# B Pressure vessel

1. Dec 13, 2017

### 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 space ship 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 space ship 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 space ship 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 space ship I can figure out precisely what the pressure is in the space ship vessel

2. Dec 13, 2017

### Orodruin

Staff Emeritus
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.

3. Dec 13, 2017

### Staff: Mentor

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.)

4. Dec 13, 2017

### Staff: Mentor

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

5. Dec 13, 2017

### Ross Arden

what in gods name is a stress energy tensor?

6. Dec 13, 2017

### 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

7. Dec 13, 2017

### Ross Arden

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

8. Dec 13, 2017

### Orodruin

Staff Emeritus
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.

9. Dec 13, 2017

### Ross Arden

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?

10. Dec 13, 2017

### Orodruin

Staff Emeritus
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 im 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).

11. Dec 13, 2017

### Orodruin

Staff Emeritus
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.

12. Dec 13, 2017

### Orodruin

Staff Emeritus
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.

13. Dec 13, 2017

### Ross Arden

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

14. Dec 13, 2017

### Ross Arden

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

15. Dec 13, 2017

### Orodruin

Staff Emeritus
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 space ship 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 eachother.

16. Dec 13, 2017

### Orodruin

Staff Emeritus
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.

17. Dec 13, 2017

### Ross Arden

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

18. Dec 13, 2017

### Ross Arden

Im happy with a yes no answer

19. Dec 13, 2017

### Orodruin

Staff Emeritus
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.

Last edited by a moderator: Dec 13, 2017
20. Dec 13, 2017

### Orodruin

Staff Emeritus
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

Last edited by a moderator: Dec 13, 2017