# Lorentz contraction of box filled with gas

• Xeinstein

#### Xeinstein

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:

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?

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:
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 w 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.

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

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)

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 stress-energy tensor. For introductory pedagogical purposes, I think the simpler treatment is all that is necessary.

Last edited:
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 stress-energy tensor. For introductory pedagogical purposes, I think the simpler treatment is all that is necessary.

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).

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).

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:
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:
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.

Correct, both kev and I have shown you this , mathematically.

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,

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:

Pr'=Pr

Please review the mathematics posted by kev and I, they both show where you are going wrong in your reasoning.

Last edited:
This compression is resisted by pressure, and the energy required to compress the gas has to come from somewhere.

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.

Measuring the pressure in some other frame is possible, but would require a detailed discussion of the stress-energy tensor.

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.

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.

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:
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).
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.

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.

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

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...

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

Doc,

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:
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...

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:
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.

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

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

\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. ;)

\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.

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:
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.

It is imaginary from the point of view of an observed measurement outside the frame. It doesn't really contract physically.

It is imaginary from the point of view of an observed measurement outside the frame. It doesn't really contract physically.

More correctly said:

-the modern view is that the contraction is not physical, it is just a geometric (trigonemetric) artifact of the Lorentz-Einstein transforms : http://en.wikipedia.org/wiki/Length_contraction#A_trigonometric_effect.3F

-we do not have any experimental confirmation to the contrary :
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Length_Contraction

More correctly said:

-the modern view is that the contraction is not physical, it is just a geometric (trigonemetric) artifact of the Lorentz-Einstein transforms : http://en.wikipedia.org/wiki/Length_contraction#A_trigonometric_effect.3F

-we do not have any experimental confirmation to the contrary :
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Length_Contraction

If the contraction is "Not physical", then can you tell us why the thin thread connected between two spaceships break in Bell's spaceship paradox?

If the contraction is "Not physical", then can you tell us why the thin thread connected between two spaceships break in Bell's spaceship paradox?

Firstly, "Bell's paradox" is a thought experiment, not a real one, so it has no bearing on my statement pertaining to the absence of experimental confirmation for length contraction.

Secondly, here is a very good explanation. The calculations do not use length contraction.

More correctly said:

-the modern view is that the contraction is not physical, it is just a geometric (trigonemetric) artifact of the Lorentz-Einstein transforms

The "modern view" surely is the common sense view anyway. Why would anyone think there was a physical contraction unless they put the 'proof' of mathematical equations above common sense.

Firstly, "Bell's paradox" is a thought experiment, not a real one, so it has no bearing on my statement pertaining to the absence of experimental confirmation for length contraction.
While I agree that there's no direct evidence for length contraction (kind of hard to set up those experiments!), there is a ton of evidence for special relativity, and length contraction is a necessary consequence of SR.

Secondly, here is a very good explanation. The calculations do not use length contraction.
That site uses space-time diagrams, which include not only length contraction but simultaneity and time dilation as well. And length contraction is most certainly mentioned. There's no escaping it.

The "modern view" surely is the common sense view anyway. Why would anyone think there was a physical contraction unless they put the 'proof' of mathematical equations above common sense.
The "modern view" is that length contraction is quite "real" (albeit a kinematic/geometric effect of space-time), as is time dilation and the relativity of simultaneity. This is not the first time that "common sense" has been put to shame by careful mathematical argument and--more importantly--experimental evidence.

While I agree that there's no direct evidence for length contraction (kind of hard to set up those experiments!), there is a ton of evidence for special relativity, and length contraction is a necessary consequence of SR.

I think we are arguing about semantics: we agree on both of the above sentences.
At issue is whether length contraction has a physical manifestation. To date, none has been shown via experiment.

That site uses space-time diagrams, which include not only length contraction but simultaneity and time dilation as well. And length contraction is most certainly mentioned. There's no escaping it.

Please follow the math carefully, it uses relativity of simultaneity only. The article is an aggregate (like all wiki articles), this is why it is important to follow the math and to ignore the surrounding text. The equations do not use length contraction.

Last edited:
The "modern view" is that length contraction is quite "real" (albeit a kinematic/geometric effect of space-time), as is time dilation and the relativity of simultaneity. This is not the first time that "common sense" has been put to shame by careful mathematical argument and--more importantly--experimental evidence.

No one denies that length contraction is "real".
At issue is that no physical manifestation has ever been confirmed experimentally. It is indeed strage that there are so many experiments that test SR and none of them concerns itself with length contraction. http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Length_Contraction

I don't get it. On the one hand you say that we agree, and yet on the other hand you seem to think that somehow there's an "explanation" of the Bell spaceship paradox that doesn't imply length contraction. The distance between the ships (once they reach final speed) is $L_0 \gamma$, but in the Earth frame it remains $L_0$. That's length contraction, all right.

I don't get it. On the one hand you say that we agree, and yet on the other hand you seem to think that somehow there's an "explanation" of the Bell spaceship paradox that doesn't imply length contraction. The distance between the ships (once they reach final speed) is $L_0 \gamma$, but in the Earth frame it remains $L_0$. That's length contraction, all right.

There are many,many explanations of the "Bell paradox". Some include length contraction. You are using one of the explanations (the more simplistic one). Have a close look at the wiki detailed explanation, as I pointed out, it does not use length contraction. As you can see, the calculations prove that it isn't the string that contracted but the distance between the rockets that increased resulting into stretching the string.
Length contraction is not intrinsic for the explanation of Bell's paradox.

Last edited:
The "modern view" is that length contraction is quite "real" (albeit a kinematic/geometric effect of space-time), as is time dilation and the relativity of simultaneity. This is not the first time that "common sense" has been put to shame by careful mathematical argument and--more importantly--experimental evidence.

But not "real" in a physical sense. How can it honestly shrink physically simply because of its speed... What is measured or observed from a different frame is not the reality of the physical object.

It also wouldn't be the first time that "careful mathematical argument" had been put to shame because of blind faith over common sense. Show me an experiment where it is proved that an object physically contracts with an increase in speed as measured in the frame of the object itself...

There are many,many explanations of the "Bell paradox". Some include length contraction. You are using one of the explanations (the more simplistic one).
Oh really? What explanation am I using?
Have a close look at the wiki detailed explanation, as I pointed out, it does not use length contraction.
The explanation given is the standard one that I use. The key--as in most relativity "paradoxes"--is the relativity of simultaneity.
As you can see, the calculations prove that it isn't the string that contracted but the distance between the rockets that increased resulting into stretching the string.
Length contraction is not intrinsic for the explanation of Bell's paradox.
Yes, the distance between the rockets does increase, which is what breaks the string. This is the same thing I've said several times when describing this "paradox". But length contraction applies here--as always. The calculations on that site--the very same ones I would use--apply the Lorentz transformations to figure out that the distance between the rockets in their own frame is $L_0 \gamma$. Knock knock... who's there? Lorentz contraction, as always.

If you are arguing against some strange idea where just moving past a string magically reaches out and puts stress on it: I agree, that's pretty silly. But if you agree with that wiki site, which uses the Lorentz transformations (which imply length contraction and all the rest), then you must conclude that length "really" is contracted.

But not "real" in a physical sense. How can it honestly shrink physically simply because of its speed... What is measured or observed from a different frame is not the reality of the physical object.

It also wouldn't be the first time that "careful mathematical argument" had been put to shame because of blind faith over common sense. Show me an experiment where it is proved that an object physically contracts with an increase in speed as measured in the frame of the object itself...

"Lorentz Contraction of Flux Quanta Observed in Experiments with Annular Josephson Tunnel Junctions", A. Laub et al., Phys. Rev. Lett. 75, 1372 - 1375 (1995).

Zz.

Show me an experiment where it is proved that an object physically contracts with an increase in speed as measured in the frame of the object itself...
I guess it's pretty clear that you have no idea what length contraction (much less the subtler issue of the relativity of simultaneity) is all about.