Bilge wrote:

SAL wrote:
2) OK, now something less obvious: I have a large balloon
filled with low-pressure gas, and I weigh it (in a vacuum, of
course). Now, I compress the gas into a small tank. I let it
cool off so it's the same temperature it was in the balloon. I
weigh it again. It weighs more, due to the pressure terms in
the stress-energy tensor ... right? Or does it? (I added
"free energy" when I compressed it, which presumably is where
the extra mass/energy -- if any -- came from.)

Bilge:
Yes. The stress energy tensor is of the form:

T^ab = (\rho + P)U^a U^b + P g^ab

where \rho is the density, P is the pressure, U^a and U^b are
four velocities and g^ab is the metric tensor. Since the initial
and final temperatures are the same and the velocities are
prortional to kT, the initial and final stress-energy tensors
differ only in the second terms. The change in energy is then due
to the additional force required to confine the gas to a smaller
volume.

SAL:
Ah hmm. This is what I think I've read, and it's sure how the
stress energy tensor looks. But it seems to lead to something
unpalatable, which implies that there's something more going on
than I've grokked as yet.

Specifically, this appears to violate conservation of energy.

Bilge:
How so? By your own construction, you added energy to compress the
gas. So, the process isn't adiabatic.

SAL:
If I put a compressor, battery, tank, and gas supply in a sealed
laboratory, and put the whole thing on a spring, then when the
compressor runs and the gas is compressed into the tank, if the
weight (mass/energy) increases then the spring will be compressed.

Bilge:
Why is that? If the laboratory is sealed, then all of the energy that
compresses the gas has to come from inside the laboratory, so
whatever energy is transferred to the gas, comes from something else
in the lab. That's a different question than before where you were
only talking about the gas and adding and subtracting energy.

That's right, it is. In the first case the system was "open"; I'm
attempting to close it so I can understand where the energy is coming from.


I can use the motion of the laboratory as the spring is compressed
to do work. If I then let the gas out of the tank again, the lab
gets lighter, the spring relaxes a bit, and I can use the upward
motion of the laboratory to do more work. The total mass/energy of
the universe has increased.

Again, why would that happen? The energy you release from the gas,
can't go anywhere but inside the laboratory.

Ah. This is the point I'm trying to understand -- if we account for all
the energy, rather than just part of it, does everything balance? A
closed system (a sealed laboratory) is just an attempt to force an
accounting.



That seems wrong -- it seems like there must be a balancing effect
as well.

Now, no mass/energy entered or left the sealed laboratory.
However, the amount of Gibbs free energy in the lab decreased as
the battery's charge was drained off and was converted to heat.

The heat goes into the kinetic energy of the molecules inside the
lab.

Yes, that's right. All mass-energy is retained inside the lab.
However, the overall "free energy" decreased -- it's not conserved and
some was used up running the compressor.


Is _that_ the solution to the conundrum?

The reason I told you to decide what happens at each step was to
avoid doing just what you've done. You've strung together a lot of
processes without analyzing (or even carefully specifying) how each
process takes place.

Yes, you are correct; that's exactly what I've done. Does that actually
make it impossible to draw any conclusions about what happens? Consider:

If this were a problem in classical mechanics, we could close the system
and then say, "Energy and mass must both be conserved within the closed
system!" and we could draw conclusions from that, without knowing
exactly what happened inside the box.

If this were a problem in thermodynamics we could, again, draw
conclusions from the fact that the system was closed, without analyzing
everything which happens inside the system in detail.

But this is a problem in GR and I don't know the rules. So, I'm asking
you: If we close the system, then must the total mass/energy of the
system be conserved? Alternatively, can we state, in advance of
analyzing every process in the closed system, that the spring-balance
weight of the system as a whole must remain constant?

(Come to think of it the system isn't really closed, after all -- energy
can still leak out as gravity waves. But at any rate, nothing's going in.)



3) I have a tank of gas with a membrane down the middle. All
the gas is squeezed into one half of the tank. I rupture the
membrane. The gas expands to fill the tank. The pressure in
the tank drops; the temperature doesn't change, since the gas
did no work while expanding. Does the tank get lighter? (Note
that no energy -- or anything else -- flowed between the tank
and the outside world.)

Yes, it did. Before rupturing the membrane, there were stresses
in the membrane that confined the gas to one region in the tank.
By allowing the pressure to equalize on both sides, you removed
the force acting to confine the gas.

Hmmm.

In this case it seems that we have a flat violation of mass/energy
conservation.

Why is that? You obviously believe there is stored energy in a
compressed (or stretched) spring. That's exactly what your membrane
is.

But !

I've got a sealed tank sitting on a spring balance. Nothing goes into
it, nothing comes out of it. The membrane in the tank pops; still,
nothing goes in or comes out. Yet, the thank becomes lighter -- the
needle on the balance moves.

In the universe as a whole, _something_ just decreased.



But it's a one-time thing so you can't build a perpetual motion
machine out of it: there's no way to put the gas _back_ in the
smaller volume without doing work on it. Is that a fair statement?

Why does that matter?

In other words, you can destroy free energy, and that reduces the
overall mass/energy of the universe. But you can't create more
free energy, so when it's gone, it's gone -- there's no way to run
it in a cycle. Right? Close? Totally off-base?

I think you first need to specify each step of the "experiment" well
enough to answer the question "what happened to the energy".

Perhaps I didn't phrase this properly. In elementary thermodynamics a
half-filled tank of gas with a barrier in it is a standard "gedanken"
item. If the gas is allowed to expand into the whole tank _without_
doing any work, the temperature of the gas doesn't change, and no
(classical) energy enters or leaves the system. That's all I'm trying
to portray.

But when the gas expands, we lose the potential to do work with the gas,
so something does go away. As far as I know, what goes away is "Gibbs
free energy" -- which is exactly the capacity to do work.


I have one additional question to add to the list (one which I
asked of PMB earlier). Since the answers to the others were
surprising, perhaps this will be, too...

We have two particles at relative rest, separated by distance L.
They're held at rest relative to each other by a massless rigid
rod, or

First of all, a massless, rigid rod violates relativity, so using you
shouldn't be surprised if you get a non-sensical answer when using
such a construct.

OK, if the rod must not be massless and totally rigid, make it very
light and give it a very high spring rate. Then it can enter into the
analysis. Does that affect the overall conclusion? (I'm not looking
for a contradiction -- I'm just trying to understand what happens!)

For that matter, I'm also happy to assume the masses are momentarily at
rest. In that case they'll be accelerating, so they'll be emitting
gravitational radiation but I don't think that affects the overall
conclusion.


they're only momentarily at rest, or they're both suspended from
strings -- I don't care which. I measure the distant gravitational
field of the pair.

Now, I move them together to distance L/2, and assure that they're
again at relative rest (stick them to a shorter rod, or do whatever
else is necessary).

The distant gravitational field of the pair of particles should
_decrease_, right? The reason is that the total energy of the
system decreased, due to the loss of potential energy -- to bring
the particles to rest a second time energy had to be removed from
the system. Yes?

OK, this isn't really a relativity question, since potential energy
is not really a relativistic idea.

I beg your pardon!

It certainly is a relativity question! I'm describing a situation and
asking what relativity predicts would happen to the gravitational field.
I used some phrases from classical mechanics in an effort to describe
the situation clearly and to try to see if I understand the principles
involved, but I'm not asking about classical physics here. I know
perfectly well that the far field of two masses in Newtonian mechanics
doesn't vary as the masses move together or move apart. My question is
what GR predicts happens.

If the description I gave is inadequate to answer the question, then I'd
like to clean up the description and try again. So, what else is needed
to come to a conclusion?



In newtonian mechanics, lowering
the potential energy means making the value a
_larger_negative_number.

Yes, I know that; it's not what the question concerned. The question
concerned GR. I have read that in GR the "potential energy" also
contributes to the gravitational field. So, I'm asking you -- in this
case, does the far field of two masses increase, decrease, or stay the
same as they move closer together, and their classical potential energy
drops?

All my talk of rigid rods was just an attempt to make it clear that as
they move closer together, I'm pulling kinetic energy out of the system,
so in classical terms _just_ the potential energy changed.




First cousin to these issues is a much messier question: Does the
distant gravitational field of a star increase, decrease, or stay
the same when it collapses into a black hole? But that's a
question I'm happy to leave for next year...

If all of the mass falls into the black hole, then the distant
gravitational field doesn't change.

Well! This is certainly consistent with a decrease in the far field in
the much simpler case I tried to describe above.

--
To email me directly, take out nospam and put back foobox.