Much later, after erg and Oz have gotten the Wiz to stop talking about the test, and many hours of fascinating conversation have taken them past sunset, the moon is beginning to rise. The view from the cave entrance is spectacular.
Some smoke from the campfire drifts in - and assumes a vaguely human form.
"Damn," says Oz! "It's vaguely human!"
"Vaguely human!" agrees G. Wiz. "It must be a sci.physics reader. I wonder what it wants? What's your name, smoke?"
"Bhv," says the smoke.
"Bhv?" asks Oz. " That sounds like the noise my straw mattress makes when I crash down on it after a long tiring day of sweeping the floor. Curious name! "
"Sorry I couldn't make it in person", says bhv, "My sending will have to do. Besides, in this form, the worst thing that can happen to me is being dispelled - which is much less stressful than experiencing in person lightning bolts and fireballs and whatnot that seem to fly around here!"
"The worst thing, eh?" asks the wizard, who knows a lot about flaming. "Have no fear, ask away."
"Anyway, I have some ideas. It seems to me that pressure should cause things to expand - if you have a ball of a perfect gas, and it has a positive pressure, it should want to expand, right?"
"And gravity should want to make things contract!" The wizard nods, "Ah, well, looking ahead into the future I sense a forthcoming proliferation of possibilities for sign errors, but I feel pretty sure that pressure creates gravity which makes things want to CONTRACT. Of course, pressure makes gases such as yourself expand, for the usual reasons; that has nothing to do with general relativity. But the reason why black holes must form under certain circumstances is that, no matter how much a star tries to resist collapse with pressure, that pressure merely generates more gravity, which makes it collapse all the more. Ironic, ain't it?"
"Now if we finally decided that
R00 = T00 - .5 Tcc ...
Drat - I need a corrected version of the classnotes. Why G. Wiz likes those dratted Rab and Tab I'll never understand, he should stick to Rab and Tab if you ask me. Not that he will, of course. Use the dratted gab for the dratted kronecker delta function if need be! Oh well, if I was a powerful wizard like G. Wiz, I might not listen to a ball of smoke, either."
"I listen to all sorts of hot air, here on sci.physics" notes the Wiz. "In your present form, you fit right in."
The smokes swirls about in amusement and says, "Anyway this means that R00 = .5(T00 - T11 - T22 - T33) so that R00 = .5(density - 3*pressure) if we have a perfect fluid."
"And d2V/dt2 was proportional to -R00. Which means we've just proved that positive density causes things to contract and positive pressure causes them to expand!" Oz hears a buzzing swarm of minus signs in the distance, heading for the campfire.
"This is a nice start, but I was hoping that pressure would contribute to the energy somehow. I sure can't see why a sufficient ppressure can't make R00 negative (which indicates expansion) with this formulation!" With a buzzing roar, the swarm of signs reaches the campfire. Oz and the Wiz can barely hear over the noise of the signs. They swat this way and that, to no avail.
"Oh well, back to the drawing board. Maintaining this sending isn't free, I'll stop back in a bit to see what happens".
The puff of smoke dissipates. But alas, the minus signs he appears to have brought with him remain.
"Hmmm - I don't see why R00 can't be negative either", said Oz. Do you think that R00 was negative for that puff of smoke, erg?"
"Ack!" says the wizard, swatting madly. "It's these damn signs. It's impossible to think when they're around."
"Let's just wait" said Oz, "until the smoke returns. After all, if it really wants to know the answer, it'll come back. Why waste our time fighting these damn signs?"
"Gee whiz!" said G. Wiz. "A great idea."
So they hung out by the campfire and talked about other subjects until the signs, seeing there would be no more food, buzzed off. Eventually, hours later, the smoke returned!
"I hate those signs too", said the smoke. "I've found that raising and lowering indices serves as a breeding ground for them, which is why I suggest this process be avoided whenever possible!! That's why I recommend using Tab, Gab and Rab"
"Well," said the wizard, "that's fine as long as you don't ever need to use Rab, but typically you do. One needs to battle signs occasionally in this business... perhaps your method would help mimimize it, but not eliminate it."
"I am now getting:
Tab =
-E 0 0 0 0 P 0 0 0 0 P 0 0 0 0 Pi.e. T00 = -E, Tii = P (pressure) for i=1,2,3."
The wizard said "Good, that's what I get."
"Did you really work it out," asked the smoke, "or are you just taking my word for it? I have a feeling you're not all that eager to do this sort of calculation."
The wizard smiled. "You're right. But I think this is consistent with what I said about the stress-energy tensor earlier. "
The smoked approximated a nod. "Furthermore, I find that R = Rcc = -Tcc
i.e. R= -(T00 + T11 + T22 + T33) = E-3P"
"Yes, that's what I got too," said G. Wiz.
Since Rab = Tab + .5 R gab . (gab = 1 iff a=b, 0 otherwise)
R00 = .5^(T00 - T11 - T22 - T33) = .5(E+3P)
"This indicates that P makes things contract! (Or at least go in the same direction as E.) This seems weird, but" the cloud turns slightly pinkish, "we are talking about d2 V/dt2, not dV/dt. It's obvious that dV/dt is positive when a high pressure area is in empty space, but that says nothing about d2V/dt2"
"Wait a minute!" said G. Wiz. "Your formula for R00 agrees with the stuff in the course outline, and you are right that it means the gravitational effect of pressure is to cause things to contract. In layman's lingo: like energy, pressure causes an attractive gravitational force."
Oz added "I thought you said gravity wasn't considered a `force' in general relativity."
"It's not," admitted the wizard, "that's why I said, `in layman's lingo'."
"BUT," added G. Wiz, "the formula relating R00 to d2V/dt2 only applies to a ball of particle following geodesics! That is, particles in free fall, feeling no force other than gravity (which, ahem, is not really a force either). In reality, when we have a gas with different pressures at different regions, the atoms of the gas *do* feel forces other than gravity. So pressure does have the usual effect of making stuff want to expand, in addition to the special GR effect of making it want to contract. Just don't mix them up; they are very different."
"Something else that is interesting" says bhv. "If we take a simple particle with no pressure term, the stress energy tensor is"
T00 = -E
all other terms are zero."
"If we take this and boost half of it in the -x direction at some velocity v, (energy E/2) and boost half of it in the +x direction at the same velocity v so that the average momentum/velocity is zero and add, we get
T00 = -gamma2E
T11 = v2gamma2E
where gamma = 1/sqrt(1-v2) and c is assumed to be 1, as usual around here."
"The other terms all cancel out." The cloud swirls as it attempts to cross its nonexistent fingers as it makes this remark.
"This gives us some insight into the pressure term, the above anisotropic case has a positive pressure in the x direction."
"It seems odd at first that T00 scales as gamma2, but it's energy / volume, not energy, and volume is definitely not Lorentz invariant!"
"Hmm," said the wizard. "Interesting. I'll have to ponder that."