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Old Feb 12th 2018, 08:54 PM   #1
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Reference frames

This is a message I got from TemporalMechanic:

A fundamental concept of Special Relativity is relative motion. Assume two inertial systems S and S' are moving in completely arbitrary paths. The instinct is to say that the relative motion is 3-dimensional. One could not be faulted for that view, but relative motion is a 1-dimensional effect with two degrees of freedom.

Draw a straight line segment between S and S' and call that distance d. There are many paths the systems can take where d is invariant. There are other paths where d is variable. It is this variation of d that is the essence of relative motion, it occurs in a single dimension, the line segment connecting the two origins.

Within that dimension the translation of S and S' is one degree of freedom, the rotation of S and S' is the second degree of freedom.

Does anyone see difficulty with this perspective of relative motion, a one-dimensional effect with two degrees of freedom?
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Old Feb 13th 2018, 06:14 AM   #2
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Do we only have one rotational degree of freedom?
I have not thought heavily about this,
but it seems to me that while the distance might require only one value
the orientation of S to S' will require more angles (3?).

The rotation angles I am thinking of are; one where the the axis of rotation is aligned with the vector "d"
and then two where the axes of rotation are perpendicular to the vector "d" (and to each other).

Note that this assumes a "standard" 3D space.
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Old Feb 13th 2018, 07:27 AM   #3
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I'm not very good at relativity and frames of reference, so perhaps I'm not best to reply to this one, but the initial sentences sounded a bit funky to me:

A fundamental concept of Special Relativity is relative motion. Assume two inertial systems S and S' are moving in completely arbitrary paths. The instinct is to say that the relative motion is 3-dimensional. One could not be faulted for that view, but relative motion is a 1-dimensional effect with two degrees of freedom.
If the two reference frames are inertial, that means that the relative motion between them is not completely arbitrary... Perhaps the OP should replace "arbitrary" with "arbitrary rectilinear motion".

Then comes this bit...

Draw a straight line segment between S and S' and call that distance d. There are many paths the systems can take where d is invariant. There are other paths where d is variable. It is this variation of d that is the essence of relative motion, it occurs in a single dimension, the line segment connecting the two origins.

Within that dimension the translation of S and S' is one degree of freedom, the rotation of S and S' is the second degree of freedom.
I think it's correct, but is it important to think of the system in terms of degrees of freedom required to explain the relative motion of the frames? I would say that it's more important to consider whether the frames are inertial or non-inertial in terms of acceleration (for example). Consider two cases:

1. Two inertial frames, S is stationary, S' has a constant velocity of 1 m/s to the right relative to S.
2. One inertial frame and one non-inertial frame; S is stationary, S' is initially stationary, but then accelerates to the right at 1 m/s^2

In the first case, the line connecting two stationary points in the frames is determined by a single vector (one degree of freedom) with increasing size. This is also true for case 2, but now the laws of physics in S' requires fictitious forces in order to fully characterise the motion within that frame. You could argue that the proper "second degree of freedom" is the acceleration, but that's one way of determining whether a frame is inertial or not anyway...
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Last edited by benit13; Feb 13th 2018 at 07:37 AM.
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Old Feb 13th 2018, 05:10 PM   #4
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A fundamental concept of Special Relativity is relative motion. Assume two inertial systems S and S' are moving in completely arbitrary paths.
That's not possible. Special relativity is the relativity of inertial frames only. As such S and S' are in relative motion where the origin of one traces out a straight line and moves uniformly, just as benit13 described.

The instinct is to say that the relative motion is 3-dimensional.
The term "dimension" is defined as the amount of numbers required to uniquely specify an element of the space. A line is a 1-dimentional entity. Speaking of "relativity motion" as having a dimension is not meaningful

One could not be faulted for that view, ..
On the contrary, I just did.

Draw a straight line segment between S and S' and call that distance d. There are many paths the systems can take where d is invariant.
That's a vague statement. You can only speak of the distance between two points so you must be thinking of the distance between the origins of S and S.

It seems from what you wrote after that means that by "invariant" means "does not change in time" and as such not possible if S and S' are moving.


I'm sorry but there's a lot of meaningless comments in that quote Dan.
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Old Feb 14th 2018, 05:28 PM   #5
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Originally Posted by Pmb View Post
I'm sorry but there's a lot of meaningless comments in that quote Dan.
Yes, I see that now. I was running short on time and he sent the message to me using the Visitor message system. As I never answer Physics questions this way I didn't look too hard at the post. It seems that was an error.

-Dan
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