[SOLVED] Energy conservation in an expanding universe

Will Kastens

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).

However, I am very interested in learning more about this field. It
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
generally accepted? I haven't read his explanation but I've read many
comments about his explanation along with short quotes. Some comments
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
PO Box 378, Madang, PNG
Tel. (675) 852-3673, -2962, -2909
Fax (675) 852-3289

Phillip Helbig---remove CLOTHES to reply

In article <000001c7528c$ed4a5840$f2cd5fca@PNGIMR.local>, "Will Kastens"
<wkastens@datec.net.pg> 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!

carlip-nospam@physics.ucdavis.edu

Will Kastens <wkastens@datec.net.pg> 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?

It's trickier than that. Start with Newtonian gravity. For energy
to be conserved, one must clearly include gravitational potential
energy -- otherwise, the increasing kinetic energy of, say, a falling
apple would violate conservation. But in general relativity, there
can be no local definition of gravitational potential energy, since
for a small enough region one can always switch to a freely falling
reference frame, in which gravity locally disappears. So you can look
at the problem as not so much one of energy conservation as one of
even defining energy.

GR guarantees energy conservation locally in one, specific sense: if
in such a freely falling frame, in a region small enough that both
internal gravitational interactions and tidal forces from the outside
can be neglected, GR reduces to special relativity, in which energy is
conserved.

In an asymptotically flat spacetime -- that is, a spacetime in which
matter is all concentrated in a finite region, and the spacetime
becomes flat at large distances from that region -- one can also
define a *total* energy in GR ("ADM mass"), essentially because distant
observers see a flat, SR-like spacetime. This does not hold in our
Universe, of course, but it allows for good approximations. Spacetime
around the Solar System is not quite asymptotically flat -- eventually
you'll hit other stars -- but it's flat enough far enough out that the
ADM mass of the Solar System makes sense as a very good approximation.

> 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?

The argument I gave above implies that there is no good local definition
of gravitational energy. There are, however, a number of somewhat
different definitions of conserved "quasilocal energy" in GR. Quasilocal
energy describes the total energy in a finite region; it typically depends
on the location, shape, and motion of the boundary of the region, but is
independent of coordinates on the inside. The various proposals for
quasilocal energy don't always agree, and I think it's safe to say that
none is generally accepted as "the" right definition, but several are
useful for capturing particular aspects of what we usually mean by energy.
Quasilocal energy can be thought of as being conserved, but this requires
a definition of "energy flux" through the boundary of the region; such
a thing can be defined in a way that I think is sensible, and that
reduces to what it ought to in flat spacetime, but you could argue --
not entirely incorrectly -- that it's just invented to save conservation.

> 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?

I'm not sure what this one means.

> 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 Universe is nearly *spatially* flat at a fixed time, but the
*spacetime* is not at all flat.

Conservation laws in physics are associated, via Noether's theorem, with
invariances of laws of nature. Energy conservation, in particular, is
associated with time translation invariance, that is, with the fact that
the Universe acts the same at all times. This is true in SR, but not in
GR, and certainly not in our Universe, which is expanding, and definitely
not the same today as it was yesterday.

> 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? [...]

Conservation laws are derived properties of underlying laws of physics.
Their origin has been understood since Emmy Noether's work of 1918; as
I said above, they are a consequence of invariances of the laws of
nature. The Universe is observably changing in time, so there is no
particular reason to expect energy conservation to hold. If you wanted
to throw out GR and keep conservation, you would need to find a theory
that reintroduced time translation invariance [a technicality: as a
physical symmetry, not just a coordinate invariance]. This seems unlikely
to happen.

(I should add that I know of no theory that keeps energy conservation,
throws out GR, and gets gravity or cosmology close to being right.)

> 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?

Dark energy is not relevant. Your intuition about gravitational
potential energy is roughly right; the trouble is that there is no
local gravitational potential energy possible in GR, or in any theory
in which the equivalence principle holds at a fundamental level. As
I said above, one can define, in various ways, "quasilocal energy" in
a finite region, which does include a sort of gravitational potential
energy. As far as I know, this has mainly been applied to systems like
black holes; I don't know of much that's been done in cosmology, where
some of the definitions may start to get weird.

Steve Carlip

ebunn@lfa221051.richmond.edu

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
>answers (Steve, Ted, John, Igor,...).

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.]

Igor Khavkine

On 2007-02-18, Will Kastens <wkastens@datec.net.pg> 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?

Both Phillip and Steve Carlip have already given good answers to most of
your questions. I want to clarify notion of local energy conservation in
GR, which seems to lie at the root of most of them.

First, how is energy usually represented in a relativistic context?
Well, since, according to special relativity, the separation of energy
and momentum is coordinate dependent, we must keep both together. On the
other hand, once we adopt a continuum (as a fluid) as opposed to a
particle-like description of matter, we also have to consider local
energy flux, pressure, and shear stresses. Carefully examining how these
quantities transform under Lorentz transformations, we find that the
energy fluxes mix with pressures and shear stresses. So, we must keep
them together as well. Putting all of the above in one gadget, we get
the stress-energy tensor:

[ rho g ]
T = [ ] .
[ g t ]

This is a symmetric second rank tensor (in any given coordinate system,
you can think of it as a symmetric 4x4 matrix). The time-time component
rho corresponds to the local energy density (which includes mass as well
as kinetic energy). The time-space components g give the momentum
density at a given point. The space-space components t are a 3x3 matrix,
whose diagonal entries correspond to pressures and off-diagonal entries
to shear stresses.

Local energy conservation is expressed as

div T = 0.

The divergence of T corresponds to taking the divergence of each row of
T, which leaves a column vector behind. The time component of this
vector is roughly the continuity equation

d(rho)/dt = div g,

which states that the momentum density is the energy flux. The spatial
components of div T give a similar conservation law, but for momentum
density this time, identifying pressures and shear stresses as momentum
fluxes. These equations basically reduce to the Navier-Stokes equations.
Note that the correspondence is only rough because the exact expression
for the divergence involves the metric and so doesn't have exactly the
same form as in flat Minkowski space-time.

So far so good, the equation div T = 0 seems to capture what we normally
understand by local energy and momentum conservation. Note however, that
T describes only matter, but not gravity. So, how can we have
conservation of energy without including a "gravitational potential"?
The main difficulty is formulating what "gravitation potential" is in
GR, as Steve Carlip already discussed. But to see that we can still do
without it, one need only realize that in a small enough region of space
we can go to a locally inertial coordinate syste, thereby eliminating
gravity alltogether. So, again locally, we need not even take gravity
into account to get energy and momentum conservation.

So, now we know how energy and momentum move around locally? What about
the total total amount of them? First, one has to find a sensible way to
add up the total amount of energy in some spatial region. Even doing
that in a manner independent of a choice coordinates is non-trivial: a
first hint that things are going wrong. But suppose that can be done,
and we can calculate a quantity called the "total energy" of a spatial
region. Do we expect it to be conserved with time? A naive answer would
be Yes, since we see energy conservation in all familiar garden variety
cases.

However, it is important to remember that energy conservation holds only
for *closed* systems, those that are not under the influence of external
forces. This property is exactly what gives the system time translation
invariance (you can't tell what time it is just by looking at a motion
of the system and knowing nothing else), from which follows energy
conservation through Noether's theorem. However, if an external force is
present, this property no longer holds. Case in point is a forced
harmonic oscillator. But looking at the motion of the oscillator, you
can tell when the driving force reaches a positive or negative extreme,
for instance by looking at the oscillator's acceleration. Not
surprisingly, the energy of an externally driven oscillator is not a
constant; it changes with time.

Now, lets go back to the energy-momentum tensor T in GR. Recall that T
only contains contribution from matter and radiation, but not gravity.
So, even if we could construct a "total energy" quantity out of T, it
would not be conserved as long as the gravitational field is
time-dependent (as in an expanding universe). Hence, it is not expected
for this "total energy" to be conserved.

> (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?)

It's no good to cling to principles without keeping in mind why the hold
true. Energy conservation fails even in a situation that might be very
familiar to you, a system held at constant temperature. Such a system
actively exchanges energy with its environment. Energy conservation
holds only on average. Its total energy fluctuates with time. Closure is
the key condition that ensures energy conservation.

The lesson to remember here is that the well defined notion of energy
and momentum (the energy-momentum tensor T) that can be defined in GR
does not involve gravity itself, and hence does not describe a closed
system. And, as Steve Carlip pointed out in his reply, there is no
unique well defined notion of energy associated with the gravitational
field. These obstacles prevent the usual notion of total energy
conservation from holding true.

Hope this helps.

Igor

a student

On Feb 19, 8:56 am, "Will Kastens" <wkast...@datec.net.pg> 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 ?

Tom Roberts

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

Oh No

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

Igor Khavkine

On 2007-02-22, a student <of_1001_nights@hotmail.com> wrote:
> On Feb 19, 8:56 am, "Will Kastens" <wkast...@datec.net.pg> 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

island

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.

a student

On Feb 26, 5:22 pm, Igor Khavkine <igor...@gmail.com> 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.

ebunn@lfa221051.richmond.edu

In article <JAsMgjExlr3FFwGr@charlesfrancis.wanadoo.co.uk>,
Oh No <NotI@charlesfrancis.wanadoo.co.uk> 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.]

Oh No

Thus spake ebunn@lfa221051.richmond.edu
>In article <JAsMgjExlr3FFwGr@charlesfrancis.wanadoo.co.uk>,
>Oh No <NotI@charlesfrancis.wanadoo.co.uk> 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

ebunn@lfa221051.richmond.edu

In article <FX5CVdlh$34FFw+i@charlesfrancis.wanadoo.co.uk>,
Oh No <NotI@charlesfrancis.wanadoo.co.uk> 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.]

jacques.fric@neuf.fr

On 18 fev, 22:56, "Will Kastens" <wkast...@datec.net.pg> 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
+++

Oh No

Thus spake ebunn@lfa221051.richmond.edu
>In article <FX5CVdlh$34FFw+i@charlesfrancis.wanadoo.co.uk>,
>Oh No <NotI@charlesfrancis.wanadoo.co.uk> 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

Phillip Helbig---remove CLOTHES to reply

In article <a3h0t5Zppb5FFw3i@charlesfrancis.wanadoo.co.uk>, Oh No
<NotI@charlesfrancis.wanadoo.co.uk> 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.)