Pressure Vessel Length Contraction and Pressure Comparison in a Spaceship

[Ross Arden]

I have 2 identical pressure vessels P1 and P2, exactly 1000 cc each. I pressurise them both to exactly $1000 lbs/in ^2$. I place one on a spaceship and keep the other beside me. The ship blasts off.

As the ship whizzes past me, as a result of length contraction, I see that the vessel on the spaceship is no longer 1000cc, but some value less than 1000cc, as its length has contracted but its width remains the same. Therefore I must conclude that the pressure in the vessel on the spaceship is higher than the pressure in the vessel at rest wrt to me...agree?

If I remotely measure the dimensions of the vessel on the spaceship I can figure out precisely what the pressure is in the spaceship vessel

 

[Orodruin]

As the ship whizzes past me, as a result of length contraction, I see that the vessel on the spaceship is no longer 1000cc, but some value less than 1000cc, as its length has contracted but its width remains the same. Therefore I must conclude that the pressure in the vessel on the spaceship is higher than the pressure in the vessel at rest wrt to me...agree?

Disagree. You are treating pressure as a scalar with a formula that is only valid in the rest frame when it really should be treated as part of the stress-energy tensor.

 

[PeterDonis]

I must conclude that the pressure in the vessel on the spaceship is higher than the pressure in the vessel at rest wrt to me...agree?

Yes. What you are seeing here is a portion of the relativistic transformation of the stress-energy tensor of the vessel. The other portion is that the energy density in the vessel also increases. IIRC the energy density and pressure both go up like $\gamma^2$; roughly, one factor of $\gamma$ comes from the increase in speed, the other comes from length contraction. (Mathematically, the two factors of $\gamma$ are there because the SET is a two-index tensor, so Lorentz transforming it requires two transformations, one for each index.)

 

[PeterDonis]

Disagree.

Qualitatively it is true that the pressure goes up, though, correct?

 

[Ross Arden]

what in gods name is a stress energy tensor?

 

[Ross Arden]

Yes. What you are seeing here is a portion of the relativistic transformation of the stress-energy tensor of the vessel. The other portion is that the energy density in the vessel also increases. IIRC the energy density and pressure both go up like $\gamma^2$; roughly, one factor of $\gamma$ comes from the increase in speed, the other comes from length contraction. (Mathematically, the two factors of $\gamma$ are there because the SET is a two-index tensor, so Lorentz transforming it requires two transformations, one for each index.)

OK so you are saying the energy density increases as does the pressure ? makes sense as $E = MC^2$, the number of atoms of gas per unit volume increases so the energy density must increase

 

[Ross Arden]

OK so you are saying the energy density increases as does the pressure ? makes sense as $E = MC^2$, the number of atoms of gas per unit volume increases so the energy density must increase

"one factor comes from the increase in speed" - what does that mean ?

 

[Orodruin]

Qualitatively it is true that the pressure goes up, though, correct?

It depends on what meaning you put into ”pressure”. Treating the gas as an ideal fluid, there is a Lorentz scalar pressure that is equal to the rest frame pressure, i.e., the metric portion of $T^{\mu\nu} = (\rho_0 + p) U^\mu U^\nu - p \eta^{\mu\nu}$.

Just looking at the stress part of the stress-energy tensor, it will no longer be isotropic in a frame that is not the rest frame. Taking the isotropic part (i.e., the part of the symmetric representation of rotations that is a singlet - essentially the trace), I do believe it will increase, yes, but not for the reasons that the OP imagines.

 

[Ross Arden]

It depends on what meaning you put into ”pressure”. Treating the gas as an ideal fluid, there is a Lorentz scalar pressure that is equal to the rest frame pressure, i.e., the metric portion of $T^{\mu\nu} = (\rho_0 + p) U^\mu U^\nu - p \eta^{\mu\nu}$.

Just looking at the stress part of the stress-energy tensor, it will no longer be isotropic in a frame that is not the rest frame. Taking the isotropic part (i.e., the part of the symmetric representation of rotations that is a singlet - essentially the trace), I do believe it will increase, yes, but not for the reasons that the OP imagines.

you have 1 billion atoms of oxygen in 1000cc...if you put 1 billion atoms of oxygen in a volume less than 1000cc,Ceteris paribus, the pressure must be higher? sorry at room temperature and 1 atmosphere

I thought pressure in relation to gas meant the number of gas molecule for a given volume?

 

[Orodruin]

what in gods name is a stress energy tensor?

It is an object with components that describe the energy density, momentum densities, and stresses (out of which pressure is a part) of a matter distribution. When you switch reference frame (ie, do a Lorentz transformation) all of those components mix I am a way similar to how ghe Lorentz transformation mixes space and time. In other words, as an example, the energy density in the new frame generally depends on all of these quantities in the original frame.

In general relativity, it is this object that is the source of gravitation, not just mass (although mass contributes to the components).

 

[Orodruin]

you have 1 billion atoms of oxygen in 1000cc...if you put 1 billion atoms of oxygen in a volume less than 1000cc,Ceteris paribus, the pressure must be higher? sorry at room temperature and 1 atmosphere

No, you are wrong. Again, you are trying to use a relation that is only valid in the rest frame of an object to an object moving at relativistic speeds.

 

[Orodruin]

I thought pressure in relation to gas meant the number of gas molecule for a given volume?

No. Pressure (a type of stress) is related to force, ie, momentum transfer, per area.

In the rest frame of the gas it will relate to the number of molecules, the volume, and the temperature according to the ideal gas law. For a more general fluid you may have a different equation of state.

 

[Ross Arden]

It is an object with components that describe the energy density, momentum densities, and stresses (out of which pressure is a part) of a matter distribution. When you switch reference frame (ie, do a Lorentz transformation) all of those components mix I am a way similar to how ghe Lorentz transformation mixes space and time. In other words, as an example, the energy density in the new frame generally depends on all of these quantities in the original frame.

In general relativity, it is this object that is the source of gravitation, not just mass (although mass contributes to the components).

sorry "pressure" is how often, and the speed at which, gas molecules collide with the walls of a container

 

[Ross Arden]

No, you are wrong. Again, you are trying to use a relation that is only valid in the rest frame of an object to an object moving at relativistic speeds.

so are you saying in a moving frame you can have the same amount of gas, in a smaller volume, and the pressure will go down?

pressure being the number and velocity of collisions of gas molecules with the walls of a container

or are you saying pressure will no longer be evenly distributed across the walls of the container? The "pressure " on the leading edge will be less than the pressure on the trailing edge

 

[Orodruin]

An important part of post #9 is the incariant pressure I talked about. It describes a fundamental property of the gas and will tell you how someone at rest with respect to the gas will find the surrounding pressure. Also note that this is nothing strange - in the rest frame of the space ship, the spaceship cylinder is not length contracted. It is a common misconception that an object’s properties (mass, proper length, etc) somehow change when it is in motion - they do not. The object remains just the same. Things like length contraction are based on how space and time relate to each other.

 

[Orodruin]

so are you saying in a moving frame you can have the same amount of gas, in a smaller volume, and the pressure will go down?

I am sorry, but your view is too simplistic. In order to really explain what is actually going on would require concepts that are at least I level. At B level I can only tell you that what you learned about pressure and gases in the gas rest frame simply does not generalise in the way you suggest to the relativistic case.

 

[Ross Arden]

An important part of post #9 is the incariant pressure I talked about. It describes a fundamental property of the gas and will tell you how someone at rest with respect to the gas will find the surrounding pressure. Also note that this is nothing strange - in the rest frame of the space ship, the spaceship cylinder is not length contracted. It is a common misconception that an object’s properties (mass, proper length, etc) somehow change when it is in motion - they do not. The object remains just the same. Things like length contraction are based on how space and time relate to each other.

is it possible to conduct an experiment to determine if length contraction was actual or not?

 

[Ross Arden]

I am sorry, but your view is too simplistic. In order to really explain what is actually going on would require concepts that are at least I level. At B level I can only tell you that what you learned about pressure and gases in the gas rest frame simply does not generalise in the way you suggest to the relativistic case.

Im happy with a yes no answer

 

[Orodruin]

is it possible to conduct an experiment to determine if length contraction was actual or not?

It cannot be ”actual” in the way you imagine it. That would directly violate one of the the fundamental assumptions behind special relativity, the special principle of relativity, as you would be able to define and determine absolute motion. Note that this principle is also embedded in classical mechanics, the new thing in SR is that the speed of light is invariant.

For this reason, length contraction (and time dilation) must be reciprocal, ie, A is length contracted in the rest frame of B and B is length contracted in the rest frame of A.

Length contraction is real. It is just not what you are imagining that it is and it does not work in the way you are imagining that it does. In order to understand it properly, you would need to go deeper into the theory and study it at advanced undergraduate level.

 

[Orodruin]

Im happy with a yes no answer

And I am telling you that such an answer would be open to interpretation and just lead to giving you new misconceptions. Do you want to build an understanding based on your own misconceptions obtained by stringing pieces together in a way that seems convenient to you or do you want to understand what the theory actually says?

That you do not understand the theory does not mean that factors are randomly thrown around. SR rests on a very solid framework with countless verified predictions. Nothing is anywhere close to what you want to convey

 

[Ross Arden]

And I am telling you that such an answer would be open to interpretation and just lead to giving you new misconceptions. Do you want to build an understanding based on your own misconceptions obtained by stringing pieces together in a way that seems convenient to you or do you want to understand what the theory actually says?That you do not understand the theory does not mean that factors are randomly thrown around. SR rests on a very solid framework with countless verified predictions. Nothing is anywhere close to what you want to convey.

well some ppl are saying length contraction is real and some are saying its not, its analogous to a measurement error. Obviously if it is not real then it can be ignored

if a spaceship whizzes past me has it actually contracted ie the distance between the atoms that make up the spaceship decreased, or not, or the distance between the quarks that make up the atoms decreased, or not. "Length" being the average distance between atoms that make up the space ship? or we don't know?

there are 4 choices
yes it has contracted
no it has not contracted
we don't know
it depends

 

[Orodruin]

well some ppl are saying length contraction is real and some are saying its not, its analogous to a measurement error. Obviously if it is not real then it can be ignored

”Some people” is not a valid reference. Please link to these statements. It is also likely that the statements have all been true, but that your incomplete understanding of SR have led you to conclude that they are contradictory.

Will an object be shorter than its proper length in a frame where it is moving? Yes. This is what length contraction means.

Does this mean that the proper length of the object changes when it is moving? No. The object remains the same and looks no different in its rest frame just because it happens to move at relativistic speeds relative to some arbitrary observer.

An object’s length is not the same in all frames.

 

[Ross Arden]

”Some people” is not a valid reference. Please link to these statements. It is also likely that the statements have all been true, but that your incomplete understanding of SR have led you to conclude that they are contradictory.

Will an object be shorter than its proper length in a frame where it is moving? Yes. This is what length contraction means.

Does this mean that the proper length of the object changes when it is moving? No. The object remains the same and looks no different in its rest frame just because it happens to move at relativistic speeds relative to some arbitrary observer.

An object’s length is not the same in all frames.

so the answer is we don't know as there is no experiment that can be carried out to determine it if actually contracts, as opposed to "appearing" to contract as a result of the fact that we have nothing faster than the speed of light to measure with. It is an artifact of the measuring equipment. For instance lest say in the future physicists discover a previously unknown phenomena that travels 100 trillion times faster than light, would SR would cease to exist?

The fact that it appears to be all different lengths shouldn't we be blaming the equipment rather than assuming space and time somehow warp

 

[Orodruin]

so the answer is we don't know as there is no experiment that can be carried out to determine it if actually contracts, as opposed to "appearing" to contract as a result of the fact that we have nothing faster than the speed of light to measure with.

No, this is an incorrect representation of what I said. I said that ”actual” contraction as in an object getting a shorter proper length would fundamentally violate not only a basic principle of SR but also classical mechanics. Having access to things faster than light to measure with has nothing to do with it. Signal delay due to a finite speed of light is easily taken into account.

You are currently moving (and hence length contracted) at very close to the speed of light relative to a cosmic proton hitting the atmosphere. Do you feel any different just because I told you that?

It is an artifact of the measuring equipment. For instance lest say in the future physicists discover a previously unknown phenomena that travels 100 trillion times faster than light, would SR would cease to exist?

No. This assertion is just absurd. Please learn the actual theory and the predictions it makes (this does not mean to learn about the theory through popular science presentations) if you want to be able to draw your own accurate conclusions about what it implies.

 

[Ross Arden]

No, this is an incorrect representation of what I said. I said that ”actual” contraction as in an object getting a shorter proper length would fundamentally violate not only a basic principle of SR but also classical mechanics. .

so what you are saying here is your measuring equipment tells you it contracted, but in reality it didnt? as a result of the limitations of the measuring equipment

 

[PeterDonis]

Just looking at the stress part of the stress-energy tensor, it will no longer be isotropic in a frame that is not the rest frame.

Ah, of course. So the idea that length contraction makes the pressure go up is at best a heuristic; it can be a good thought starter but it won't give you a correct analysis once you dig into the details.

 

[PeterDonis]

so what you are saying here is your measuring equipment tells you it contracted, but in reality it didnt? as a result of the limitations of the measuring equipment

There is no point in continuing this discussion if you are going to continue to misunderstand and misrepresent what we say.

As @Orodruin commented earlier, this topic is really not a "B" level topic; it requires at least an "I" level background to be able to go into details. The best "B" level answer that can be given is what has already been said: there is a tensor called the stress-energy tensor that represents energy, momentum, and stress; the components of this tensor do change when you change frames; but the change cannot be simply described as "pressure goes up because of length contraction"; it's more complicated than that, and the complications require an "I" level discussion.

Thread closed.