Defining a rest frame in the real world

[analyst5]

Hey guys, as we know the concept of the rest frame is one of the most famous concepts in any kind of relativity, because it must be known in which frame the body is at rest.

My question is how do we define a rest frame on a solid object that has atoms vibrating, the body clearly does not have motion as a whole, it remains in place in some kind of way, but its atoms are vibrating and it isn't at rest in a different way. So what is different between standard analogy in SR between frames that are mutually at rest, and in real life when bodies are much more complex? What are the criteria?

 

[phinds]

Hey guys, as we know the concept of the rest frame is one of the most famous concepts in any kind of relativity, because it must be known in which frame the body is at rest.

My question is how do we define a rest frame on a solid object that has atoms vibrating, the body clearly does not have motion as a whole, it remains in place in some kind of way, but its atoms are vibrating and it isn't at rest in a different way. So what is different between standard analogy in SR between frames that are mutually at rest, and in real life when bodies are much more complex? What are the criteria?

A rest frame is a mathematical construct that does not exist in reality, so your problem isn't a problem. You can pick an XYZ system based however you like and that is your frame of reference. If atoms then jiggle in that frame of reference, then so be it. You could pick a frame of reference based on the the center of the nucleus of one atom and then that atom as a whole would be at rest in that frame of reference but its electrons would not.

For most issues in SR, you're getting too far down in the weeds. Just pick a frame of reference and then assume that the objects you are talking about (a ball, for example) act like a point vis a vie that frame of reference.

 

[jtbell]

If the net external force on an object is zero, its center of mass moves inertially (constant velocity), regardless of the individual motions of its component atoms. The object's "rest frame" is the inertial reference frame in which the center of mass is at rest.

 

[phinds]

If the net external force on an object is zero, its center of mass moves inertially (constant velocity), regardless of the individual motions of its component atoms. The object's "rest frame" is the inertial reference frame in which the center of mass is at rest.

How is that different than what I said?

 

[Dale]

So what is different between standard analogy in SR between frames that are mutually at rest, and in real life when bodies are much more complex? What are the criteria?

As phinds mentioned, rest frames are not part of "the real world", they are part of an analysis. Now, if your analysis assumes that a given object is at rest in an inertial reference frame (e.g. because you have excluded external forces or because you have used an accelerometer to measure the acceleration and it is measured as 0) and in fact there is some residual undetected acceleration, then you can mathematically analyze how much of an impact that error will cause in your final results. If that error is less than the precision of your measuring instruments for your experimental setup then the deviations don't matter and the assumption is valid.

 

[WannabeNewton]

Hey guys, as we know the concept of the rest frame is one of the most famous concepts in any kind of relativity, because it must be known in which frame the body is at rest.

It's only "famous" because bad SR textbooks make it seem way more important than it actually is

Anyways, I seem to recall this being explained to you multiple times in the past. You cannot define a rest frame for a non-rigid body, no matter its state of motion. A body must be rigid (rotating or non-rotating) in order for a rest frame for the entire body to even make sense.

 

[DrStupid]

how do we define a rest frame on a solid object that has atoms vibrating

p=0 and maybe L=0

 

[russ_watters]

As phinds mentioned, rest frames are not part of "the real world", they are part of an analysis.

As an engineer, I'm not sure I agree - maybe it is just a different way of thinking or an issue of theory vs practice:

If an architect designs a house, they have a model that exists only mathematically, in a computer. But when it gets built, you can then point to the real point in space that corresponds to the points in the model. It is diffult for me to conceive of why a real object can't be said to have a real reference frame.

 

[jkl71]

why a real object can't be said to have a real reference frame.

Suppose you have a meter stick and you are in its real rest frame, along with observers on each end of if. Now you accelerate in the direction along it. You’ll observe it to contract. The two ends get closer, from your perspective the observers on each end are in different rest frames, the observers on each end would both say they haven't changed frames and are still in the real rest frame.

To me the basic point is just that picking a reference frame in special relativity is analogous to picking a coordinate system in Newtonian mechanics, there’s nothing special or fundamental about it.

 

[Dale]

As an engineer, I'm not sure I agree - maybe it is just a different way of thinking or an issue of theory vs practice:

If an architect designs a house, they have a model that exists only mathematically, in a computer. But when it gets built, you can then point to the real point in space that corresponds to the points in the model. It is diffult for me to conceive of why a real object can't be said to have a real reference frame.

I see your point, this may be a personal prejudice of mine, but, as you know, a reference frame corresponds to a set of orthonormal basis vectors defined at each event in spacetime. At a whim, the researcher can use a different set of basis vectors and get all of the same physical results.

I do like the blueprint analogy. Many features of a blueprint, such as the size of a window, cannot be changed without changing the "real world" thing that is built. But a blueprint also has features, such as the font, which can be changed without changing the "real world" thing. I would put things like the invariant mass in the "window size" category and things like the reference frame in the "font" category. Whatever you might claim about the "real world" status of the windows in a blueprint, surely you would not claim the same for the font.

 

[analyst5]

@Wannabe Newton, you're right that we've had a discussion regarding this and I apologize if I'm bringing old stuff up, but it's clear that I still have some work to do in understanding.

This was you original post from the last thread:

Well for starters this assumes the extended object has a rest frame to begin with. If the constituents of the object are moving radially relative to one another, such as in an arbitrary fluid, then it doesn't make much sense to talk about the rest frame of the object itself, right? In fact in general the best we can do is talk about the individual rest frames of the constituents of the objects. In a fluid these individual rest frames would correspond to those of the individual fluid elements. When can we talk about the global rest frame of an extended object itself? Precisely when the object is rigid.

But the definition of rigidity in relativity is quite different from the definition of rigidity in Newtonian mechanics. The precise definition requires some mathematical machinery but intuitively rigidity in relativity, often termed Born rigidity, means that given any constituent of the extended object, the spatial distances in the rest frame of this constituent to all neighboring constituents of the object remain constant i.e. there are no relative radial velocities between the neighboring constituents of the object. Note that this definition still allows a rigidly rotating object to have an extended rest frame.

Now coming back to your question, imagine we have a rod at rest in an inertial frame and say we want to accelerate the rod along its length. For the reasons explained above, if we even want to talk about the rest frame of the entire rod itself during the acceleration phase, we better make sure the acceleration is done in a Born rigid manner i.e. the rod must remain rigid during the acceleration phase. Then we can indeed talk about the global rest frame of the rod i.e. an extended frame in which all points of the rod are at rest. In order to do this, each point of the rod must receive a different proper acceleration-in fact we must impart a proper acceleration to each point of the rod inversely proportional to the fixed spatial location of the point in the rest frame of the rod. Now obviously in this case if all points of the rod are accelerated simultaneously in the inertial frame in which the rod is initially at rest then all points of the rod will accelerate simultaneously in the rest frame of the rod by construction*

On the other hand, say we don't accelerate the rod Born rigidly. Say we impart to all points of the rod the same proper acceleration and do so simultaneously in the inertial frame in which the rod is initially at rest. Then in fact it no longer makes sense to talk about the extended rest frame of the rod itself because in the individual rest frames of the points of the rod, the neighboring points will have radial velocities. Furthermore, now if we go to the rest frame of any given point of the rod, then relative to this frame the other points of the rod will have actually accelerated at different times i.e. even though all points were accelerated simultaneously in the initial inertial frame, because we gave all points the same proper acceleration we find that in the rest frame of any given point of the rod the other points will not have accelerated simultaneously. This is in fact the content of the famous Bell spaceship paradox: http://en.wikipedia.org/wiki/Bell_spaceship_paradox

*We have implicitly assumed here that we can actually talk about global Einstein simultaneity in the extended rest frame of the rod. Rigidity is actually not enough to guarantee this. The reason we can make this assumption is due to the irrotationality of the rod motion.

I've red everything you wrote here, and it's a great post to begin with, but it's clear as I understand, that the definition of rigidity in SR is different than in classical mechanics. So you wrote here that we can talk about rest frames of the whole object, except in the case when the object isn't undergoing Born rigid acceleration and accelerates in a different manner.

 

[WannabeNewton]

...but it's clear as I understand, that the definition of rigidity in SR is different than in classical mechanics.

Yes, it is very different indeed. At least within the confines of SR, Born rigid motion is in fact a much more restrictive notion of rigidity than the Newtonian one, particularly in the context of rotational motion. This is because of a rather deep theorem in SR which states that the world-lines of any rotational Born rigid motion are necessarily flow lines of flat space-time isometries (i.e. those transformations which leave invariant the Minkowski metric).

 

[pervect]

If I might suggest, as far as vibration goes, just cool everything down low enough so it stops, then it's not a "real world issue". Then you analyze everything relative to the cold, non-vibrating frame.

I seem to recall that vibration of the atoms affects the rates of atomic clocks, so the most accurate ones need to be cooled, so atomic vibration issue is actually an issue if you want the maximum achieveable accuracy in timekeeping.

 

[analyst5]

Yes, it is very different indeed. At least within the confines of SR, Born rigid motion is in fact a much more restrictive notion of rigidity than the Newtonian one, particularly in the context of rotational motion. This is because of a rather deep theorem in SR which states that the world-lines of any rotational Born rigid motion are necessarily flow lines of flat space-time isometries (i.e. those transformations which leave invariant the Minkowski metric).

I get that WBN, thanks. But during the non-accelerated period of motion of the object, the inertial one, it seems reasonable to define a rest frame for the whole object where there are no radial velocities between its components and all parts move with the same velocity, right? In a restricted sense of rigidity of course.

 

[analyst5]

I mean, in all examples of SR there is a frame associated with the body as a whole, a global rest frame which brings up all the points that are at rest together when the body is undergoing inertial motion and no acceleration is measured. And it seems that a body maintains the distances between its points after each point has accelerated, so that we can define a rest frame. Even the terms proper length and proper time are associated with the rest frame of the object as a whole, but I understand why the object 'loses' its rigidity during acceleration if it's not done in a Born rigid manner. So what is the answer? Some discutants clearly assign a rest frame to the body as a whole, and we can then indeed measure the proper length of the object at least in some phases of its motion (except acceleration). Note that I'm coming to this question from more of a philosophical standpoint, than from a point of measurement, even though of course the measurements are priority and they are the basis of this topic and physics in global. And assigning things like rest frames also.

 

[WannabeNewton]

So what is the answer?

What exactly is the question? As an aside, the answer to your question in post #14 is "yes".

 

[analyst5]

What exactly is the question? As an aside, the answer to your question in post #14 is "yes".

That's enough then, I was confused with your first post on this thread. Thank you very much