This is a page about modern introductions to unidirectional motion. It's only a stub at present. Look for more here soon.
To whet your whistle in the meantime you might enjoy the summary below of a possible writeup, and the discussion of related student exercises below that. At the bottom you'll also find some more technical notes, for which some background in 4-vector analysis may come in handy. Extensive notes on use of this strategy with intro-physics classes are already available on the campus wiki for those with a campus ID.
Scientific objectives: The objective here is to examine advantages of a metric-first approach to the description of motion, starting as early as a student's first introduction to unidirectional kinematics. Like the metric equation itself, the approach starts from the vantage point in flat spacetime of a single map-frame of yardsticks and synchronized clocks, but it explicitly recognizes that time-elapsed on a traveler's clock must be considered separately.
Results Overview: Proper (not coordinate) time/speed/acceleration underlie dynamics when coordinate-velocity v ≡ dx/dt = w/γ ≈ lightspeed c i.e. in both relativistic (w ≈ c) and hyper-relativistic (w >> c) domains. Proper-velocities (w ≡ dx/dτ = γv) add vectorially with help from an out-of-frame magnitude-only correction. Radar-time simultaneity allows one to explore space-time curvature using accelerated perspectives in flat space-time. All reduce to the familiar coordinate relations when v << c.
Introductory physics texts often start with: (i) time as an implicitly universal-variable, and (ii) the oddly mass-independent acceleration-due-to-gravity near earth's surface, as givens with no larger context. In this short note, we take an "engineering" rather than a "physics" approach and (for those who might enjoy it) invoke modern concepts to tell a story about: (a) time as a quantity like position that depends on one's choice of yardsticks & clocks, and (b) the geometric origins of gravitational acceleration. The note is designed to tantalize students interested in the subject with predictive equations within range of their math-background, while for students otherwise interested it provides context while delivering the good-news that their course involves only low-speed approximations.
That possible writeup introduces the space & time (then velocity & acceleration) variables used to track motion with help from Minkowski's space-time version of Pythagoras' theorem. The story is more complicated than the usual one because the distinction between time-elapsed on the traveler's clock (proper-time) τ, and time elapsed on synchronized map-clocks (coordinate-time) t, gives rise to proper and coordinate versions of velocity (w≡dx/dτ and v≡dx/dt) and acceleration (α and a) as well. Thankfully, all of these reduce at low speeds to the coordinate versions (t, v, and a) actually put to use in a typical intro-physics course.
An interesting model to illustrate these concepts at both low and high speeds is that of the "constant-acceleration" round-trip. A terrestrial-version of this roundtrip is illustrated in the animation at top right. Click on it for a discussion of some activities that might be fun to try with it.
The graphs to the right illustrate the quantitative predictions of Newton's equations for such round-trips, including the way that round-trip time for a given amount of acceleration is proportional to the square-root of the distance to be traveled one-way. Hopefully, clicking on that plot will provide clues to the underlying equations, as well as what happens when traveler-velocities (both coordinate and proper) begin to approach lightspeed c.
Time-dilation of course is not the subject-matter of an intro-physics course. However we argue that it's not a bad idea to provide examples early on nonetheless for two reasons: (i) It's cool; and (ii) The idea of specifying "which clocks" goes hand-in-hand with the idea of specifying "which yardsticks" i.e. your choice of coordinate-system.
This is therefore a sneaky but hopefully-fun way to deconstruct the notion of "universal time" otherwise implicit in some ancient worldviews (including Newton's). The figure at right is example of the kind of problem students might enjoy thinking about, even if it is not going to help them on their intro-physics exams.
Although time-dilation across the breadth of an accelerated object is fun (and potentially confusing) to think about, we don't have tons of everyday applications to offer in this area. However, purely kinematic-acceleration as a source of space-time curvature in an accelerated-traveler's coordinate-system (particularly with help from radar-time definitions of simultaneity) is we feel an underutilized pedagogical tool since it can be addressed using only calculus and a few metric equations.
The four-vectors and four-tensors of general-relativity of course are needed to address the causes of mass-related space-time curvature, and allow for much simpler "coordinate-independent" expression of important relationships. When it comes to solving "boots on the ground" problems, however, the fact that students with much less math can start playing with these phenomena quantitatively is probably worth a closer look.
Although it may also be fun to think about your head aging faster than your feet, or the earth's surface aging faster than its center, the most oft-cited practical application of time-dilation associated with gravitational-acceleration is the gravitational-part of global-positioning-system time-dilation corrections. An earlier figure on this subject is posted for exploration at right, as well.
This older (2+1)D animation of a constant acceleration round-trip from map and traveler points of view uses co-moving free-float-frame simultaneity, as distinct from the more robust radar-time simultaneity used in the illustration of a single constant-acceleration leg below that.
A (1+1)D version of a single-segment acceleration using radar-time simultaneity from both map (left) and traveler (right) perspectives is shown here by way of comparison. Note that the curvy-nature of traveler coordinates in the x-ct diagram at left arises simply from accelerated-motion in the flat-space metric.
Question: Is there a metric-equation for acceleration-related time dilation as well as for the other two time-dilations mentioned in the writeup?
Possible answer: Yes. In fact these time relations all come from three expressions for what may be the same invariant-separation δs between events:
The first is of course the flat-space expression which may only work locally. The second is the local radar-metric in terms of coordinates τα and ρα measured by an observer undergoing constant proper-acceleration α in the +x direction (within the forward light-cone of acceleration's onset & the rearward light-cone of its stop). The third metric is of course the Schwarzschild metric mentioned, i.e. the metric at finite r from a massive object in terms of far coordinates. These metrics all offer different perspectives on ways to break down the same space-time interval δs into constituent time and space parts.
Note that technically, then, the "local zone" acceleration-related radar-time dilation might be more clearly written for a fixed lag-distance L as:
Here only the last and least-accurate approximation above for L << c2/α is the square-root of 1-2×energyratio (i.e. mαL/mc2) mentioned in the writeup. Nonetheless the form has mnemonic value for comparison to the other two types of dilation treated.