Re: Energy conservation in an expanding universe

On 2007-02-22, a student <of_1001_nights@xxxxxxxxxxx> wrote:
On Feb 19, 8:56 am, "Will Kastens" <wkast...@xxxxxxxxxxxx> 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

.

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