Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: Universal Loss and the Laws of Conservation (long)



Regarding Merlin's comments about global energy nonconservation in a GR
context:

It's been awhile, during the move my email got messed up, but I'm back and
far from satisfied on a particular subject.

In reading Principles of Physical Cosmology by Peebles, I came across a
passage (and I'll be gald to reference it upon request) that explained that
at the edge of the universe, which is to say the event horizon of the
universe, depending on whether the universe is infinite yet bounded or
asymptotically flat, there is predicted a loss of energy!

The effect that Peebles discusses has nothing to do with any "edge of the
universe, which is to say the event horizon of the universe". Rather it
has to do with the (large-scale) homogeneous (throughout the universe)
cooling of the CBR as Hubble expansion of space itself proceeds. The CBR
is (to an excellent approximation) in thermal equilibrium and isolated
from interaction with the rest of the universe's degrees of freedom (i.e
it is effectively decoupled from the matter present), and therefore, its
entropy is both constant and maximal. Therefore, the expansion of the
space containing the radiation is an adiabatic and isentropic process.
Since for any volume V (proportional to a(t)^3) the radiation in that
volume (when it is in equilibrium) at temperature T has an entropy which
is proportional to V*T^3. But the (radiant) energy inside volume V is
proportional to V*T^4. Now since the expansion is isentropic this means
that the temperature must be proportional to V^(-1/3) in order to keep S
constant. This means the energy inside volume V is therefore
proportional to V*(V^(-1/3))^4 = V^(-1/3) which is proportional to
1/a(t) as the expansion proceeds. Thus the radiant energy in volume V
decreases in a way inversely proportional to the linear extent of that
region of space as that region stretches.

Here's the quote
from the text:

"...since the volume of the universe varies as a(t)3 (cubed), the net
radiation energy in a closed universe decreases as 1/a(t) as the universe
expands. Where does the lost energy go? Since there is no pressure gradient
in the homogeneously distributed radiation (Cosmic Background Radiation,
CBR) the pressure does not act to accelerate the expansion of the
universe...The resolution of this apparent paradox is that while energy
conservation is a good local concept and can be defined more generally in
the special case of an isolated system in asymptotically flat space, there
is not a general global energy conservation law in general relativity
theory."

Needless to say I was shocked. The Laws of Conservation are one of a few
select group of concepts that survive the transition from Newtonian to
Relative physics. A few select Laws that even the most advance and cutting
edge research strive to adhere to, and yet here is one of the most respected
cosmologists saying these Laws don't hold. It's a new one to me that there
isn't a conservation of energy law in general relativity (GR). I've seen
entire explanations of GR based on conservation!

No need to be too shocked. Energy conservation still works just fine
in situations where the Hubble expansion of space is irrelevant (as in
all laboratory situations confined to our local group of galaxies). A
major reason why global energy conservation (including the energy in
matter and radiation, but not gravitation) fails is that as far as the
matter and radiation are concerned, a spacetime which is time-dependent
causes time-dependent "boundary conditions" to appear in the Hamiltonian
for the matter and radiation, since that Hamiltonian depends on the
metric (or geometry) of spacetime, and that geometry provides a
background host or 'container' for the matter and radiation degrees of
freedom present. Noether's Theorem tells us that the energy will be
conserved *if* the Hamiltonian has no explicit time-dependence, *but*
with a time-variable 3-geometry of space, the matter/radiation
Hamiltonian *is* explicitly time-dependent, and total energy (the value
of the Hamiltonian) is, correspondingly, not conserved. In such a
situation it is probably best to not imagine that the lost energy in the
expanding universe is going anywhere; rather the numerical value of
the Hamiltonian is just decreasing. The energy isn't 'stuff' that has
to be accounted for in the first place. It's the numerical value of
the Hamiltonian function of the system's microstate.

If we wish to make an analogy with other systems with explicitly time-
dependent Hamiltonians and account for the energy loss by interactions
with an external forcing agent, this can be done (sort of) in GR as well.
Suppose we have a swinging simple pendulum where the length of the cord
between the pivot and the bob has a time-dependence due to a spool at the
pivot connected to an electric motor which winds up the pendulum cord as
time goes on. Suppose we make the pendulum perfectly frictionless and
perfectly isolated (other than the connection via the spool to the motor)
from the rest of the universe. If we set this pendulum swinging we will
notice that the total energy (potential + kinetic) in the swinging mode
of the pendulum is not conserved as the motor shortens the cord. In this
case the motor does work on the pendulum system as it shortens the cord
and this adds energy from the motor to the now-externally-forced
pendulum. The value of the restricted Hamiltonian for the pendulum
degree of freedom alone keeps changing (increasing) as the motor works
the system. Here we can say that the motor and its battery or alternator
power supply was the source for the pendulum's increased energy. If we
included the degrees of freedom of the motor and its power supply into
the system's Hamiltonian, then the total energy of the composite system
would remain constant (*if* this composite system could legitimately be
considered isolated from the rest of the universe).

The analogy with the universe is that the pendulum is sort of like the
matter/radiation in the universe, and the motor is sort of like the
geometry of spacetime (i.e. gravitation)--except that in the cosmology
case the gravitation is the *sink* for the lost matter energy, whereas
the motor and its power supply is the *source* of extra energy for the
pendulum. So, crudely, we can say that the energy lost in the Hubble
expansion of the universe went into raising the curvature/strain energy
of spacetime (aka gravitational potential energy) of the universe. The
problem with this accounting technique is that once we try to include the
energy of gravitation into our system Hamiltonian we are faced with
ambiguities in that just how this is to be done depends on the coordinate
system one uses. There is no unique coordinate system-independent way to
account for the 'energy of gravitation' when the spacetime is not
asymptotically flat with all the matter, radiation, spacetime curvature
and geometric distortions localised in an otherwise infinite and empty
universe. In a globally finite or other non-asymptotically flat
spacetime different coordinate systems will give different energy
accounting schemes--no one of which can legitimately be thought of as
*the* total energy of the universe, which, necessarily, remains an
ambiguous concept.

Another, somewhat weird, point is that for a normal strained elastic
medium the elastic strain energy goes up as the distortion goes up,
but for gravitation this is the opposite. As the curvature of
spacetime (geometric distortion) goes up the gravitational "energy"
due to this distortion is *negative* and the the gravitational
energy decreases with increasing distortion. (We recall that
the Newtonian gravitational potential is negative and decreases the
closer we go toward a gravitating mass where the curvature is greater.)
(Effectively we can think of spacetime as *sort of* having a negative
Young's modulus.)

But *if* the expansion of the universe *was* like that of an ordinary
explosion where a finite amount of matter and radiation move outward from
some center-of-explosion in a pre-existing infinite, otherwise empty,
asymptotically flat spacetime, then, in this special circumstance, it
is possible to define a well-defined total energy (including gravitation)
of the universe (as J. Epstein reported about Penrose doing in the case
of a Black hole geometry), and the energy in this case would be
conserved where the energy of the matter & radiation in the explosion
decreases in exact concert as the (potential) energy of gravitation
increases, so that there is a net flow of energy from the matter/radiation
degrees of freedom to to gravitational field's energy. This version of a
well-defined energy is defined with respect to the flat asymptotic
background metric and the localized gravitational effects (and their
corresponding energy) are measured in terms of the local deviation from
the flat background geometry. *But*, it is thought that this geometric
situation does *not* describe our universe. Rather, our universe is
thought to follow a Hubble expansion following some variety of a
Friedmann-Robertson-Walker metric. Such global geometric situations
(where the spacetime is globally time-dependent and not asympotically
flat) make any accounting of the total energy of the universe much more
problematic.

I know we've discussed this before but I must reapply the question. There
would seem to be a lack of research in this area. Certainly a global general
loss of energy has effects on the local scale!

Yes, locally the energy density decreases. If you defined a small local
region of space which participates in the Hubble expansion (by having
fixed spatial comoving coordinates) then the energy in that region
mysteriously seems to decrease such that energy seems to be destroyed.
If, OTOH, you defined a sufficiently small local region of space which
does not participate in the Hubble expansion, say, by having a fixed
proper volume, then the energy in that region decreases in a
conservative way because all the energy loss in that volume is
accounted for by a flux of energy out through the boundary surface of
the region. As the temperature of the surrroundings is decreased the
flux of photon energy into the region from the outside is less than the
flux of photon energy out of the region, and the region's temperature
continues to decrease to match the falling temperature of the
surroundings.

Merlin
Paramedic, and amateur cosmologist.
Portland, OR

I hope this helps.

David Bowman
dbowman@georgetowncollege.edu