Hold all of your horses, everyone! Does MM still think that the Casimir force asymptotes out at one atmosphere (10^5 N/m^2)? If so, who cares whether he thinks this arises from a negative pressure or a positive pressure? Who cares whether he thinks it does or does not represent a vacuum energy density? If you're arguing with a Kennedy assassination theorist who thinks that Kennedy died in Ford's Theater while watching "Our American Cousin", you don't start by explaining the detailed bolt mechanics of the Carcano rifle.
Let's start over, MM.
Anyway. That's negative pressure for you. Discuss.
Let's start over, MM.
- You want to model the Casimir force as an external "gas". OK, that's a hypothesis we can work with. Pick an equation of state and we'll go with it.
- Calculate the energy density of your gas: pick a small box of it and calculate the integral of P*dV like you would for any gas. You get some number in energy/volume units. Good so far.
- Now that you have an *energy density*, you have an alternative way of calculating the pressure: use that energy density to define a Hamiltonian, and now the pressure has to be the generalized force which corresponds to "volume" as the generalized coordinate. Try this out on your box of ideal gas and you'll see that it works. This is a very good definition of pressure.
- There's one thing about your hypothetical gas which makes it inappropriate for the real-world Casimir force: its pressure decreases as the gas expands. If you filled the Universe with your hypothetical gas, then let this gas "drive" your hypothetical Casimir-like force, you'll find that the Casimir force was twice as strong 1 Gy ago (since the Hubble flow indicates that that gas was packed into a smaller volume)---and this hypothetical Casimir force will get weaker in the future as the Hubble flow makes your gas expand further. So the "naive" gas hypothesis is wrong, but maybe we've learned something that will let us generate a new hypothesis.
- Your new hypothesis is an ideal gas with P = constant or perhaps n/V = constant---it basically tries to keep the Casimir force equation independent of the Hubble Constant. Do whatever you need to to make this new equation-of-state work. But you will write down an equation-of-state for this new hypothesis.
- Take this new, Casimir-like equation of state and plug it through the Hamiltonian definition of pressure. What do you get? Negative pressure. Uh oh.
- The easiest way out is to argue that this must be the *wrong* definition of pressure and it's just a mathematical trick. But you will look into a GR textbook and find that, no, the Hamiltonian generalized-pressure is the *only* way that pressures ever find their way into GR.
- The next way out is to argue that maybe the Casimir force isn't constant in time. But then you're talking about something with no relation to the QED *vacuum* calculations which yield that force. You can certainly hypothesize about some new particle-force which isn't constant in expanding space, but not here.
Anyway. That's negative pressure for you. Discuss.
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