Definition of Frame of Reference

[Chestermiller]

What is your criterion for something being "in your frame of reference"? Is it only objects that are at rest relative to you? That seems like a very restrictive definition, which basically makes the concept of "frame of reference" useless, as ghwellsjr pointed out earlier. But alternatively, if objects that are moving relative to you can still be in your frame of reference, why can't *all* objects be in it?

No. I tried to explain what I'm saying, but it looks like I didn't do a very good job. I realize that if I only paid attention to objects that are at rest relative to myself, spacetime would be a very boring place. What I'm saying is that objects may be flying all over the place in spacetime, and that I would be able to observe and track them, but I only count the objects and coordinate systems that are at rest with respect to myself as part of my frame of reference (and, of course, I am part of their frame of reference). All other objects and coordinate systems occupy other frames of reference that, of course, share spacetime (i.e., overlap) with my own.

 

[PeterDonis]

What I'm saying is that objects may be flying all over the place in spacetime, and that I would be able to observe and track them, but I only count the objects and coordinate systems that are at rest with respect to myself as part of my frame of reference (and, of course, I am part of their frame of reference). All other objects and coordinate systems occupy other frames of reference that, of course, share spacetime (i.e., overlap) with my own.

I'm afraid I still don't understand how this system of yours is supposed to work. Or, rather, I can't tell whether or not it's just the standard system described in non-standard terminology. When you say that all objects "share spacetime" with you, and that you can observe and track them regardless of their states of motion, how is that different from saying they are all in your frame of reference? You can certainly *describe* their motions using coordinates in which you are at rest; and you can therefore compute any physical result you like using those coordinates. What's left over that makes you say the objects are nevertheless not in your frame of reference? It seems to me like that's just a fancy way of saying they are moving relative to you, and nothing else.

 

[PeterDonis]

Instead of saying that "an object is moving in my frame of reference," I would prefer to say that "an object is moving relative to my frame of reference."

This makes me think that your system *is* the standard system, just described in non-standard terminology. To me, both of these statements say the same thing.

 

[Darwin123]

That's already very general. Indeed it does not need to be a material "frame", it can be imaginary; it can refer to a collection of spatial coordinate systems in which no physical object is in rest - compare "ECI frame".
https://en.wikipedia.org/wiki/Earth-centered_inertial

Thus, what is essential is the "R" of "reference", and not physical objects. We only need such physical objects as rulers to define the spatial coordinates, and even so a ruler may be composed of a laser with a mirror and a clock.

However, next you write:

That does not follow from your OP... A collection of coordinate systems is not generally limited in space; just as the ECI frame, they can be of infinite extension. As a matter of fact, standard "inertial frames" are defined like that, so that all objects "reside" in all inertial frames.

Not all objects "in a reference frame" are "part of a reference frame." A reference frame can be considered a set of bodies that are stationary relative to each other at all points of time. Now, a body that is moving relative to these objects can be said to be within the reference frame. However, it is not part of the reference frame.
I include as part of my reference frame a set of identical clocks and a set of identical rulers that are stationary relative to my left eye. I will confine my measurements to these measuring instruments and none other.
There may be similar clocks and rulers moving near the speed of light relative to my left eye. These rulers and clocks are not part of my reference frame. However, they are inside my reference frame in a geometric sense. If measurements are done with these instruments, one can't consider the measurements as having been done in my reference frame. No measuring instrument that moves at high velocity relative to my left eye can be considered part of my reference frame.
If there is no external force acting on my left eye, my reference frame is also an inertial frame. Special relativity is applicable to all measurements made with instruments that are part of the same inertial frame.
All bodies everywhere are "geometrically within" ever frame everywhere. However, a frame is defined in terms of a set of bodies that do not move relative to each other.
An inertial reference frame specifies measuring instruments with a very specific state of motion. None of the instruments are moving relative to each other. None of the instruments are accelerating in a dynamic sense.
There is more than one inertial frame. One inertial frame can be moving with respect to the other. However, every body that is part of the inertial frame is moving as though it is attached to a rigid frame with no force acting on it.
You can reside in a house without being part of it. The pieces of house are stationary relative to each other. If one can move from room to room, then one is not part of the house.

 

[ghwellsjr]

Can you please expound on this statement. How does "my frame of reference" provide any advantage in being able to observe anything over any other frame of reference?

There's nothing particularly special about my own personal frame of reference. Taking as the definition of a frame of reference that given in my original posting just seems to conceptually simply things for me, and I thought it might also simplify things for others. Instead of saying that "an object is moving in my frame of reference," I would prefer to say that "an object is moving relative to my frame of reference." I suppose that it's not such a big deal, but it's more to my personal taste, and, as we said, it shouldn't lead to any errors in analyzing SR problems. Maybe I'm making a mountain out of a molehill.

Chet

You keep changing your definition of a Frame of Reference.

In your first post, you made no mention of it being "your personal frame of reference". Rather, a collection of physical objects at rest with one another constituted a single FoR.

Then in post #16, you added that this single FoR was your own FoR implying that you were one of the physical objects in mutual rest.

Finally in post #36, you disclosed that each of the objects in mutual rest with you has their own FoR.

Added to that is your statement that all the objects that are not at rest with respect to you have their own FoR and your preferred wording in post #35 leads me to the conclusion that you are envisioning a whole bunch of mutually overlapping and (possibly) relatively moving Frames of Reference, one for each object, correct?

Here's what's wrong with your idea: it's too limiting. There are an infinite number of Frames of Reference, all mutually overlapping and possibly moving with respect to one another. Why do you limit the number of them to the number of physical objects? You can use the Lorentz Transformation process to create as many new Frames of Reference based on your own personal FoR as you want, each moving at a slightly different speed with respect to your own personal FoR. And each one of them is just as valid as any of the others. This is a fundamental point of relativity. I emphasize the word one, because you need to use just one FoR when describing a scenario and when analyzing what is happening in that scenario.

Please go back in read my post #3 where I explained all this and see if it makes more sense to you now.

 

[Vandam]

Not all objects "in a reference frame" are "part of a reference frame." A reference frame can be considered a set of bodies that are stationary relative to each other at all points of time. Now, a body that is moving relative to these objects can be said to be within the reference frame. However, it is not part of the reference frame.
I include as part of my reference frame a set of identical clocks and a set of identical rulers that are stationary relative to my left eye. I will confine my measurements to these measuring instruments and none other.
There may be similar clocks and rulers moving near the speed of light relative to my left eye. These rulers and clocks are not part of my reference frame. However, they are inside my reference frame in a geometric sense. If measurements are done with these instruments, one can't consider the measurements as having been done in my reference frame. No measuring instrument that moves at high velocity relative to my left eye can be considered part of my reference frame. ...

Good job, Darwin. I would like to elaborate on this.

It made me think more specifically about the layman's definition of a world (everything that happens simultaneously at a specific moment in time)...
What you call 'in' a reference frame, I would rather call this 'part' of a world.
Let me explain this.

When I first learned about SR I interpreted the reference frame as a 'the world of simultaneous events at a specific time', but strictly spoken that's not correct if one considers that a moving object 'in' that reference frame' is not 'part' of that reference frame. However, the event of the moving object that is 'in', but not 'part' of my reference frame is indeed part of my world of simultaneous events at that moment !

A diagram shows this better:

154271[/ATTACH]"]

Event 'e' (car clock physically on 0,7) is in a green reference frame, but is not part of that reference frame.
However, event 'e' is part of the green world B ! And event 'e' is also part of red world R.

The O,7 car is definitely part of my green world of sim events. And the car clock ticks slow in my world relative to my reference frame. But the car clock is not part of my reference frame. The O,7 car clock is not slower in the green reference frame, but rather relative to my green reference frame.

Of course the car driver will have a similar but different story to tell... ;)

V.

 

[Chestermiller]

I'm afraid I still don't understand how this system of yours is supposed to work. Or, rather, I can't tell whether or not it's just the standard system described in non-standard terminology. When you say that all objects "share spacetime" with you, and that you can observe and track them regardless of their states of motion, how is that different from saying they are all in your frame of reference? You can certainly *describe* their motions using coordinates in which you are at rest; and you can therefore compute any physical result you like using those coordinates. What's left over that makes you say the objects are nevertheless not in your frame of reference? It seems to me like that's just a fancy way of saying they are moving relative to you, and nothing else.

As I said, maybe I've been making a mountain out of a molehill, and all we're really talking about is semantics.

Chet

 

[Chestermiller]

You keep changing your definition of a Frame of Reference.

In your first post, you made no mention of it being "your personal frame of reference". Rather, a collection of physical objects at rest with one another constituted a single FoR.

Then in post #16, you added that this single FoR was your own FoR implying that you were one of the physical objects in mutual rest.

Finally in post #36, you disclosed that each of the objects in mutual rest with you has their own FoR.

My comments in posts #16 and #36 are just examples. My OP is what I really intend to stand by.

Added to that is your statement that all the objects that are not at rest with respect to you have their own FoR and your preferred wording in post #35 leads me to the conclusion that you are envisioning a whole bunch of mutually overlapping and (possibly) relatively moving Frames of Reference, one for each object, correct?

Yes, but not only physical objects; Coordinate systems (which may or may not include physical objects) as well.

Here's what's wrong with your idea: it's too limiting. There are an infinite number of Frames of Reference, all mutually overlapping and possibly moving with respect to one another. Why do you limit the number of them to the number of physical objects? You can use the Lorentz Transformation process to create as many new Frames of Reference based on your own personal FoR as you want, each moving at a slightly different speed with respect to your own personal FoR. And each one of them is just as valid as any of the others. This is a fundamental point of relativity. I emphasize the word one, because you need to use just one FoR when describing a scenario and when analyzing what is happening in that scenario.

I agree with all of this. In my original post, I did include coordinate systems as well as objects.

Please go back in read my post #3 where I explained all this and see if it makes more sense to you now.

In my original post, I guess I should have allowed for coordinate systems that do not contain any physical objects as valid candidates for frames of reference. Is that what you are driving at?

Chet

 

[ghwellsjr]

In my original post, I guess I should have allowed for coordinate systems that do not contain any physical objects as valid candidates for frames of reference. Is that what you are driving at?

Yes and also physical objects that are moving in the reference frame. Here's your original definition:

According to my understanding, a FoR is a collection of physical objects (and possibly 3D spatial coordinate systems) which are all at rest relative to one another.

Note how you said the frame of reference is the collection of physical objects and the 3D coordinate system is optional.

Here's my correction:

A Frame of Reference is, as you say, a 3D spatial coordinate system, along with a 1D temporal coordinate added. Einstein describes this in the first section of his 1905 paper. It's very important to recognize how the time coordinate is established to make the coordinate system 4D. Once you get the concept down, you don't even have to think of the coordinate system as having any physical objects at rest either with one another or with respect to the coordinate system. For example, you could consider a FoR in which one observer/clock is traveling at 0.6c in one direction and another observer/clock is traveling a 0.6c in the other direction.

Note how I said the frame of reference is a 4D coordinate system and the physical objects (moving or stationary) are optional.

I then added:

The other aspect of a FoR is the ability to take any event (the 4 coordinates describing a location at some time) and transform it into a new FoR moving with respect to the first one and get a new set of coordinates. This is done with the Lorentz Transformation process.

An event in SR is a particular set of the four coordinate values in a particular coordinate system. It doesn't matter if there is a physical object or anything happening at that place at that time and there are an infinite number of events along each of the coordinates.

Even when Einstein built the concept of a FoR using rigid rulers connecting synchronized clocks at various locations, he made it clear that these were imaginary (as harrylin pointed out in post #34). If you required physical objects (physical clocks connected by physical rigid rulers) filling the universe to define a FoR, there would be no room for any other physical objects to move around and what about an event that was at the middle of a rigid ruler where there was no clock or an event that was in the middle of a clock? How would you define those events? And think about what happens when you take the coordinates of an event in one FoR and convert them into the coordinates of the same event in a second FoR moving with respect to the first one, the calculation can result in an arbitrary location where there may not be a clock if you follow the physical idea of a FoR.

To restate: a Frame of Reference is defined by Einstein as a 4D coordinate system empty of any physical objects and then you add in whatever physical objects you want and describe their positions and motions in terms of events, precise and exact with infinite resolution. Not only do you not require any "observers" you don't require anything at the origin or anywhere else or at any particular time. You do what you want.

 

[Darwin123]

Yes and also physical objects that are moving in the reference frame. Here's your original definition:

Note how you said the frame of reference is the collection of physical objects and the 3D coordinate system is optional.

Here's my correction:

Note how I said the frame of reference is a 4D coordinate system and the physical objects (moving or stationary) are optional.

I then added:

An event in SR is a particular set of the four coordinate values in a particular coordinate system. It doesn't matter if there is a physical object or anything happening at that place at that time and there are an infinite number of events along each of the coordinates.

Even when Einstein built the concept of a FoR using rigid rulers connecting synchronized clocks at various locations, he made it clear that these were imaginary (as harrylin pointed out in post #34). If you required physical objects (physical clocks connected by physical rigid rulers) filling the universe to define a FoR, there would be no room for any other physical objects to move around and what about an event that was at the middle of a rigid ruler where there was no clock or an event that was in the middle of a clock? How would you define those events? And think about what happens when you take the coordinates of an event in one FoR and convert them into the coordinates of the same event in a second FoR moving with respect to the first one, the calculation can result in an arbitrary location where there may not be a clock if you follow the physical idea of a FoR.

To restate: a Frame of Reference is defined by Einstein as a 4D coordinate system empty of any physical objects and then you add in whatever physical objects you want and describe their positions and motions in terms of events, precise and exact with infinite resolution. Not only do you not require any "observers" you don't require anything at the origin or anywhere else or at any particular time. You do what you want.

At this point, I am starting to lose you. Perhaps I should point out what the experimenter who validates relativity is doing. We have been talking about what the theorist considers a reference frame. What does the experimenter consider a reference frame?
An experimenter working on relativity usually has an array of detectors that determine when certain events occur. Almost always, these detectors have a zero velocity relative to each other. These detectors do not have to been near each other. In fact, they are often far away from each other. However, they are usually stationary with respect to each other.
This set of detectors determines a reference frame. I will refer to this as the first reference frame.
Suppose the experimenter has a second set of detectors that are moving at a a single velocity relative to the first with respect to the detectors in the first reference frame. It is easy to prove that if all these detectors are moving at the same nonzero velocity relative to any detector in the first reference frame, then they are not moving with respect to each other. This second set of detectors determines a second reference frame.
The experimenter can have any number of detectors. Every set of detectors that are not moving relative to each other determines a reference frame. If there is a set of detectors which are stationary with respect to each other, then any detector moving at a nonzero velocity is not part of this reference frame.
Relativity produces theoretical results that are specific to a reference frame. The experimenter validating relativity does relativity calculations specific to each reference frame in his experiment. If he does calculation for one reference frame, then he examines the detectors that are stationary in that reference frame. He does not include measurements from detectors that are part of a different reference frame.
Sometimes, the experimenter tries to compare the difference between measurements of two detectors that are in different inertial frames. For example, suppose the experimenter wants to compare two clocks that are moving at different velocities.
Suppose the experimenter is comparing measurements of two similar detectors that are moving at high velocity relative to each other. Because of signal delay, the results will be ambiguous unless the two detectors happen to be located at the same place at the same time. Einstein presented the hypothesis that two events can only be proven simultaneous unless they are coincident in space.
The ambiguity in simultaneity is a practical and unavoidable problem in engineering. Therefore, experimenters don't compare detectors corresponding to different inertial frames unless the two detectors are very close together.
In experimental practice, the reference frame is really determined using the detectors. The difference between the experimenters reference frame and the theorists reference frame is in number. The experimenter always works with a finite number of detectors.
Einstein's hypothetical reference frame had an infinite number of detectors. However, infinity is just a limit. What this means is that the theory can work with any number of detectors. In practice, one works with a finite set of detectors that are close to stationary with respect to each other. As long as the detectors are sufficiently stationary with respect to each other, they are close enough to being a reference frame.
An inertial frame is a type of reference frame. An inertial reference frame is a reference frame where there is no force applied to any of the detectors. As long as the total force acting on any detector is zero, the reference frame is an inertial frame.
Thus, the state of the detectors unambiguously determines what the experimenter decides is the reference frame, and whether it is inertial. If there is any "art" involved in experimentation, it is in making sure that the detectors approximately satisfy these approximations. As in all physics, reality is always slightly different from the theory. However, the criteria for reference frames is logically unambiguous when one considers the detectors.

 

[Chestermiller]

To restate: a Frame of Reference is defined by Einstein as a 4D coordinate system empty of any physical objects and then you add in whatever physical objects you want and describe their positions and motions in terms of events, precise and exact with infinite resolution. Not only do you not require any "observers" you don't require anything at the origin or anywhere else or at any particular time. You do what you want.

Yes, I can see evolving to this definition, but along the way, the clocks, meter sticks, and train examples are very helpful to the learning process. I can also see that the Lorentz Transformation represents the geometric relationship between two 4D coordinate systems that are offset from one another.

 

[harrylin]

Not all objects "in a reference frame" are "part of a reference frame." [..]

Yes indeed. Why did you add that as comment to my post? The point that I and others brought up is that the OP seems to think that a standard reference frame such as used in SR is of limited extension, which is wrong.

By the way (and this has nothing to do with your post), the coordinate systems that Einstein describes in his 1905 paper are very standard, made up of 3 coordinate axes ("each of three rigid material lines"), and so he can place a watch at the origin of such a system. By means of rulers and clocks he defines standard reference systems with "co-ordinates and times". And while according to his definition the related reference clocks must be at rest in the coordinate system, this is not the case in the ECI frame, for purely practical reasons.

 

[ghwellsjr]

At this point, I am starting to lose you. Perhaps I should point out what the experimenter who validates relativity is doing. We have been talking about what the theorist considers a reference frame. What does the experimenter consider a reference frame?

You're right, this forum is to help people learn the theories of relativity (SR and GR). We assume they have already been validated. I doubt that any experimenters who are still trying to validate them are coming here to learn how to perform their experiments.

But since you have pointed out that you are talking about something different, I don't know why you are losing me. Just remember, I'm trying to help people who don't understand Special Relativity learn what the theory is, not how to do any experiments to validate (or invalidate) it.

For that reason, I don't have any comments on the rest of your post.

 

[ghwellsjr]

Yes, I can see evolving to this definition, but along the way, the clocks, meter sticks, and train examples are very helpful to the learning process.

I agree and would even go back to what scientists believed prior to Einstein (ether) so that the learner can understand what the problem was that SR solved. They believed in a single absolute ether rest state in which light propagated at c and since the Earth and all of us observers were moving in that "frame", we must be experiencing length contraction and time dilation, although by an unknown amount, since the stationary state of the ether was indeterminate. Einstein said you could pick any inertial state in which you presume that light propagates at c and in which you presume that stationary objects do not experience length contraction or time dilation and from this you can build a Frame of Reference.

I can also see that the Lorentz Transformation represents the geometric relationship between two 4D coordinate systems that are offset from one another.

The geometric relationship between two 4D coordinate systems (in the standard configuration) is simply a constant speed between the x-axes. The Lorentz Transformation is how you convert the specific coordinates of one frame into the corresponding coordinates of a second frame.

 

[Chestermiller]

I agree and would even go back to what scientists believed prior to Einstein (ether) so that the learner can understand what the problem was that SR solved. They believed in a single absolute ether rest state in which light propagated at c and since the Earth and all of us observers were moving in that "frame", we must be experiencing length contraction and time dilation, although by an unknown amount, since the stationary state of the ether was indeterminate. Einstein said you could pick any inertial state in which you presume that light propagates at c and in which you presume that stationary objects do not experience length contraction or time dilation and from this you can build a Frame of Reference.

The geometric relationship between two 4D coordinate systems (in the standard configuration) is simply a constant speed between the x-axes. The Lorentz Transformation is how you convert the specific coordinates of one frame into the corresponding coordinates of a second frame.

I never mentioned the Standard Configuration in any of my posts. I don't know where you guys got the idea that I was restricting my thinking to the Standard Configuration. Also, I might mention that, even in the Standard Configuration, the orientation of the x-axis in 4D spacetime is not the same as the orientation of the x' axis. I've been working with, and deriving, much more-advanced forms of the Lorentz Transformation than the simple form specific to Standard Configuration.

Chet

 

[ghwellsjr]

I never mentioned the Standard Configuration in any of my posts. I don't know where you guys got the idea that I was restricting my thinking to the Standard Configuration. Also, I might mention that, even in the Standard Configuration, the orientation of the x-axis in 4D spacetime is not the same as the orientation of the x' axis. I've been working with, and deriving, much more-advanced forms of the Lorentz Transformation than the simple form specific to Standard Configuration.

Chet

I was using the definition of the Standard Configuration presented in the wikipedia article on the Lorentz transformation which is identical to the one that Einstein presented in section 3 of his 1905 paper so I don't know why you state that it is something different. In any case, my point was that the geometric relationship between two Frames of Reference is not the same as the transformation process. I'm sure that each of the more advanced forms that you are working with starts with a description of the geometric relationship and then determines the transformation process to convert the coordinates of events from one to the other.

 

[Vandam]

I really do not understand why you guys are still debating this topic. My post with diagram shows and explains it all.

 

[Chestermiller]

... and what about an event that was at the middle of a rigid ruler where there was no clock or an event that was in the middle of a clock? How would you define those events? And think about what happens when you take the coordinates of an event in one FoR and convert them into the coordinates of the same event in a second FoR moving with respect to the first one, the calculation can result in an arbitrary location where there may not be a clock if you follow the physical idea of a FoR.

I'm sure you've heard the term Linear Interpolation. We engineers use it all the time in analyzing problems numerically, so that our finite difference grid or finite element grid does not have to be infinitely fine (and completely fill every iota of space). But wait, ..., is the functionality we are dealing with in the SR situation we are examining linear? Why yes it is.

Chet

 

[ghwellsjr]

I'm sure you've heard the term Linear Interpolation. We engineers use it all the time in analyzing problems numerically, so that our finite difference grid or finite element grid does not have to be infinitely fine (and completely fill every iota of space). But wait, ..., is the functionality we are dealing with in the SR situation we are examining linear? Why yes it is.

Chet

OK, now you're coming full circle. As long as you're going to allow for the determination of coordinates to include events for which there is no physical object at the time and place of the event, then you're simply imagining what a physical clock and ruler would determine the coordinates of the event to be, correct? And as long as you recognize that there is a 3D finite element grid with a previously synchronized clock at each finite element (as opposed to a small number of randomly located physical objects all at mutual rest) and you can interpolate anywhere else, then all you have to do is imagine that you have a very course grid with elements only at the extremities and you can interpolate to find the coordinates of any event anywhere you want. The final step is to simply imagine that even the elements at the extremities are also imaginary.

But I don't want to lose sight of the import of this thread which is your claim that only objects that are at mutual rest with a Frame of Reference belong to that Frame of Reference. I want to make sure you recognize that Einstein's definition of a Frame of Reference does not contain any physical objects, but an infinite number of imaginary objects (rigid rulers and synchronized clocks in the finite element grid) and you use these to specify the positions and motions of any number of physical objects. (Of course, in our thought problems, even these "physical objects" are imaginary, but we image that they are actually physical, whereas we image that the coordinate rulers and clocks are actually imaginary.)