Register to reply 
Lorentz contraction of box filled with gas 
Share this thread: 
#1
Jan2308, 06:32 PM

P: 91

Consider what happens when we accelerate a box filled with gas. We have to expend a certain amount of energy to accelerate the box, 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, Do we have to spend some extra energy because the box contracts and its pressure goes up? How does the box know it's moving? 


#2
Jan2308, 11:05 PM

P: 3,967




#3
Jan2408, 11:16 AM

P: 321

I am getting the same result with a slightly improved mathematical formalism. In the frame of the box, the mass of the gass is m_0 and the speed of the molecules is w so, the momentum is p=m_0*w The force exerted by molecules is F=dp/dtau=m_0*dw/dtau where tau is the proper time as measured in the box frame The crossection of the top of the box is A=a*b The pressure in the box frame is: Pr=F/A In the observer frame , assuming the crossection is: A'=a'*b'=a/gamma*b=A/gamma where a is the dimension of the box side moving along the box movement, b is the dimension perpendicular on the movement, gamma=1/sqrt(1(v/c)^2) and v is the box speed wrt the observer. Lorentz transforms say that the molecules move with speed w'=w/gamma p'=gamma*m_0*w'=gamma*m_0*w/gamma=m_0*w=p! (no real surprise here, it is quite intuitive) But: F'=dp'/dt=m_0*dp/dt=m_0*dp/dtau*dtau/dt=F*dtau/dt dt=gamma* dtau (time dilation) so dtau/dt=1/gamma so: F'=F/gamma Pr'=F'/A'=F/A=Pr (Q.E.D) 


#4
Jan2408, 05:42 PM

Emeritus
Sci Advisor
P: 7,633

Lorentz contraction of box filled with gas
There is really not much need for calculation, *if* you measure the pressure in the reference frame of the box itself. In that case, the pressure will not be affected by the velocity of the box. (This should be obvious from the fact that velocity is relative and not absolute).
The box does not contract in its own frame, and the pressure in its own frame does not increase. In fact, the box cannot tell if it is moving or not. If this is not obvious, it might be helpful to watch, for example http://www.onestick.com/relativity/ And note that Al cannot tell if his train is moving, or if he is moving. Measuring the pressure in some other frame is possible, but would require a detailed discussion of the stressenergy tensor. For introductory pedagogical purposes, I think the simpler treatment is all that is necessary. 


#5
Jan2408, 06:22 PM

P: 321




#6
Jan2408, 10:57 PM

P: 3,967

On the subject of kinetic energy of the gas (that both the OP and 1effect alluded to) it is interesting to look at the classical gas law PV/T = P'V'/T'. In the frame moving wrt the box, the box has contracted in volume (V) by gamma but the pressure (P) is unchanged. If the classic gas law holds in the relativistic context then the temperature (T) must have cooled by a factor of gamma. Temperature is classically related to average kinetic energy of the gas particles. This implies a loss of energy. However it is not too surprising if we compare it to a flywheel that is moving with relativistic speed wrt to us. The flywheel has to slow down by a factor of gamma (it is after all a simple form of clock) so the flywheel's angular kinetic energy must have reduced from our point of view. On the other hand the kinetic energy of the box or flywheel due to its linear motion relative to us has increased. Presumably if we factor in the energy used to accelerate the box (or flywheel) and the momentum of particles ejected by a rocket used to accelerate the box, then the overall energy and momentum of the two reference frames is conserved. [EDIT] Perhaps I should add that to an observer in the reference frame of the box, would of course not detect any change in volume, pressure or temperature of the gas. 


#7
Jan2708, 05:30 PM

P: 91

In the labframe (moving wrt the box), the Lorentz contraction is real and inevitable: the faster the box goes, the shorter it gets. But this shorting 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) that 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



#8
Jan2708, 06:46 PM

P: 321

This results into F'=F/gamma and that results, in turn, into: Pr'=Pr Please review the mathematics posted by kev and I, they both show where you are going wrong in your reasoning. 


#9
Jan2708, 07:50 PM

Mentor
P: 17,322

What about a solid rod of steel? Even fairly small changes in length (strain) of a steel bar result in enormous changes in pressure (stress) within the bar. At relativistic speeds the stress and strain would be far beyond the failure point of the steel. Since you cannot have something failing in one frame and being unstressed in another frame then you must come to the conclusion that Lorentzcontraction does not cause material stress (pressure) in general. 


#10
Jan2708, 08:03 PM

P: 321

Just to take this discussion on a more interesting tack: how would you use the stressenergy tensor in order to do the calculations for a steel rod. I'd love to see the equations. 


#11
Jan2708, 08:04 PM

P: 321




#12
Jan2708, 08:25 PM

Mentor
P: 17,322

I don't have any equations for you, but here is a handwaving analysis. Since relativity is based on EM phenomenon you know that the EM field around an isolated atom will lengthcontract as it attains relativistic velocities. The unstressed length of a piece of metal is determined by the spacing of atoms that yields the lowest energy state, which is in turn determined by the fields generated by the atoms. If the field lengthcontracts then the lowest energy state spacing will be correspondingly smaller and the unstressed length will also be correspondingly smaller. Thus you have physical length contraction without any material stress. 


#13
Jan2708, 08:50 PM

P: 321

I cannot parse without some equations to look at :) Let's hope that pervect can come up with the math. BTW, I doubt that the distance between atoms decreases, there is no direct test for length contraction to date: http://www.eduobservatory.org/physi...th_Contraction 


#14
Jan2808, 04:57 AM

Mentor
P: 41,475

You seriously think that length can be contracted without the distance between atoms decreasing? Or did you mean something else?



#15
Jan2808, 06:14 AM

P: 26

if you measure the pressure in the reference frame of the box itself then this doesn't change. What is calculated from any other frame is purely just a calculation and surely incorrect if it doesn't come to the same answer, it's incorrect because of an observed length contraction, not a real one...



#16
Jan2808, 09:42 AM

P: 321

I meant exactly what I wrote. There is no experimental evidence that the distance between atoms contracts, nor is there any evidence that the atoms' radius contracts either. Now, I used length contraction in my detailed post, just as a convenient mathematical tool. 


#17
Jan2808, 10:11 AM

P: 3,967

Time dilation is considered real because we place two clocks that had relative motion alongside each other and see that different times have elapsed on the two clocks. When we place two rulers that had relative motion alongside each other we do not see a difference in length and this leads some people to conclude that length contaction is imaginary. Here is a thought experiment that might demonstrate length contraction is real. Imagine a wheel with a spike on its perimeter. A narrow tape is fed to the wheel at the same speed as a point on the perimeter of the spinning wheel so that a hole is punched in the tape every time the the wheel completes one rotation. When the wheel stops spinning we can directly measure the distance between the holes on the stationary tape. We would find that the holes are spaced at intervals (2*pi*r*gamma) that are greater than the rest perimeter of the wheel. This is because the holes were spaced at intervals of 2*pi*r from our point of view when the tape and wheel were moving. [EDIT} Perhaps I should make it clear that the wheel is spinning but not moving linearly with respect to us. Somebody at rest with the tape when the wheel is spinning would see the wheel as rolling (without slipping) along the tape, which is the statinary "road" in his frame. 


#18
Jan2808, 10:23 AM

P: 321

L/c  L/c =0 Only the explanation in the frames moving wrt the lab use length contraction. Length contraction alone is not sufficient for the explanation: time dilation and the aberation of the light path are also necessary. http://en.wikibooks.org/wiki/Special...ley_Experiment 


Register to reply 
Related Discussions  
Verification of the Lorentz contraction of the space  Special & General Relativity  7  
Lorentz contraction problem.  Introductory Physics Homework  5  
A Lesson on Lorentz Contraction  Introductory Physics Homework  4  
Lorentz Contraction  General Physics  13 