High School Am I understanding the concept of proper frame of reference?

[mcastillo356]

TL;DR

A proper reference frame in the theory of relativity is a particular form of accelerated reference frame, in which an accelerated observer can be considered as being in rest. Well, the context isn't Special Relativity, this is, inertial frames of reference?

MOVING CLOCKS

In this section, we show that clocks moving at high speeds run slowly.
We construct a clock, called a light clock, using a stick of proper lenght $L_0$, and two mirrors. The two mirrors face each other, and a pulse of light bounces back and forth betweem them. Each time the light pulse strikes one of the mirrors, say the lower mirror, the clock is said to tick. Between successive ticks the light pulse travels a distance $2L_0$ in the proper reference of frame of the clock. Thus, the time between ticks $T_0$ is related to $L_0$by

$$2L_0=cT_0\quad{R_1}$$

Next, we consider the time between ticks of the same light clock, but this time we observe it from a reference frame in which the clock is moving perpendicular to the stick with speed $v$. In this reference frame, the clock moves a distance $vT$ between ticks and the light pulse moves a distance $cT$ between ticks. The distance the pulse moves in travelling from the bottom mirror to the top mirror is

$$\displaystyle\sqrt{L_0^2+\bigg (\frac{1}{2}vT\bigg )^2}$$

The light pulse travels the same distance in traveling from the top mirror to the bottom mirror. Thus,

$$2\displaystyle\sqrt{L_0^2+\bigg (\frac{1}{2}vT\bigg )^2}=cT\quad{R-2}$$

Because the speed of light is the same in all inertial reference frames, we have the same symbol $c$ for the speed of light in Equations $R-1$ and $R-2$. Solving Equation $R-1$ for $L_0$ and substituting into Equation $R-2$ gives

$$\displaystyle\sqrt{\bigg (\frac{1}{2}cT_0\bigg )^2+\bigg (\frac{1}{2}vT\bigg )^2}=\frac{1}{2}cT$$

Solving for T gives

$$T=\displaystyle\frac{T_0}{\sqrt{1-(v^2/c^2}}\quad{R-3}$$

According to Equation $R-3$, the time between ticks in the reference frame in which the clock moves at speed $v$ is greater than the time between ticks in the proper reference frame of the clock.

Attempt at a solution

The proper reference frame both observers can be considered as being in rest.

Post without preview.

 

[anuttarasammyak]

Your thought about two IFRs seems OK. What’s the problem?

 

[pervect]

It looks more like you are trying to understand the relationship between two inertial frames of reference rather than the concept of a "proper reference frame".

My graduate level textbook, Misner, Thorne, Wheeler "Gravitation", has a discussion of proper reference frames of possibly accelerated observers (in section $13.6, pg 337), but it's not at high school level. But fortunately, your post doesn't really seem to be about proper reference frames or accelerated frames, but more about the the relationship between "stationary" and "moving" inertial frames.

The Lorentz transform is a more general relationship for the relation between a "stationary" frame S, and a "moving" frame S'. It provides a map between the coordinates (t,x,y,z) in some inertial frame S, to the coordiantes (t', x', y', z') in some other inertial frame S', which is usually taken to be moving with respect to S, and has the property that at t=0, x'=x, y'=y, z'=z. You can find the exact equations in many many posts.

 

[PeterDonis]

@mcastillo356 what is your question?

 

[mcastillo356]

@mcastillo356 what is your question?

@pervect has answered my doubts. Thanks a lot.

 

[pervect]

I barely said anything, but if you have a chance and ability to read the reference I cited, it's the best treatment of accelerated frames of reference that I know of. I will add that it starts out with a good discussion of why said "frame of reference" is limited in extent.

I'll add that from an informal point of view, the main feature of the (local) frame of reference of an accelerated observer is time dilation in such a frame which depends on what one might call "altitude".

 

[mcastillo356]

I`m working on the right meaning of the concept "proper". I think it doesn't refer to any acceleration, as the source I read says.
At this moment it's just an intuition.

 

[PeterDonis]

I`m working on the right meaning of the concept "proper"

I'm not sure what you mean by "the right meaning".

the source I read

What source is that?

At this moment it's just an intuition.

This sounds like you are trying to develop your own personal interpretation. Please be aware that discussion of personal theories is off limits here.

 

[mcastillo356]

The source is Wikipedia.

This sounds like you are trying to develop your own personal interpretation. Please be aware that discussion of personal theories is off limits here.

Thanks indeed. I'm reading a pdf of the Uned, a Spanish reliable University..
Anyhow, thank you for the advice.

 

[PeterDonis]

The source is Wikipedia.

Please give a specific link; Wikipedia is a very big site.

 

[mcastillo356]

https://en.wikipedia.org/wiki/Proper_reference_frame_(flat_spacetime)

 

[PeterDonis]

https://en.wikipedia.org/wiki/Proper_reference_frame_(flat_spacetime)

Ok. As you yourself note in the TL/DR to this thread, this article defines a proper reference frame as a particular kind of accelerated reference frame. So this statement by you is obviously false:

I think it doesn't refer to any acceleration, as the source I read says.

Which again makes me ask, as I asked in post #8:

I`m working on the right meaning of the concept "proper".

What do you mean by "the right meaning"? The meaning given in the Wikipedia article is a definition; you can't argue with a definition. It is what it is.

 

[FactChecker]

As the article you reference says: a proper reference frame for an observer (accelerated or not) is "a reference frame in which an accelerated observer can be considered as being at rest." That is a proper reference frame for that observer. The definition is as simple as that. The consequences when the measurements in a proper reference frame are observed by an inertial reference frame are what gets complicated.

 

[PeterDonis]

a proper reference frame for an observer (accelerated or not)

The parenthetical statement here is not correct; the article specifies an accelerated observer as the one that is at rest in that observer's proper reference frame.

 

[mcastillo356]

This sounds like you are trying to develop your own personal interpretation. Please be aware that discussion of personal theories is off limits here.

Really true. But I won't try any self made explanation. Thank you, @PeterDonis.

 

[Ibix]

Note that the English meaning of "proper" has drifted over time to mean "correct". In this context, though, it's being used in an older sense similar to "property" (it shares a root in Latin, and also with propio en español), simply meaning a thing that is one's own.

Proper time is the time your own clock measures. Proper acceleration is the acceleration measured by an accelerometer in your pocket. A proper reference frame is one where the origin is attached to you.

 

[FactChecker]

The parenthetical statement here is not correct; the article specifies an accelerated observer as the one that is at rest in that observer's proper reference frame.

Is it not true that once an observer is specified, he is the center of a proper reference frame that follows him? And isn't that true through his path, whether he is accelerating or not in any part?

 

[Ibix]

Is it not true that once an observer is specified, he is the center of a proper reference frame that follows him? And isn't that true through his path, whether he is accelerating or not in any part?

I would have thought so, but the wiki article specifies it as only for an accelerated observer. I suspect they're meaning "an observer who is non-inertial at least once", since an eternally inertial observer could just use a global Minkowski frame, but I don't see the point of such an exclusion immediately. For example, radar coordinates are a perfect example of proper coordinates, but there's nothing stopping an eternally inertial observer using radar coordinates - in fact they reduce to Minkowski coordinates in that case, and that is one of their selling points.

 

[anuttarasammyak]

A proper frame of reference (FR) that is inertial (IFR) does not depend on the observer’s location or size; the observer may be regarded as ubiquitous. However, this is not true for a proper FR that is non-inertial.　For example, in an accelerating rocket with multiple floors, observers on the first and second floors experience different proper times.　Ideally, an observer defining a proper FR should be modeled as a point particle with zero spatial extent （and no intrinsic spin）.

 

[Ibix]

A proper frame of reference (FR) that is inertial (IFR) does not depend on the observer’s location or size; the observer may be regarded as ubiquitous. However, this is not true for a proper FR that is non-inertial.　For example, in an accelerating rocket with multiple floors, observers on the first and second floors experience different proper times.　Ideally, an observer defining a proper FR should be modeled as a point particle with zero spatial extent （and no intrinsic spin）.

But observers on the same floor of the rocket have the same relationship between their proper and coordinate times. So I suppose the rigorous definition is something like that a proper coordinate system is one where at least one "at rest" worldline has a 1:1 relationship between proper and coordinate time and orthonormal coordinate basis vectors along its length, and at least one does not.

If that's the definition then that's the definition. It can exclude whatever the people who framed it want to exclude. It just seems odd to me (and apparently @FactChecker) to frame it to exclude Minkowski coordinates just because all, rather than some, "stationary" observers have that "proper" relationship with coordinate differences.

 

[anuttarasammyak]

It just seems odd to me (and apparently @FactChecker) to frame it to exclude Minkowski coordinates

It seems odd to me too because I think Minkowski coordinate belongs to SR or IFR where we find no problems as for Proper FR.

 

[FactChecker]

It seems odd to me too because I think Minkowski coordinate belongs to SR or IFR where we find no problems as for Proper FR.

Regardless of the wording of the definition given in the Wikipedia article, I believe that the term "proper reference frame" has been used often to refer to situations where the reference frame was not always accelerating. The Twin Paradox is one example. The traveling twin can be in inertial reference frames for much of his departure and return trips. I wouldn't take the wording of the Wikipedia article as a formal definition. In fact, I would assume that the proper frame of an object can be completely inertial.

 

[PeterDonis]

A proper frame of reference (FR) that is inertial (IFR)

There is no such thing according to the Wikipedia article that has been referenced.

Regardless of the wording of the definition given in the Wikipedia article, I believe that the term "proper reference frame" has been used often to refer to situations where the reference frame was not always accelerating. The Twin Paradox is one example.

Please give a specific reference where the term "proper reference frame" is used this way. I get that Wikipedia by itself is not an authoritative source in general, but the specific Wikipedia article that was referenced here references multiple textbooks in the field, all of which, as far as I know, use "proper reference frame" in the specific way defined in the article. That's much stronger evidence of standard usage in the field, which is what we go by here.

 

[anuttarasammyak]

Proper time is the time your own clock measures. Proper acceleration is the acceleration measured by an accelerometer in your pocket. A proper reference frame is one where the origin is attached to you.

Indeed. Next I survey my environment to construct my PFR which might not be defined unikely. That will be a tough work to do.

 

[FactChecker]

There is no such thing according to the Wikipedia article that has been referenced.


Please give a specific reference where the term "proper reference frame" is used this way. I get that Wikipedia by itself is not an authoritative source in general, but the specific Wikipedia article that was referenced here references multiple textbooks in the field, all of which, as far as I know, use "proper reference frame" in the specific way defined in the article. That's much stronger evidence of standard usage in the field, which is what we go by here.

If you are saying that, as far as you know, the term "proper reference frame" has always been used for a frame that is always accelerating and never has a zero acceleration, even for an instant, then I will take your word for it. But I am skeptical.

 

[PeterDonis]

If you are saying that, as far as you know, the term "proper reference frame" has always been used for a frame that is always accelerating and never has a zero acceleration, even for an instant, then I will take your word for it.

No, that's not what I said. What I said is that, based on the multiple textbooks referenced in the Wikipedia article, standard usage in the field is that "proper reference frame" refers to a frame in which an accelerated observer is at rest. "Standard" is not the same as "nobody has ever used the term any other way, anywhere".

However, the claim of yours that I responded to was not that the term has been used at least once to refer to an inertial frame--it was that the term has been used that way "often". That is a claim you will need to back up with references.

 

[FactChecker]

No, that's not what I said. What I said is that, based on the multiple textbooks referenced in the Wikipedia article, standard usage in the field is that "proper reference frame" refers to a frame in which an accelerated observer is at rest. "Standard" is not the same as "nobody has ever used the term any other way, anywhere".

However, the claim of yours that I responded to was not that the term has been used at least once to refer to an inertial frame--it was that the term has been used that way "often". That is a claim you will need to back up with references.

Ok. That could allow the case that a non-accelerated observer is also at rest and can still be called a proper reference frame. So it is more flexible. That would make sense to me.

 

[PAllen]

No, that's not what I said. What I said is that, based on the multiple textbooks referenced in the Wikipedia article, standard usage in the field is that "proper reference frame" refers to a frame in which an accelerated observer is at rest. "Standard" is not the same as "nobody has ever used the term any other way, anywhere".

However, the claim of yours that I responded to was not that the term has been used at least once to refer to an inertial frame--it was that the term has been used that way "often". That is a claim you will need to back up with references.

Well, MTW calls it the proper reference frame of an accelerated observer, implying that a standard inertial frame is the proper reference frame of a non accelerating observer (in flat spacetime). The discussion in the section of this name (p.327 in my edition) clearly includes zero acceleration and rotation as a special case.

 

[PeterDonis]

MTW calls it the proper reference frame of an accelerated observer, implying that a standard inertial frame is the proper reference frame of a non accelerating observer (in flat spacetime).

I'm not sure I see this implication, since the section that discusses this for curved spacetime (13.6) refers to Exercise 6.8 which defines an accelerated rotating frame for flat spacetime, and says that this section will do the same for curved spacetime. There is no reference anywhere that I can see for a use of the term "proper reference frame" to describe a standard inertial frame in flat spacetime.

That said, I agree that zero acceleration and zero rotation can be plugged into the equations as a special case, and yield a standard inertial frame (in curved spacetime, a local inertial frame), and that MTW does include that special case.

 

[PAllen]

I'm not sure I see this implication, since the section that discusses this for curved spacetime (13.6) refers to Exercise 6.8 which defines an accelerated rotating frame for flat spacetime, and says that this section will do the same for curved spacetime. There is no reference anywhere that I can see for a use of the term "proper reference frame" to describe a standard inertial frame in flat spacetime.

That said, I agree that zero acceleration and zero rotation can be plugged into the equations as a special case, and yield a standard inertial frame (in curved spacetime, a local inertial frame), and that MTW does include that special case.

The wording “proper reference frame of an accelerated observer” implies, in ordinary English, the notion of proper reference frame of a non accelerated observer. MTW, so far as I see, does not use “proper reference frame” by itself, to imply a mandatory accelerated observer.