So, my professor posted answers for an assignment and I don't understand his solution. I would ask him but he takes a while to answer emails and I would rather know how to do this sooner rather than later. I took a screenshot of the question and his answer that you can check out here: The question

What I want to know is his solution correct? He assume proper length to be in the observer on Earth's reference frame, but then his proper time is assumed to be in the particle's reference frame.

If his solution is correct, then does the proper length and proper time not have to be in the same reference frame? Also, should the proper length always be longer than the other reference frame's length or does that not matter?


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  • $\begingroup$ Proper time and proper length (which is essentially the same as proper time with an opposite sign) are the length and time measured in the rest frame of the moving particle. $\endgroup$ – K7PEH Apr 15 '17 at 17:44
  • $\begingroup$ And, every observer and every moving particle has its own proper time. How these various proper times relate to each other is the mathematics part of Special Relativity. $\endgroup$ – K7PEH Apr 15 '17 at 17:46
  • $\begingroup$ so then proper time and length should be calculated in the same reference frame, correct? $\endgroup$ – mhold Apr 15 '17 at 17:47
  • $\begingroup$ In the rest frame, your velocity is zero. That is, the particle is at rest with zero velocity in its rest frame, and the observer in his rest frame are at rest with zero velocity. $\endgroup$ – K7PEH Apr 15 '17 at 17:48
  • $\begingroup$ Okay, so are you saying each reference frame in a way can be the rest frame? If that's the case, how do I know which frame to assume has the proper length and time for calculations? $\endgroup$ – mhold Apr 15 '17 at 17:53

I believe that what you are confused on is the symmetry of time dilation and length contraction. Meaning that if you observe an object to be moving relative to you, it is also appropriate to consider that you are moving relative to it, meaning that special relativity applies "both" ways. Thus, its critical to define what reference frame is being considered.

First lets define who measures the proper length and time:

  1. Proper length is the length measured by the observer for whom the endpoints of the length remain fixed in time and space (in the referred to reference frame)

  2. Proper time is the time interval measured by the observer who sees the two events happen at the same position in space (in the referred to reference frame)

The question refers to the reference frame of the scientist, consequently in this reference frame he is motionless on the ground, and the muon is moving relative to him, and from this we can conclude two things:

  1. Since the scientist is motionless in such a reference frame and the muon is moving relative to him, he is the observer with the "privilege" of measuring the endpoints of the altitude to be fixed, since no matter what time of the event you take the endpoints will remain in the same position relative to him (unlike the muon). Thus, he measures the proper length.

  2. Moreover, since we are referring to the scientists reference frame we know that the muon is moving relative to him, thus muon's position relative to him is a subject of time, however for the muon in such a reference frame we know that from its "perspective" it sees the events as happening in the same position relative to it, thus it measures the proper time, with respect to the scientists reference frame

In conclusion an object which is in motion relative to another object in a certain reference frame always measures the proper time, as well as the dilated length, in that reference frame.


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