# Does pressure resist acceleration?

1. Feb 7, 2008

### Xeinstein

Suppose we have a box at rest that is filled with a uniform gas. We denote the volume by V and the pressure by p. Suppose next that we apply a small force to the box and accelerate it until it has a speed v. Once it is at speed v, the gas in the box has acquired a kinetic energy, so one might think that the total energy that we had to add to the box in order to accelerate the gas in it would have been equal to its kinetic energy. But this is not the whole story, because the Lorentz contraction has shortened the length of the box and therefore changed its volume. Making a box smaller when it contains a gas with pressure requires one to do work on it, in other words to put some energy into the gas. This extra energy represents the extra inertia of the gas.

The key question is: Is it harder to accelerate the gas because it takes work not only to accelerate the existing energy but also to compress the gas as the Lorentz contraction demands? In other words, the moving box will contract but the gas in it will resist the Lorentz-contraction of the box

Last edited: Feb 7, 2008
2. Feb 7, 2008

### country boy

Didn't you just do this one?

On 23 Jan 08, "Lorentz contraction of box filled with gas."

3. Feb 7, 2008

### Staff: Mentor

In its own frame, the box has not shrunk.

4. Feb 7, 2008

### pervect

Staff Emeritus
There are not any issues related to the Lorentz contraction of the volume of the box. This was addressed in the previous thread, and that issue should continue to be discussed in that thread.

The idea of "resisting acceleration" is essentially the idea of mass. The short answer is that pressure does not affect the mass (which is how I interpret the concept "resisting acceleration") of an isolated system of a box containing a pressurized gas.

Using the SR defintion of mass, pressure does NOT contribute to mass, ever. In some GR applications, however, pressure does contribute to mass. There are actually several concepts of mass in GR, one of the most relevant here is the concept of Komar mass. Note that the term "mass" actually has several different meanings! (We have the SR defintion, and we have several different GR definitions). This is not widely appreciated and occasaionally causes confusion, angst, and long arguments.

Is the SR mass of a box containing a pressurized gas different than the GR Komar mass? The answer to this question is no. The reason that the answer is no is that while the pressure in the interior of the box does contribute to the Komar mass, making it higher, the tension in the walls of the box also contributes to the Komar mass, making it lower. The net effect is that there is no change in the mass of the box due to the pressure terms.

This assumes that the box is an isolated system. If the box is not an isolated system, the answer gets more complicated.

5. Feb 7, 2008

### Xeinstein

That one has been hijacked
The question is this: The moving box will contract, but how about the gas, will the gas in it contract also? Will the gas resist the contraction of the box?

6. Feb 8, 2008

### 1effect

The box does not become smaller in the frame comoving with it, so there is no work expanded in compressing the gas.
The total energy for the system box+gas at rest is

$$E_0=m_0c^2$$

where $$m_0$$ is the proper mass of the box+gas.
The total energy when the system has stopped accelerating and has reached cruising speed $$v$$ is :

$$E_1=\gamma m_0c^2$$

If the system is accelerated slowly, there is no energy radiation, hence the same value $$m_0$$ is used in both expressions. This was also pointed out earlier by pervect.

The total work is:

$$W=E_1-E_0=(\gamma-1)m_0c^2$$

Last edited: Feb 8, 2008
7. Feb 8, 2008

### pervect

Staff Emeritus
I gave my answer, a good one, in #4 of that thread:

https://www.physicsforums.com/showpost.php?p=1583385&postcount=4

The short answer is that the moving box does not contract in its own frame, and that relativity is frame independent.

A fuller answer can be found in

I get the impression that the simple answer is not satisfying you, and you either missed the longer answer above in the flurry, or that it is going over your head.

Perhaps we need to take a step back.

Do you understand "frame independence"?

Do you understand that the box does not contract in its own frame? Do you understand the logic of working the problem in the simplest possible frame, and then applying the above principle of "frame independence" to get the correct answer in other frames?

I mentioned "invariant mass" before. Do you understand that concept, and its relevance to the answer to your question?

I have the feeling that communication just isn't happening here, and I'm afraid I don't know how to fix that.

8. Feb 8, 2008

### Xeinstein

Finally, I figure out what's wrong with your argument.
I can give you a hint: you jump around in different frames
In both the initial lab-frame and final co-moving frame, the box is at rest,
Then can we say the total work is zero? No, of course not
So it's clear that we must stay in one single frame, i.e. the initial lab-frame in which the box is moving at end of acceleration. In this frame, the moving box does indeed contract and compress the gas in it

Last edited: Feb 9, 2008
9. Feb 8, 2008

### 1effect

No, everything is calculated from the frame the box was originally at rest. So , you continue to write nonsense.

Not really, the box gets accelerated up to speed $$v$$ in the lab frame, as per your description. So, this is the second nonsense.

Exactly as I did.

Now, you are indeed mixing frames. Some moderator ought to close this thread. Pervect?

10. Feb 9, 2008

### Xeinstein

I don't think you understand relativity at all and continue to write nonsense.
It's clear from the similar thread, titled "Lorentz contraction of box filled with gas", that you are wrong about Lorentz-contraction. Your biggest problem is that somehow, you believe Lorentz-contraction is an illusion, Not real. In fact, most people in the forum of that thread disagree with you and you couldn't explain, in the lab-frame, why the thin thread between two spaceships breaks in the Bell spaceship paradox.

I think it's obvious that you jump around in different frames, from the initial lab-frame to the final co-moving frame. You said that the box does not become smaller in the frame co-moving with it, I think that's well-known and we all agree with it. But you can't extrapolate that and claim Lorentz-contraction of moving object is Not physically real. So the question is: does the box become smaller in the lab-frame (in which the box is moving)? If yes, will it compress the gas in it? you can find the answer in Schutz's book: Gravity from the Ground Up, at page 194; This is the kind question that really test your understanding of Lorentz-contraction.

Would you be surprised if I tell you that Schutz disagree with you and claim "pressure does resist acceleration"? Since Schutz is such a big shot in relativity, I don't think you have a chance....

In his book "Gravity from the Ground Up", page 194, in the box of 15.4 Investigation: How pressure resists acceleration" It says: it is harder to accelerate the gas because it takes work not only to accelerate the existing energy but also to compress the gas as the Lorentz-contraction demands.

Here is the link to that page in the book: Gravity from the Ground Up
I hope you can learn something about Lorentz-contraction from Schutz's book

Last edited: Feb 9, 2008
11. Feb 9, 2008

### 1effect

I think that anybody that knows physics realized already that both $$E_0$$ and $$E_1$$ are calculated in the same frame. I cannot help you if you don't know elementary relativity.

The box does not become smaller. A moving observer will measure it thru light signals as being smaller.

Yes, I read the book. It is interesting to note afew things:

-what you did not know, and this is really interesting for the mainstream people watching this thread, is that Schutz made a mistake on the page you are citing (p.194). It happens to the very best, even to famous professors :-)

Can you spot the mistake?

12. Feb 11, 2008

### Xeinstein

Do you mean that the length of the box is "frame independent"? It's the same in all frames? I think only the space-time interval is "frame-independent", but the distance or length is Not.
Yes, I do understand "frame independence" and the box does not contract in "its own frame". I think that's well-known and we all agree with. So that's not the question.
The question is this: Does the box contract in "the lab-frame" in which the box is moving? If yes, will it compress the gas in it? That's all we need to know.

Last edited: Feb 11, 2008
13. Feb 11, 2008

### 1effect

The answers are "no" and "no".

14. Feb 11, 2008

### pervect

Staff Emeritus
The length of the box depends on who observes it. Length is therefor not a frame-independent quantity.

Whether or not you consider length to be "real" depends on your philosophy. "Real" is a very vague term. "Independent of the observer" is a much more precise term. When you say "real" I think you probably mean "independent of the observer", but it's hard to be sure. If you mean something else by *real*, please give a short example of what you mean by "real", and some examples. (Are bricks real? How do you know they are real). Please try not to get too far into the philosphy in this forum, however - but sometimes a little philosphical disussion is unavoidable in order to answer non-philoophical questions.

The length of a physical object is thus not a property of the object alone. It is depends both on the object, and on the observer - specifically on the frame in which the observer resides.

For an example of a property of a physical object that is a property of the object alone, consider the Lorentz interval, or proper length, of the object.

Now, onto energy, mass, and momentum.

The box also has energy, momentum, and mass. The energy and momentum of the box depend on the observer. This is the same in relativity as it is in Newtonian mechanics - the kinetic energy of an object depends both on the object and who observers it.

In relativity, the energy and momentum together comprise the energy-momentum 4-vector. The energy-momentum 4-vector, however, is not a property of the box alone - like length, it depends on both the box and the observer. (But it transforms in standard and well understood manner, making it very useful and fundamental).

The invariant mass of the box, however, is a quantity which depends only on the box, *if* we assume that the box is an isolated system (as it is in this example). This invariant mass can be computed from the energy momentum 4-vector. To make the equations simple, chose units such that c=1. Then the invariant mass of the box is E^2 - |p|^2, where E is the energy of the box, and |p| is its momentum

I haven't responded to the rest of your post, because you're getting off on the wrong track :-(.

Last edited: Feb 11, 2008
15. Feb 11, 2008

### marcus

Does pressure resist acceleration?

Pressure contributes inertia. Inertia resists acceleration.

However this does not have much to do with the Lorentz contraction and that box example. The best thing to do, I think, is to forget about Lorentz contraction, and the box of gas, and special relativity

I think other people have pointed out that the Lorentz contraction does not change the pressure measured in the box. So this line of reasoning seems fruitless.

Instead, think of the sun. I believe that part of the gravitational attraction of the sun, which the earth feels, is due to the pressure in the core (as a simple consequence of the GR equation). Part of it is due to the kinetic energy associated with the temperature at the core. Besides the obvious contribution of particle masses, the gravitational mass of the sun is compounded of several things, including pressure.

The gravitational mass and the inertia of the sun are the same. Therefore a portion of the inertial mass of the sun is contributed by the pressure-----the pressure everywhere in the sun contributes somewhat, but I mentioned the core pressure especially because I suspect it is high enough to have a noticeable effect.

It would be fun to know the percentage of the mass of the sun which is attributable to internal pressure. Maybe someone more knowledgeable than I has a source for this. Please correct me if I am wrong. I'm just reasoning from the Einstein field equation of GR, which in effect features both energy density and pressure on the right hand side.

Last edited: Feb 11, 2008
16. Feb 12, 2008

### Xeinstein

I don't think it has anything to do with pressure change or not. If you check Schutz's calculation in his book, the pressure p is a constant but the volume of the gas does change (page 194 in his book: Gravity from the Ground Up). So this is not the question

Hello Marcus,

So we both agree "pressure resists acceleration", but how?
Since you are one of the knowledgeable "Astro/Cosmo Guru", so let me ask you an honest question, The question is this: Does the box contract "in the lab-frame" in which the box is moving? If yes, will it compress the gas in it?
Please Note: If both the box and the gas contract at the same time, then there is No compression. If somehow the box contracts but the gas does Not, then the box has to compress the gas in it. In other words, will the gas resist the contraction of the box, as Lorentz-contraction demands. That's all we need to know.

Last edited: Feb 12, 2008
17. Feb 12, 2008

### Xeinstein

>>> because you're getting off on the wrong track
Why? Just because I asked you an honest question.

Last edited: Feb 13, 2008
18. Feb 12, 2008

### pervect

Staff Emeritus
It seems to me that you are pushing your own point of view, which as nearly as I can tell is incorrect, rather than asking questions and trying to understand the answers.

I really *am* trying to get you on the right track, right now in my opinion you are going off on a non-productive tangent and basically "not getting it".

Some of the issues involved here are subtle. So let's start with some of the issues that are not.

What I'd ideally like for you to come away with from this thread is an understanding of the concept of invariant mass in special relativity (and if not that much, at least a desire to learn more about it), and how this concept relates to "resistance to acceleration".

If we could accomplish that much, it might be time to get into the GR issues, which are also present.

19. Feb 14, 2008

### Xeinstein

Hello russ_wallers,

We all agree that "in its own frame", the box has not shrunk.
So that's Not the question. The question is this: Does the box contract "in the lab-frame" in which the box is moving? If yes, will it compress the gas in it? In other words, will the gas resist the contraction of the moving box, as Lorentz-contraction demands? That's all we need to know, thanks....

Last edited: Feb 14, 2008
20. Feb 14, 2008

### 1effect

"No" and "no". How many times do u need to ask the same question?

21. Feb 14, 2008

### matheinste

Hello all.

I have been following this thread and in general i am out of my depth but it does throw up a question which seems relevant to me.

The box is contracted in the "moving" frame as observed from the "stationary" frame. Are the sizes of molecules of the gas and their seperation distances also affected by the relative motion.

At a more fundamental level is everything affected dimensionally by the relative motion.

Matheinste.

22. Feb 14, 2008

### 1effect

No, they are not. Uniform relative motion does not affect either the size of the molecules, nor the separation distances between molecules. The box does not contract, either in the proper frame , nor in the observer frame, so the pressure remains constant.

The above is not true for the general case of non-uniform (accelerated) motion. In this situation, the outcome depends heavily on the way of accelerating the objects.

Last edited: Feb 14, 2008
23. Feb 14, 2008

### Staff: Mentor

You seem to have a misconception about the nature of the Lorentz contraction. You seem to be thinking of it as an active process, as opposed to a transformation of measurements between frames. Example: Imagine a can of beer sitting on the table. A rocket ship goes by at high speed and--amazingly enough--observers on the rocket are able to perform measurements of the beer can as they fly by. Is the can and its contents Lorentz contracted? Of course! Does that somehow increase the pressure in the can? Do you expect anything unusual to happen to the can of beer? Do you think that pressure can somehow increase so as to burst the can as viewed in the frame of the rocket? I'm being a bit facetious, but I hope you see the point.

Of course, subtle things can happen when you accelerate an object. (See all the discussions about the Bell spaceships.) But your first sentence said "We all agree that "in its own frame", the box has not shrunk." That implies that you have accelerated the box in such a way as to preserve its proper length (as opposed to stretching or squashing it), thus you have introduced no stresses on the box or its contents whatsoever.

24. Feb 14, 2008

### Staff: Mentor

Of course. As measured from the moving frame.

As measured from the "stationary" frame, the separation distances between molecules will be contracted.

Last edited: Feb 14, 2008
25. Feb 14, 2008

### 1effect

Sure, as a measurement process, not as an active process (see your previous post to Xeinstein).
Interestingly enough, this is the error in the Schutz text, Schutz takes the contraction as an active proces when he calculates the mechanical work $$p\Delta V$$ resulting from such contraction. This type of contraction, due to measurement effects, cannot be responsible for any mechanical work. Subtle are the ways of relativity.