's explanation for why energy is not conserved
assert that Harrison's analysis demonstrates a perpetual motion machine,
even if only in principle. Not too surprisingly, I've seen a book on
the web by someone who has figured out how we can harvest the "free"
vacuum energy. And one comment has Martin Rees countering that the
"free energy" comes at the expense of the expansion. Has consensus
emerged on Harrison's analysis and is it worth buying one of his books
to get straight on these concepts?

10) Is it widely accepted that dark energy density is the same
everywhere in space, or does the accepted model allow for local variance
with constant average density only on very large scale? If the density
of dark energy might vary on small scales, and the dark energy density
varied as a function of baryonic matter density or dark matter density,
would it have been clearly detected by now?

Thanks for any insights you can offer about the current state of the art
in these topics.

Will Kastens
PNG Institute of Medical Research &
CWRU Center for Global Health & Diseases
Tel. (675) 852-3673, -2962, -2909
Fax (675) 852-3289

 In article <000001c7528c$ed4a5840$f2cd5fca@PNGIMR.local>, "Will Kastens" writes: > I've been looking through several old threads that address the issue of > cosmic red shift, dark energy and conservation of energy in an > accelerating, expanding universe. Since I'm a total outsider to the > field of cosmology I have no way to determine the relative merits of the > (seemingly) opposing assertions made except intuitions based on style or > process (which probably isn't a very good method of determining truth). First, consider that it as theoretically possible that a universe without dark energy exists. (Indeed, many folks thought for a long time that this was our universe.) Thus, studying energy conservation independently of dark energy is a good idea. I hope that the GR experts will comment on (or correct) my terse answers (Steve, Ted, John, Igor,...). > 1) Is it generally accepted that GR guarantees conservation of energy > (or read "energy + momentum" wherever I write only "energy") locally, > but not globally (in the universe as a whole) in an expanding universe? Yes. > 2) Whether energy is conserved universally, does it depend on how one > defines "energy" and "conserved"? No. Obviously, if you can define the terms any way you want, you can get almost any answer you want, but within standard usage, no. > Are there generally accepted > definitions of those terms such that *most* physicists would answer the > question of conservation in a particular way? Yes. > 3) Does it also depend on which model of space-time geometry is > *postulated* (not sure if that's the right word)? No. > Is there general > agreement on which model of space-time geometry is the closest > approximation to the universe we observe? Yes. > 4) I've read that energy is strictly conserved in SR due to an > underlying assumption of flat space-time but not in GR due to an > underlying assumption of curved space-time. I've also read that > numerous observational methods show that the universe we observe is, or > is very close to, spatially flat. Does that mean that even if GR does > not conserve energy in principle, the real universe should still come > very close to conserving energy? No. > 5) In the case that it is accepted that energy is *not* conserved in GR, > does it necessarily follow that either GR or the Conservation Law must > be abandoned or is it still plausible that reasonable modifications of > definitions or other underlying assumptions will result in an accurate > cosmological model that preserves both? We know GR is not a complete theory (no quantum effects), but not for this reason. Most folks would say the conservation law is not absolute. Perhaps it is possible to preserve both by juggling the terminology, but is it worth it? Note that conservation laws go hand in hand with symmetries, as first shown by Emmy Noether. For conservation of energy, it is the homogeneity of time. However, note that in GR cosmology, "cosmic time" exists. The experts can comment on how these are related. > 6) In the case that energy *is* conserved by GR in an expanding > universe, I've seen 2 explanations: a) the "lost" radiant energy of > red-shifted photons (due to expansion of space-time) is not actually > lost but converted to gravitational potential energy (which also changes > due to the expansion of space-time), and b) as total dark energy > increases (dark energy *density* staying constant) it is offset by the > negative work it does on surrounding volumes. Are either of these > explanations generally accepted? No, though one does run across them even in serious literature. Harrison takes pains to demonstrate that the universe is not like a steam engine. > 7) In the case that both of the above explanations are accepted as valid > interpretations, are the 2 explanations considered equivalent? That is, > describing the same physical phenomenon but in different terms? (To be > more explicit: I've seen suggestions that dark energy *is* the energy > "lost" by the cosmic red shift, but I've also seen that idea summarily > dismissed as ridiculous. It is "not even wrong". > Yet if increasing total dark energy is the > *cause* of the cosmic expansion No: the cause is the initial conditions. Consider that one can have an expanding universe with no dark energy. > and the cosmic red shift is the > *consequence* of that expansion, That bit is true. > 8) Have the explanations offered in #6 been experimentally tested? It > seems that we have good quantification of energy "lost" by photons due > to cosmic expansion (at least for individual photons, but I'm not sure > if that necessarily leads to accurate estimation of the average > *density* of the "lost" energy) It follows directly. > 9) Is Edward Harrison's explanation for why energy is not conserved > generally accepted? I have never seen anything in a refereed journal which refutes it. (Some folks not very knowledgeable about cosmology might contradict it in passing when talking about something else.) > I haven't read his explanation but I've read many > comments about his explanation along with short quotes. My first piece of advice to everyone interested in cosmology: READ HARRISON'S BOOKS!

## [SOLVED] Energy conservation in an expanding universe

In article <erd43l$sn7$1@online.de>,
Phillip Helbig---remove CLOTHES to reply <helbig@astro.multiCLOTHESvax.de> wrote:

>I hope that the GR experts will comment on (or correct) my terse

Everything you say looks right to me. I'll just expand a bit on a few
points.

>> 4) I've read that energy is strictly conserved in SR due to an
>> underlying assumption of flat space-time but not in GR due to an
>> underlying assumption of curved space-time. I've also read that
>> numerous observational methods show that the universe we observe is, or
>> is very close to, spatially flat. Does that mean that even if GR does
>> not conserve energy in principle, the real universe should still come
>> very close to conserving energy?

>
>No.

The source of the confusion here is between the terms "flat spacetime"
and "spatially flat." Despite their similar appearance, these are
quite different things. Flat spacetime is spacetime with no gravity.
A spatially flat Universe in general does have gravity (i.e., the
density of various forms of matter and energy are nonzero). In a
spatially flat Universe, we can look at three-dimensional "slices" of
the Universe corresponding to a given moment of "cosmic time," and
those slices have a flat (Euclidean) geometry, but the entire
four-dimensional spacetime is not flat.

As Phillip notes elsewhere, the reason that global energy conservation
works in flat spacetime is that flat spacetime has a time-translation
symmetry. That is, it looks the same at one time as it does at all
other times. Noether's theorem says that that symmetry gives rise to
a conservation law, which turns out to be energy conservation. A
spatially flat expanding Universe doesn't have that time-translation
symmetry, so it doesn't have global energy conservation.

Other spacetimes besides flat ones do have a notion of global energy
conservation, as long as they have time-translation symmetry. For instance,
the spacetime around a static massive body (a star or a black hole,
for instance) does have such a symmetry. It makes sense to talk
about the global energy of test particles orbiting such a body, and
that global energy is conserved.

The lack of global energy conservation in the expanding Universe isn't
really as much of a blow as it seems at first. After all, we'll always
have local energy conservation. And one of the key morals of general
relativity is that you're better off thinking of physics locally rather
than globally anyway. If we lost local energy conservation, then
we'd be thoroughly flummoxed, but global energy conservation just
isn't that big a deal.

-Ted

--
[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]

 On Feb 19, 8:56 am, "Will Kastens" wrote: > 1) Is it generally accepted that GR guarantees conservation of energy > (or read "energy + momentum" wherever I write only "energy") locally, > but not globally (in the universe as a whole) in an expanding universe? Answers have been given above based on (lack of) time translation invariance. However, this raises a question about the idea that when GR is formulated as a canonical classical Hamiltonian theory (Dirac, DeWitt, Ashtekar - eg, in terms of the spatial 3-metric and its conjugate 'momentum', relative to chosen shift and lapse functions) one ends up with the 'lapse' constraint H = 0 for the Hamiltonian (as well as shift constraints). This suggests to me that one can take the total energy to be zero (I had a vague idea that this works in inflation models, where mass can be created at the expense of a large negative gravitational energy following from expansion). The constraint carries over into the quantization of the theory, and raises various issues, but I believe it arises at the classical level ?
 ebunn wrote: > The lack of global energy conservation in the expanding Universe isn't > really as much of a blow as it seems at first. After all, we'll always > have local energy conservation. And one of the key morals of general > relativity is that you're better off thinking of physics locally rather > than globally anyway. If we lost local energy conservation, then > we'd be thoroughly flummoxed, but global energy conservation just > isn't that big a deal. To me this is one of the unexplained marvels of physics -- in this and several other cases GR is basically saying that one should not attempt to discuss things which cannot be observed (measured [#]). Of course I phrased that along the lines of Bohr's dictum about quantum mechanics. The marvel is that these two theories have similar limitations with respect to "unobservables", yet are themselves incommensurable.... [#] One aspect of this is that coordinate-dependent quantities cannot be good models of physical phenomena, so one should discuss invariants; all measurements are of course invariant under coordinate transforms, and are thus in bounds. Another aspect is integrability and the difficulty of defining integrals over regions of a curved manifold (one aspect of the subject of this thread) -- if you cannot sum the parts to compute the whole you certainly can't measure it. Tom Roberts
 Thus spake ebunn@lfa221051.richmond.edu >And one of the key morals of general relativity is that you're better >off thinking of physics locally rather than globally anyway. This is at once true and false. Ted doesn't mean this to apply why thinking of global structure, e.g. solutions of the Friedmann equation. Regards -- Charles Francis substitute charles for NotI to email
 On 2007-02-22, a student wrote: > On Feb 19, 8:56 am, "Will Kastens" wrote: > >> 1) Is it generally accepted that GR guarantees conservation of energy >> (or read "energy + momentum" wherever I write only "energy") locally, >> but not globally (in the universe as a whole) in an expanding universe? > > Answers have been given above based on (lack of) time translation > invariance. However, this raises a question about the idea that when > GR is formulated as a canonical classical Hamiltonian theory (Dirac, > DeWitt, Ashtekar - eg, in terms of the spatial 3-metric and its > conjugate 'momentum', relative to chosen shift and lapse functions) > one ends up with the 'lapse' constraint > H = 0 > for the Hamiltonian (as well as shift constraints). This suggests to > me that one can take the total energy to be zero (I had a vague idea > that this works in inflation models, where mass can be created at the > expense of a large negative gravitational energy following from > expansion). > > The constraint carries over into the quantization of the theory, and > raises various issues, but I believe it arises at the classical level ? That's right, the constraint analysis is well understood at the classical level. It still causes problems when quantization is attempted. However, the idea of "taking the total energy to be zero" is not fruitful for at least one reason. The reason is that H = 0 is true over the entire constraint surface. There is an analogy in the case when the system has no constraints. Take its phase space. Then functions on the phase space represent observables. For time translation invariant systems, energy is one such function. It has the particular property that it is constant along paths representing time evolution (equivalently, time translation). But it does not assume the same value for different initial conditions (different time evolution paths). This variation is important for considerations of stability, energy dissipation, etc. Contrast this with the case of the observable corresponding to a function that constant on the entire phase space. This function is constant on any path in phase space (a time evolution or not). Therefore, it tells us absolutely nothing about the system. The same argument can be made for the constraint H, which is constant (and equal to zero) over the entire constraint surface. Hope this helps. Igor
 Energy is universally conserved in relativity if the expanding model is finite, where, and also as with; "inflation models, where mass can be created at the expense of a large negative gravitational energy following from expansion." *quotes around comment by "a student". In this model, far from equalibrium dissipative structuring serves as a natural damping mechanism which prevents the structure from evolving inhomogeneously while maximizing work per the second law of thermodynamics. Quantum mechanics depends very much on Hamiltonian mechanics, and so it isn't inherently able to describe dissipative structuring. As I understand it, this can be done, however, by way of the "Lindblad equation", which derives that flatness acts as a natural harmonic damper mechanism that keeps the imbalanced universe from evolving inhomogeneously, so this is the most natural configuration... if the universe is finite and closed... given inherent asymmetry in the energy. This will necessarily maximize the amount of work that the expansion process can produce, and that's what a flat universe accomplishes via anthropic structuring.
 On Feb 26, 5:22 pm, Igor Khavkine wrote: > However, the idea of "taking the total energy to be zero" is not > fruitful for at least one reason. The reason is that H = 0 is true over > the entire constraint surface. Thanks, that is certainly a good point! Thinking about it, the only way I can still see it of being of some interpretational value is that H = 0 could be viewed as a 'detailed energy balance' equation, holding at each point on the constraint surface. In particular, H is a specific function of the spatial 3-metric h_ij and the conjugate 'momentum' field p^ij, so that H=0 looks something like an equation for the sum of a kinetic and a potential term, i.e., of the form G_{ijkl} p^ij p^kl + V = 0, where G_{ijkl} and V are functions of h_ij and its derivatives (G is the deWitt supermetric, and V incorporates matter and curvature terms). The two terms do vary individually over the constraint surface, and so one has in effect a detailed energy balance equation (providing one can actually give an interesting physical meaning to the kinetic and potential terms!). The Ashtekar formalism would give something formally similar.
 In article , Oh No wrote: >Thus spake ebunn@lfa221051.richmond.edu >>And one of the key morals of general relativity is that you're better >>off thinking of physics locally rather than globally anyway. > >This is at once true and false. Ted doesn't mean this to apply why >thinking of global structure, e.g. solutions of the Friedmann equation. Well, I admit it's an oversimplification, like most morals. But not all that much of one. Even when talking about solutions to the Friedmann equation, it's rarely necessary or fruitful to think much about truly global properties. For instance, is the Universe spatially finite or infinite? That's a global question, and it's one that lots of people find interesting to think about. But it has remarkably little relevance to anything that we can test observationally. It's much more important to think about observables that live in small to medium-sized volumes rather than global properties. The biggest exception in cosmology is searches for nontrivial global topology, but frankly that's a bit of a sideshow. (I feel like I've earned the right to say this, since I've written papers on the subject.) -Ted -- [E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]
 Thus spake ebunn@lfa221051.richmond.edu >In article , >Oh No wrote: >>Thus spake ebunn@lfa221051.richmond.edu >>>And one of the key morals of general relativity is that you're better >>>off thinking of physics locally rather than globally anyway. >> >>This is at once true and false. Ted doesn't mean this to apply why >>thinking of global structure, e.g. solutions of the Friedmann equation. > >Well, I admit it's an oversimplification, like most morals. But not >all that much of one. Even when talking about solutions to the >Friedmann equation, it's rarely necessary or fruitful to think much >about truly global properties. > >For instance, is the Universe spatially finite or infinite? That's a >global question, and it's one that lots of people find interesting to >think about. But it has remarkably little relevance to anything that >we can test observationally. It's much more important to think about >observables that live in small to medium-sized volumes rather than >global properties. I don't know about that. I find it philosophically important. But the place to take about philosophical issues in physics is sci.physics.foundations, not s.p.r. > >The biggest exception in cosmology is searches for nontrivial global >topology, but frankly that's a bit of a sideshow. (I feel like I've >earned the right to say this, since I've written papers on the >subject.) I would have said that WMAP which gives our best measure of spacial flatness was actually quite important, and that its fairly relevant in the study of supernova redshifts also. But apart from those two, I agree, I can't think of any other measurements we can make where we can look far back enough, and clearly enough, that global properties come in to play to any great extent. That may change with the next generation of very large telescopes. We are already seeing galaxies so far back in time that we don't know how to explain how they evolved so quickly after the big bang. If that trend continues it would throw the consistency of the standard model into doubt. Regards -- Charles Francis moderator sci.physics.foundations. substitute charles for NotI to email
 In article , Oh No wrote: >Thus spake ebunn@lfa221051.richmond.edu >>Well, I admit it's an oversimplification, like most morals. But not >>all that much of one. Even when talking about solutions to the >>Friedmann equation, it's rarely necessary or fruitful to think much >>about truly global properties. [...] >I would have said that WMAP which gives our best measure of spacial >flatness was actually quite important, and that its fairly relevant in >the study of supernova redshifts also. Agreed. But I'm not sure what your point is. Curvature is a local property of spacetime, not a global one. -Ted -- [E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]
 On 18 fev, 22:56, "Will Kastens" wrote: > > 1) Is it generally accepted that GR guarantees conservation of energy > (or read "energy + momentum" wherever I write only "energy") locally, > but not globally (in the universe as a whole) in an expanding universe? +++ The Einstein Field Equation involves covariant "conservation" of the stress energy tensor (covariant divergence vanishes). So, locally, you have to take into account a eventual change in local geometry, which is the case in "expanding universes". Let's consider a comobile (at rest) free falling observer in expanding FLRW universe who performs energy mesurement, he lives in the (Minkowkian) tangent space time. He does not get what is gone in curvature changes. So, locally the energy is not conserved for such comobile free falling observer. He will measure light redshifted, proper speed of galaxies slowed down. This is what we measure, eventhough we are not exactly free falling observers in our expanding universe, but we know how to correct this. There is no time Killing vector in such expanding universe but you have a Killing tensor which can be used for demonstrating the local loss in energy. As energy is associated to time, according to Noether theorem, energy is conserved when space time is stationnary. In other cases, even the definition of energy may be tricky. And even in case of stationnary space time, some solutions such as static black holes are not so simple with energy, as you cannot use stress energy tensor which is not defined in such space time (either zero or infinite). In that cases, you may use the flow of the time Killing vector through a 2-sphere at infinity (Komar integral) (or some ADM equations). Is the energy conserved globally? As far as I know, GR, which is based on local equations, does not answer to this question. Answer comes more often from thermodynamic considerations. The universe is considered as an isolated system, so as it does exchange anything with some hypothetical exterior, the energy should be conserved within the expanding spatial geometry. Jacques +++
 Thus spake ebunn@lfa221051.richmond.edu >In article , >Oh No wrote: >>Thus spake ebunn@lfa221051.richmond.edu > >>>Well, I admit it's an oversimplification, like most morals. But not >>>all that much of one. Even when talking about solutions to the >>>Friedmann equation, it's rarely necessary or fruitful to think much >>>about truly global properties. > >[...] > >>I would have said that WMAP which gives our best measure of spacial >>flatness was actually quite important, and that its fairly relevant in >>the study of supernova redshifts also. > >Agreed. But I'm not sure what your point is. Curvature is a local >property of spacetime, not a global one. > Hmmm. Perhaps it is merely a semantic issue, or a distinction without a difference. I would have said that when we talk about Omega_k =0 we are saying space is globally flat, meaning the net curvature in a global sense, under the assumptions of homogeneity and isotropy built into the Friedmann equation. Regards -- Charles Francis moderator sci.physics.foundations. substitute charles for NotI to email
 In article , Oh No writes: > >Agreed. But I'm not sure what your point is. Curvature is a local > >property of spacetime, not a global one. > > > Hmmm. Perhaps it is merely a semantic issue, or a distinction without a > difference. I would have said that when we talk about Omega_k =0 we are > saying space is globally flat, meaning the net curvature in a global > sense, under the assumptions of homogeneity and isotropy built into the > Friedmann equation. If space is locally flat, and if space is, on average, everywhere the same, then it is flat everywhere. But the latter statement is an extrapolation based on the assumption that it is the same everywhere, not an observation. What the large-scale topology (that would be a truly global characteristic) is another issue, about which the traditional cosmological parameters say nothing. (There are, however, ways to observe it.)