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What does "In the reference frame of" exactly mean?

For example, imagine a man is standing on a rocket moving at a speed $0.8c$. The man is stationary in the reference frame of the rocket. If we want to calculate the distance covered by the man in the reference frame of the rocket during some amount of time, is it just zero or $d\gamma$, where $d$ is the distance covered by the rocket in the reference frame of a stationary observer? Is "distance covered in the reference frame of something" the same as "distance travelled relative to something"? I'm kinda confused by the reference frame stuff.

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  • $\begingroup$ When you say "stationary observer" what do you mean? What frame is this observer stationary in? $\endgroup$
    – The Photon
    Commented Sep 1, 2022 at 4:52
  • $\begingroup$ @ The Photon Thanks for your reply. I meant that the rocket is moving at 0.8c in the reference frame of the observer $\endgroup$
    – Irene
    Commented Sep 1, 2022 at 4:56
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    $\begingroup$ I'm going to let someone else post an answer, because there's probably some subtlety of this that I've forgotten. But you can think of "in the reference frame of the rocket" as a shorthand way of saying "in the reference frame in which the rocket is stationary". The distinction is that the reference frame exists even if the rocket doesn't. Which is why "distance travelled in the reference frame..." is not the same as "distance travelled relative to an object". $\endgroup$
    – The Photon
    Commented Sep 1, 2022 at 5:01
  • $\begingroup$ @The Photon I think I kinda get what you are trying to say...Thanks for your help. $\endgroup$
    – Irene
    Commented Sep 1, 2022 at 5:06
  • $\begingroup$ Just try to imagine yourself on that rocket, then obviously you will go wherever that rocket takes you, so distance travelled by you with respect to that rocket (or vice a versa) will always be zero. The distance will only be non zero for an observer outside the rocket. $\endgroup$
    – Aman pawar
    Commented Jan 18 at 9:24

2 Answers 2

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A reference frame is a physical system$^{(*)}$. In practice, everything one needs to assign values to time and position measurements. One point is insufficient because one needs to specify orientations and a distance scale. In Classical Mechanics, one assumes that a suitably extended rigid body, equipped with synchronized clocks to allow time measurements, is a convenient reference frame. Moreover, there is an often implicit assumption that the reference frame enables measurements with negligible perturbation of the measured systems.

Let me come to the example. If we can assign a value to the rocket's speed, we use a reference frame such that the rocket is moving relative to the frame. However, we can use the rocket (or part of it) as a different reference frame. In such a frame, the rocket is clearly at rest. Measures done in the rocket frame are usually different from measures done in the reference frame where the rocket is moving.

The difference between quantities measured in different reference frames depends on the rules established to assign positions and times. In Classical Mechanics, based on the hypothesis that we can synchronize clocks independently of their motion, intervals of time and distances are frame-independent quantities. Absolute values of time and positions are frame-dependent. Notice that the dependence on the frame of the positions implies that the shape of trajectories is also a frame-dependent quantity. A uniform circular motion in the reference frame where the center of the circle is at rest becomes a helicoidal motion in the reference frame where the center of the circle moves uniformly in the direction perpendicular to the circle's plane.

$^{(*)}$ In the case of the definition of reference frame, I'm afraid I have to disagree with the present status of the relevant Wikipedia page: "a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points―geometric points whose position is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers).." where most of the definition is based on ideal, geometrical, entities and not on measurable, physical objects. Moreover, no mention is made of clocks.

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  • $\begingroup$ Would you consider a physically well-defined coordinate space without distance and/or time axes, e.g. a phase space, to be a reference frame? $\endgroup$
    – g s
    Commented Sep 1, 2022 at 6:15
  • $\begingroup$ @gs If I understood well, the answer is no. Reference frames in Mechanics are useless if there is no way to assign a time to events. Phase space can be defined as soon as there is a reference frame. $\endgroup$ Commented Sep 1, 2022 at 6:55
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The underlying point is that movement means a change in position, so to quantify movement you have to be clear about what you mean by position. If you are sitting on a platform at a station, you would say that relative to the platform your position is not changing. However, relative to a passing train your position is changing at some rate. Likewise, relative to someone walking towards you along the platform your position is changing at another rate, so all motion is relative.

If you get on the train and take a seat while it is coasting along at 100m/s relative to the track, your speed relative to the train is zero while your speed relative to the track is 100m/s. If you leave your seat to walk along the train at 1m/s, your speed relative to the track will either be 99m/s or 101m/s depending on whether you are walking in one direction or the other.

So the idea of a frame is simply a reference point against which your position, and hence your movement, can be quantified. The frame can be considered as something concrete, like the reference frame of a moving train, but the mention of the train is superfluous in some respects, as you can consider a reference frame in a more abstract way. For example, you can consider two frames moving at 10m/s relative to each other along their respective x-axes. If you are stationary with respect to one of them, you will be moving in the other at 10m/s.

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