Lorentzian contraction of a box increases the pressure inside?

[Messenger]

Lorentzian contraction of a box increases the pressure inside?

I have been doing some research on gravity, and I am having a very hard time with the following example I found in a book entitled Gravity From the Ground Up by Bernard Schutz. It sounds like a 3rd grader explaining Lorentz-Fitzgerald contractions...

"The inertia of pressure can be traced to the Lorentz-Fitzgerald contraction. In Investigation 15.4 on the next page we show how to calculate the extra inertia, but even without much algebra it is not hard to see why the effect is there. Consider what happens when we accelerate a box filled with gas. We have to expend a certain amount of energy to accelerate the box, to create and maintain the force of acceleration. In Newtonian mechanics, this energy goes into the kinetic energy of the box; as its speed increases so does its kinetic energy. This happens in relativity too, of course, but in addition we have to spend some extra energy because the box contracts.
The Lorentz-Fitzgerald contraction is inevitable; the faster the box goes, the shorter it gets. But this shortening does not come for free. The box is filled with gas, and if we shorten the box we reduce the volume occupied by the gas. This compression is resisted by pressure, and the energy required to compress the gas has to come from somewhere. It can only come from the energy exerted by the applied force. This means the force has to be larger (for the same increase in speed) than it would be in Newtonian mechanics, and this in turn means that the box has a higher inertia, by an amount proportional to the pressure in the box."

I just really am not sure what to make of this statement. What the hell is so special about the box that the gas doesn't also have? I had assumed the contractions were either a dimensional change in space-time, or at the very least Lorentz's idea that it was a reduction between atoms. I have also found where the author states that the units of the cosmological constant is 1/sec^2, which the consensus elsewhere is that it is 1/length^2.

This guy knows more about cosmology than I ever will, so this logic really throws me.
Any thoughts on this, cited examples would help greatly.On second thought...maybe his example is just very poor wording. He is trying to link this pressure increase into how neutron stars work, but I think he would have been better off stating that the pressure within the walls of the box as well as the gaseous pressure increase. Thoughts still welcome on this.

 

[Passionflower]

I think Bernard Schutz is very knowledgeable but here I think he is simply wrong.

 

[Barakn]

Wouldn't time dilation come into play here? The observer in an inertial frame of reference would see the velocity of the gas particles slow down as the box accelerates away, i.e. the temperature decreases. This decrease in temperature would exactly match the decrease in volume, leaving the pressure constant. But then again, I am not Bernard Schutz.

 

[Bill_K]

Sure sounds fishy. Could you tell us what he does in "Investigation 15.4"?

 

[Elroch]

There is only more force needed due to pressure if the pressure at one end of the box is different to the pressure at the other end. Since with an accelerating box, the clocks at the two ends run at different speeds, this is not implausible.

I was looking at a quite similar situation with a perfectly reflective box filled with light and, in this instance, it seems obvious that if it is accelerated the pressure on the back end is greater than the front end (at constant speed the pressure at both ends is the same). I interpret this as inertia.

 

[Dale]

So the stress-energy tensor of an ideal gas with density $\rho$ and pressure p is:
$$\left( \begin{array}{cccc} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{array} \right)$$

Boosting that by a Lorentz transformation tensor gives (someone please check my math):
$$\left( \begin{array}{cccc} \frac{p v^2+\rho }{1-v^2} & \frac{v (p+\rho )}{v^2-1} & 0 & 0 \\ \frac{v (p+\rho )}{v^2-1} & \frac{p+\rho v^2}{1-v^2} & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{array} \right)$$

So to me it looks like pressure goes [STRIKE]down[/STRIKE] up. (See EDIT below)

Note, that this is the pressure as measured by the non-comoving observer, which is kind of a conceptually weird idea anyway. The comoving observer, of course, never measures any change in pressure.

EDIT: Oops, I caught a mistake in my code. I am not certain this is correct, but it looks better because it is symmetric now. It looks like the pressure does go up. My previous, incorrect, results were asymmetric and had the pressure going down.

 

[Elroch]

Without a Tex viewer, this is not very easy to read. What browser or add-on displays this correctly?

 

[Messenger]

Investigation 15.4 How pressure resists acceleration

We shall look at this only for slow motions, where the effects of special relativity are small. Suppose we have a box at rest that is filled with a uniform gas. We denote the volume by V, the mass density by $\rho$, and the pressure by p. Suppose next that apply a small force to the box and accelerate it until it has a speed v that is small compared to c. They key question is, how much energy did we have to put into get the gas up to speed v? For simplicity, we will only ask about the gas, not about the container: in astronomy we usually don't have containers: one part of the gas of a star is held in place by gravity and the pressure of the other parts of the gas.
Once it is at speed v, the gas in the box has acquired a kinetic energy, so one might think that the total energy that we had to add to the box in order to accelerate the gas in it would have been equal to this kinetic energy, $\frac{1}{2}$mv$^{2}$=$\frac{1}{2}$$\rho$Vv$^{2}$, where in the second expression we have used the fact that the mass m in the box is $\rho$V. But this is not the whole story, because the Lorentz-Fitzgerald contraction has shortened the length of the box and therefore changed its volume. Making a box smaller when it contains a fluid with pressure p requires one to do work on it, in other words to put some extra energy into the gas. This extra energy represents the extra inertia of the gas: it is harder to accelerate the gas because it takes work not only to accelerate the existing energy but also to compress the gas as the Lorentz-Fitzgerald contraction demands.
We only need to work out this extra energy in order to see why. The energy one has to put into it is just -p$\Delta$V, where we denote the change in volume by $\Delta$V;the minus sign is needed so that when the box contracts ($\Delta$V negative) then the energy put into the box is positive/ Using Equation 15.9 on page 192 to get the change in volume, we find that the extra energy we put in is $\frac{1}{2}$$\frac{v^{2}}{c^{2}}$pV. (to be continued)

 

[Messenger]

Investigation 15.4 continued

This energy does not just disappear; it goes into the internal energy of the gas in one form or another, depending on the details of the gas molecules. At least some of the energy goes into raising the temperature of the gas (the random kinetic energy of the molecules).
The total energy required to accelerate the gas-filled box can be written in a simple way:

E=$\frac{1}{2}$m$v^{2}$-p$\Delta$V
=$\frac{1}{2}$$\rho$V$v^{2}$+$\frac{1}{2}$$\frac{v^{2}}{c^{2}}$pV
=$\frac{1}{2}$($\rho$+$\frac{p}{c^{2}}$)$v^{2}$V.

The last expression is the one we need to examine. The energy required to accelerate the box is proportional to the sum $\rho$+$\frac{p}{c^{2}}$. This energy comes from the work done by the force we must use to accelerate the box, so the force had to be larger than might have expected. Put another way, for a given applied force, the box accelerates less than we would have expected by measuring its mass, since some of the energy we put in goes into the internal energy of the gas instead of the kinetic energy of the box. Scientists therefore say that the inertia of the box is larger than just its rest-mass, and in particular they call the quantity $\rho$+$\frac{p}{c^{2}}$ the inertial mass density of the gas. If we want to known how much force is required to accelerate a fluid we have to know the inertial mass density, not just the rest-mass density. This is purely a consequence of special relativity.

From me: I am using just standard Chrome to view and Tex seems to work fine for me.

 

[Dale]

Without a Tex viewer, this is not very easy to read. What browser or add-on displays this correctly?

I think all you need to do is enable javascript. I don't think that any add on is required.

 

[jtbell]

Yes, enabling Javascript should be all you need to do, assuming of course your browser handles Javascript properly. If your browser is really old, that might make a difference. Any further problems are probably best discussed in "Forum Feedback & Announcements".

 

[PAllen]

This may be really simplistic, but if I wanted to motivate 'inertia of pressure', I would simply start with observation from kinetic theory of gas that pressure*volume is proportional to kinetic energy of the gas molecules. Then, in a frame where the gas COM is at rest (thus net momentum is zero), invariant mass is:

(<rest energy of gas molecules> + KE of gas) / c^2 =

(RE + <constant> * pressure * volume) / c^2

Thus, pressure contributes to inertial mass of the system. Further, since invariant mass is invariant, in a boost, RE can't change, the total above can't change, so if volume decreases, pressure must increase.

[EDIT: The above has an error. After a boost, the invariant mass would be:

[ (RE + <total gas KE>) - <net gas momentum> ^2 *c^2 ] / c^2

So, to preserve the invariant, total gas KE would have to be more than rest pressure scaled by volume decrease. ]

 

[Elroch]

Yes, enabling Javascript should be all you need to do, assuming of course your browser handles Javascript properly. If your browser is really old, that might make a difference. Any further problems are probably best discussed in "Forum Feedback & Announcements".

I use the NoScripts add-on for security, but had explicitly set it to allow PhysicsForums.com. It turns out one has to allow mathjax.com as well.

Thanks. Problem solved.

 

[my_wan]

What the hell is so special about the box that the gas doesn't also have?

You are right that there is nothing special about the difference. So let's look at the same effect in terms of the kinetic energy of a mass rather than the gas pressure. Others here have come close to saying this but I think it's worth putting it in a more intuitive format.

Two spaceship doing a significant portion of the speed of light with respect to each other suffer engine failures. Hence both are dead in the water (uh.. space). Hence both are at rest in their respective frames, but on a collision course with each other neither can do anything about. Now each spaceships occupants calculates that the other spaceship is packing a whopping kinetic energy. Which spaceship really has the kinetic energy? The question makes no sense because kinetic energy only exist a relational quantity which is a property that neither ship alone posses. You can relate this energy to the degree which each ship sees the length of the other shortened, but the Lorentz contracted gas canister doesn't see itself contracted, it sees that observer claiming it is Lorentz contracted as the one that is Lorentz contracted.

If you tried to say this kinetic energy was shared equally between both it doesn't work either. It can't be twice the energy each calculates the other to have, and if you divide it between them there's is too much energy to account for the impact energy. This is due additions of velocities, which when divided gets two velocities that is greater than half the original velocity.

Hence to say that Lorentz contraction is the "cause" of the increased pressure is a misnomer. We are not even talking about a quantity that has any meaning to or property possessed by the gas box by itself. To the box of gas it it not the gas that is under increased pressure, rather it is the observer claiming the gas box has increased pressure that is Lorentz contracted. That the "inertia of pressure can be traced to the Lorentz-Fitzgerald contraction" is not a causal claim that the Lorentz-Fitzgerald contraction "caused" the pressure increase, only that it a necessary observational consequence of the fact that the relative velocity exist, but does not exist for either system alone.

In terms of "causal" disputes in this kind of claim it nearly always come down to which spaceship is really moving, which is as silly as arguing which direction in space is really up. Silly Chinese pointing up and saying down.

 

[Messenger]

Thanks everyone for your replies. I was just very worried that there was something easy I was missing. As he gets paid for understanding this it would be foolish not to give him the benefit of the doubt. I have emailed the author asking for further explanation or cited evidence that using this reasoning provides correct answers.