# 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?

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