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Lorentz contraction of box filled with gas

  1. Jan 23, 2008 #1
    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?
    Last edited: Jan 23, 2008
  2. jcsd
  3. Jan 23, 2008 #2
    The pressure inside the box does not increase. I will try and explain why. Pressure is defined as force divided by area. Force is defined as m*a = m*dv/dt = dp/dt where dp is change in momentum. If we have a box with n particles, each of mass m, conveniently bouncing straight up and down with velocity w and colliding with top of the box every t seconds then the total force of the particles colliding with the top of the box n*m*w/t. When the box is moving from left to right with velocity v with respect to us, the transverse component of the particles velocities is reduced by gamma (v) and the tranverse mass of each particle increases by a factor of of gamma(v). The time interval t also increases by gamma(v) from our point of view so that overall the force acting on the top of the box is n*(my)*(w/y)/(t*y) = (n*m*w/t)/y where y is gamma(v) or 1/sqrt(1-v^2/c^2). So the overall force is reduced by gamma. Since pressure is force divided by area and the surface area of the top of the box is also reduced by gamma (due to length contraction) the pressure is the same from our point of view as it is to to an observer that is stationary with respect to the box. You can do a similar analysis for the sides of the box and arrive at the same conclusion.
    Last edited: Jan 24, 2008
  4. Jan 24, 2008 #3
    Very nice, kev

    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


    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:


    In the observer frame , assuming the crossection is:


    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


    p'=gamma*m_0*w'=gamma*m_0*w/gamma=m_0*w=p! (no real surprise here, it is quite intuitive)



    dt=gamma* dtau (time dilation) so dtau/dt=1/gamma so:


    Pr'=F'/A'=F/A=Pr (Q.E.D)
  5. Jan 24, 2008 #4


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    Staff Emeritus
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    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


    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 stress-energy tensor. For introductory pedagogical purposes, I think the simpler treatment is all that is necessary.
    Last edited: Jan 24, 2008
  6. Jan 24, 2008 #5
    You are right , of course.It is still nice to have a formal proof, especially in the context of the question by OP. While the kinetic energy of the gas is frame dependent, its pressure is not (it is a constant in all frames).
  7. Jan 24, 2008 #6
    Yes, the proof is interesting and along the way demonstates that tranverse force is reduced by gamma (which is sometimes questioned) and that tranverse mass is increased by gamma. If we analyse the pressure on the sides of the box we find that force parallel to the motion of the box is invarient and that longitudinal inertial mass behaves as if it has increased by gamma^3. The concept of different tranverse and longitudinal inertial masses for the same object is unpleasant and this is usually wrapped up in a momentum term.

    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.
    Last edited: Jan 24, 2008
  8. Jan 27, 2008 #7
    In the lab-frame (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
    Last edited: Jan 27, 2008
  9. Jan 27, 2008 #8
    Correct, both kev and I have shown you this , mathematically.

    Incorrect: both kev and I have shown you that the speed of the gas molecules decreases. See w'=w/gamma.
    This results into F'=F/gamma and that results, in turn, into:


    Please review the mathematics posted by kev and I, they both show where you are going wrong in your reasoning.
    Last edited: Jan 27, 2008
  10. Jan 27, 2008 #9


    Staff: Mentor

    The other posters have already demonstrated the mistake here quite well, but I would encourage you to think about this further.

    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 Lorentz-contraction does not cause material stress (pressure) in general.
  11. Jan 27, 2008 #10

    Just to take this discussion on a more interesting tack: how would you use the stress-energy tensor in order to do the calculations for a steel rod. I'd love to see the equations.
  12. Jan 27, 2008 #11
    Excellent point. In other words, the distance between atoms does not decrease in the direction of motion. I asked pervect if he could show the equations (similar to the ones I showed for the gas-filled box). This would be very interesting. Can you show them? (I don't know how and I would like to learn).
    Last edited: Jan 27, 2008
  13. Jan 27, 2008 #12


    Staff: Mentor

    No, the distance between the atoms does decrease in the direction of motion, but in a way that does not stress the material.

    I don't have any equations for you, but here is a hand-waving analysis. Since relativity is based on EM phenomenon you know that the EM field around an isolated atom will length-contract 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 length-contracts 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.
  14. Jan 27, 2008 #13
    Sorry, this is indeed armwaving :-)
    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.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Length_Contraction
    Last edited: Jan 27, 2008
  15. Jan 28, 2008 #14

    Doc Al

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    Staff: Mentor

    You seriously think that length can be contracted without the distance between atoms decreasing? Or did you mean something else?
  16. Jan 28, 2008 #15
    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...
  17. Jan 28, 2008 #16

    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.
    Last edited: Jan 28, 2008
  18. Jan 28, 2008 #17
    The explanation for the famous null result of the Michelson Morley interferometer experiment is due to the length contraction of the arm parallel to the direction the interferometer is moving. I am curious if you think the length contraction of the parallel arm is is real or imaginary. I hope you agree there is something intrinsically unsatifactory about basing physcs on imaginary phenomena.

    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.
    Last edited: Jan 28, 2008
  19. Jan 28, 2008 #18
    Actually the stock SR explanation is that light speed is isotropic, as such, in the lab frame the fringe displacement is :

    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/Specia...l_analysis_of_the_Michelson_Morley_Experiment
  20. Jan 28, 2008 #19
    \yes, time dilation and aberation are required too but length contraction is an intrinsic part of the explanation for an observer moving wrt the lab. Of course in the MM experiment the lab and the earth are comoving and length contraction is not required in that reference frame. In SR we are entitled to treat our reference frame as stationary and so we can imagine the Earth is stationary and that the sun, our galaxy and the rest of the universe is rotating around the Earth. ;)
  21. Jan 28, 2008 #20
    So is aberation in the moving frame, otherwise the light beam would appear to miss the mirror at the end of the interferometer arm. All 3 effects (length contraction,time dilation and aberation) are equally essential in the explanation. Of all three, only length contraction has no experiment associated with it.
    I think that the modern view of these effects is that they are all projective artifacts when moving transferring from frame to frame.http://en.wikipedia.org/wiki/Length_contraction#A_trigonometric_effect.3F
    Last edited: Jan 28, 2008
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