Conceptual Physics: Time in Special Relativity

This is basically a quick brain dump. I’m teaching a section of conceptual physics this semester for the first time in many years. It has caused me to revisit everything about the topics I’m including in the course. When the course was assigned to me, I immediately decided to make it consist of twentieth century topics, beginning with special relativity. I see no reason not to begin any introductory physics course with either special relativity or concepts from quantum physics as these two frameworks for the foundation of all contemporary physics. The official textbook for the course is Hewitt’s Conceptual Physics, but I always begin discussion of special relativity with  chapter 36 from Arnold Arons’ Development of Concepts of Physics (1965). Arons address the underlying operational definitions of measurement and lack of absolute simultaneity in the approach to time dilation and length contraction. I have yet to see a better (non-graphical) approach.

Perhaps Arons’ most profound influence on me is that words really matter. The names we give to physical concepts and ideas can help or hinder student understanding. Time is a problematic name in that authors use it interchangeably to mean the reading on a clock at a given moment and the difference between two such readings. He advocates clock reading for the former and duration for the latter. Special relativity deals with our conceptions of space and time, but which time do we mean? For beginners, we usually mean duration, which is directly related to the rate at which time is measured by a clock. While special relativity can affect both clock readings and durations, how do we get this distinction across to students in a conceptual physics course?

I thought of a way that seemed to work. It’s really nothing but a very simple thought experiment. Consider a clock in the observer’s stationary frame to read any arbitrary clock reading; I chose three o’clock for no good reason. The clock is not running and thus is not measuring the passage of time. Now suppose the clock is in an inertial frame moving past a stationary observer at some arbitrary relativistic speed. Neither the stationary observer nor an observer moving with the clock will measure anything different about the clock. It isn’t turned on, so there’s is no discrepancy in the clock’s rate in either frame compared to the other frame. Both observer will report that clock’s reading as three o’clock.

So unless the clock is running, we see that special relativity doesn’t affect time, if by time we mean a specific clock reading. But if the clock is running, then observers in different frames will measure different durations for the clock’s tick-tock, the term I use for the fundamental period of the clock, whatever the internal periodic mechanism.

Students then naturally asked about how clocks get out of sync as in the twin paradox. I explained it by saying as a clock accelerates from rest to some final constant velocity, it goes through many different inertial frames. For each inertial frame, we can associate an instantaneous constant velocity and a unique tick-tock duration. As the clock experiences each new inertial frame, the changes in the clock’s tick-tock from being in previous frames “stick” and accumulate. This accumulated time dilation causes the clock to get out of sync from an otherwise identical clock to which it was initially synced in the rest frame. This process repeats as the clock accelerates to reverse its direction for the return trip. However, there’s something fascinating here, and that is that the time dilation accumulated by the acceleration from the outbound frame to inbound frame is not undone by reversing direction or by slowing down. The Lorentz factor, gamma, on the magnitude of the relative velocity squared, which is of course independent of direction, and time dilation happens whether the clock slows down or speeds up (we didn’t consider change in direction).

I’m aware that the simple twin paradox problem can be solved without even mentioning the accumulation of time dilation as a clock accelerates, but I wanted to go just a bit deeper without introducing unnecessary complications. So I hope I didn’t screw anything up. Did I?

Comments 8

  • Dear Prof. Heafner,

    Thanks to your recent mastodon tweet I was prompted to take a closer look at your nice blog. Looking for pages on the topic of my particular interest (foundations of relativistic geometry/kinematics) I found foremost this page expressing your appreciation of the relation between careful wording and thorough understanding (which I share), and I was very pleasantly surprised by your emphasis on the notions of “clock reading” and “duration” (i.e. terminology which I use and try to popularize, too).

    However, I must admit that I learned the utility of these notions perhaps mainly from writings of P. W. Bridgman, while I can hardly find them mentioned in chap. 36 of A. B. Arons “Development”.

    Also, this immediately brings up some (to me certainly important) questions:

    (1):
    How exactly do you call (especially in your lectures, if at all) that part (or “portion”, or “aspect”) of a clock under consideration which “is read”,
    i.e. to which there is assigned precisely one specific real number value as “its clock reading”,
    and which belongs to precisely only one specific event ?

    (Do you perhaps call that: “an instant” or “one moment” of that clock? Or: “an indication” of that clock? Or, perhaps less discerningly: “one clock time” ? …
    I tend to call that “an indication”; in translation of the German “Anzeige”, which (unfortunately, also) has several meanings, especially in colloquial use, but they surely include being understood as generalization of “Zeiger”, i.e. “pointer”.)

    And (2):
    Do you understand and treat “duration” directly as (positive) difference between two clock readings,
    i.e. as (similar to) a real number, such that any two duration value could immediately and unambiguously compared to each other
    (having a specific unambiguous real number ratio “without further ado”) ?

    Or do you instead attribute “duration” to any two indications of a clock under consideration,
    such that the ratio between two durations could only be found out through (rather involved) measurement
    (and by cross-relation to differences of the corresponding clock readings, it could then — and only then — be judged whether, or to which accuracy, a given clock had been “good”) ?

    Arons doesn’t seem to get into this much detail (and who would blame him, considering the scope of his book, and the corresponding teaching load ;).
    After all, in section 36.2 he’s indeed mentioning “local simultaneity” (a.k.a. coincidence determinations) as primitive, unanalyzable concept of RT; but then defers “a host of subtle but important philosophical-epistemological questions” in footnotes to secondary literature …
    (leaving it uncertain, whether even Arons himself had found satisfying answers there, up to 1965).

    Since you also expressed emphasis on coordinate-free presentations I might try to find in your blog some more pages to comment on;
    and, as you know, I’m happy to correspond through mastodon, too.

    Respectfully Yours,

    Frank Wappler (MisterRelativity@mathstodon.xyz)

    • Thank you for taking the time to leave a comment.

      I met Arons about one year before he died and he told me of Bridgman’s influence on him, particularly with the use of operational definitions. When I presented my copy of “Development” to him to sign he also explained that he wished he could go back and correct errors and improve wording in that book. Alas that never happened. The footnotes you cite may pertain to that.

      I don’t recall ever having addressed (1) in my teaching other than to use “clock reading” for the result of a measurement taken from the device. I am not certain whether or not that is what you are asking. Regarding (2), I always treat duration as a positive quantity. The only “goodness” I ever addressed was that the clock must have a predictable rate of advance (not necessarily uniform, but predictable to accommodate astronomical applications).

  • Joe Heafner wrote (2023-01-20 at 13:00):
    > I met Arons about one year before he died […] he wished he could go back and correct errors and improve wording in that book. Alas that never happened.

    Could you please recommend a (not necessarily “introductory”) book, or perhaps a web site, dealing with “conceptual physics” especially relating to relativity, which presents wording more in line with what Arons would have preferred ?
    (If not, that could be motivating to produce one …)

    > […] he told me of Bridgman’s influence on him, particularly with the use of operational definitions

    Did Arons mention any criticism of Bridgman’s writings ?
    (I’d hope to learn more about Bridgman’s reservations towards the notion of “coincidence” — see p. 154 of “Sophisticate’s primer”; especially related to his notion of “event”, p. 116.)

    Now, trying to further clarify question (2) of my previous comment:

    > […] that the clock must have a predictable rate of advance […] not necessarily uniform

    How exactly do you define “clock”, and “rate of advance” of a given clock under consideration ?
    — especially such that the said “rate of advance” could at least in principle, in suitable cases, be determined having been “not uniform”.

    (I also wonder how your notions of “clock” and “rate” might thereby relate to those of Laurent, or Callahan, or Misner/Thorne/Wheeler, for instance …)

    p.s.
    > The only “goodness” I ever addressed was […]

    The notions “good clock” vs. “bad clock” appear (for instance) in Fig. 1.9 of “MTW”.
    In both cases referring to “a clock” by MTW’s definition, of course.

    • I can’t recommend a book incorporating Arons’ changes because I don’t know what those changes would have been. He never told me. I know of no conceptual physics text that even addresses these issues at all. Note that Hewitt’s text is the canonical incarnation of “conceptual” physics.

      Arons also never mentioned to me any criticisms of Bridgman’s writings.

      I never defined “clock” in any physical way. I never defined “rate of advance” other than in the comparison of a sundial to a mean solar clock, and I never defined those in any detail other than the obvious realistic incarnations. I never had reason to address these issues in any more depth. Sorry.

  • Joe Heafner wrote (2023-01-23 at 12:40):
    > […] I never defined “clock” in any physical way. […]

    Well — above I had tried to ask for your definition of the notion “clock” not specificly and necessarily “in any physical way”,
    but (merely) to enable you to (subsequently) define “rate of advance [of a clock]”; and what you might then mean by “uniform[ity]” in this regard
    (cmp. your use of this terminology in your above comment, 2023-01-20 at 13:00; as I had quoted already).

    > I never defined “rate of advance” other than in the comparison of a sundial to a mean solar clock […]

    Exactly what (and how) would you suggest therefore to compare ?? …

    Anyways:
    You (and perhaps your students, too) may find sect. 2.3.2 of É. Gourgoulhon’s »Special Relativity in General Frames« instructive
    (available online e.g. in File “9783642372759-c1.pdf” as part of (zipped) supplementary material),
    where Gourgoulhon states

    – his explicit definition of »a generic ticking clock«
    (which is also suggestive of how to define “a (generic) clock”, in general),

    – his explicitc definition of »an ideal clock«
    (which closely relates e.g. to so-called “Marzke-Wheeler clocks”, even if Gourgoulhon doesn’t spell that out), and

    – his implicit definition of “(average) rate of advance of an ideal clock”
    (namely as inverse of the so-called »constant K« in his equation (2.11))
    .

    p.s.
    Searching your blog for the term “metric”, I came across this page: https://tensortime.sticksandshadows.net/archives/5207 (»A Hypothetical Introductory Chapter«).
    In the first paragraph (after the initial »tl;dr«) there it says: »Feedback is welcome«,
    but there’s (apparently, presently) no »Leave a Reply« section for public commenting (like there is on this and many other pages of your blog).

    Would you please enable commenting there ?
    Or would you mind if I gave some feedback on that page through commenting here on this page ? — Thanks again!

    • Once again, I did not get to a point where I had to define “rate of advance” or “uniformity” in any significant way other than appealing to intuition. I can’t comment on something that did not happen, as I am sure you can understand.

      I was aware of Gourgoulhon’s book only by its title but had not looked at its content until now and I agree that it looks like an excellent resource. I have a copy on order.

      That last page you found is not supposed to be visible and until now I thought it was still hidden. I will remove it since it was meant as only a very preliminary draft.

  • Joe Heafner wrote (2023-01-25 at 12:21):
    > Once again, I did not get to a point where I had to define “rate of advance” or “uniformity” in any significant way other than appealing to intuition. I can’t comment on something that did not happen […]

    Gradually it dawns on me that you’re not “only” referring to the curriculum you had been teaching, but to the full extent of your own understanding and conviction. …

    Concerning the intuitions on which the theory of relativity is based, Einstein has left us (along with precious few thought-experimental illustrations) one concise statement:
    » All our space-time verifications invariably amount to a determination of space-time coincidences.«

    From that I conclude:
    Only the notion of (being able to judge) »coincident [or not coincident]« is “left to intuition” (along with some prerequisit notions such as distinguishing “one-and-the-same” from “several distinct”). All other space-time related notions (such as “free particle”, “good clock” or at least “ideal clock”, “inertial frame”) therefore need to be defined in terms of coincidence determinations. (How to accomplish this in detail seems at least to me a main portion of what can and should be taught as the theory of relativity.)

    Had you not yet gotten to the point of reading and appreciating this statement by Einstein ?
    Or do you have arguments to reject the conclusion I described ? …

    • You keep asking me the same question over and over and I have answered it. I see no point in continuing this discussion as you seem to be trolling. Thank you.

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