1. Introduction

1.1 ...for a standard "mechanics & heat" class

Can we boost intro-physics students past the "frame-dependence abounds" ambience of special relativity (Albert Einstein preferred [1] "theory of invariants in inertial frames") using clues from the "theory of invariants in all frames" (general relativity), and at the same time empower them (optionally since there may be no time for assignments to that end in the curriculum) to do more cool stuff with familiar 3-vectors, ordinary units, and Newton's laws: (i) in accelerated frames, (ii) in gravitationally curved spacetime, and (iii) at high speeds? To that end let's start with things, like time, that everyone can talk about and agree on.

Forget "global time". Elapsed times are only defined objectively for events which take place along a world line, using the proper time elapsed on clocks traveling that line. Elapsed times between spatially separated events, on the other hand, are a matter of convention most simply defined by one's choice of map-frame i.e. of bookkeeper coordinates associated with a global metric equation. Sophisticated radar time [2] definitions of simultaneity (cf. figure at right), as well as the simpler tangent free-float-frame definition, may also come in handy on occasion but are not fundamental (except perhaps historically). There is also value in considering the proper length of an extended object to be the only robust measure, although the concept of rigidity itself may only be valid locally (i.e. over limited distances) for objects undergoing acceleration. In context of a given metric, there is only one robust (and synchrony-free) measure of velocity, namely a traveling object's momentum per unit mass or proper velocity i.e. the map-distance traveled per unit time on the traveler's clock. The mass referred to in the previous sentence is the rest mass, which also has the advantage of being frame invariant. Choice of a different metric (including the switch to the metric of a co-moving free-float-frame in flat spacetime) might be described with a 3-vector addition of proper-velocities, but only with help from adjustments to component lengths associated with changes in the clocks and metrics involved. Synchrony-free proper velocity is also more robust than coordinate velocity because the latter's requirement of an extended free-float-frame of yardsticks and synchronized clocks can rarely be met.

Because proper-times and proper-velocities are referenced to a single common map-frame (i.e. metric equation) which associates location and possible "far" time coordinates with each event in spacetime, one can always say unambiguously that a traveling object's proper-time clock slows down as its proper-velocity goes up. As in Galilean spacetime, the challenge of multiple frames (e.g. primed and unprimed coordinate systems) only arises when you want to describe motion from the vantage point of observers using different map-frames.

Regardless of metric, acceleration is only defined objectively as the proper acceleration felt by an accelerated traveler, whose magnitude is frame invariant and whose 3-vector direction is well-defined by whatever bookkeeper frame the traveler has chosen. Like time derivatives of energy at any speed, time derivatives of momentum (i.e. forces perceived externally) at high speeds are frame dependent. Hence at high speeds the relationship between frame-invariant proper acceleration and the net frame-variant force is changed a bit as well.

One advantage of this metric-first [3] or "one map two clock" approach [4] is that geometric forces like gravity and centrifugal, which are invisible to your cell phone's accelerometer since it detects only the proper accelerations to which it is subjected, allow one to extend Newton's law in its coordinate form (ΣF = ma) locally to the accelerated frames and curved spacetime that we experience every day here on earth. At low speeds in free-float (inertial) frames and in the absence of gravity, all of these quantities reduce to the coordinate forms (map-time and coordinate velocity/acceleration) first explored quantitatively long ago e.g. by Galileo.

These latter concepts will be the primary focus of this introductory "mechanics and heat" course, in which we will use our understanding of spacetime to (along with Newton) pretend that gravity is a proper instead of a geometric force. The fact that gravity's (and inertial force) effects vanish in free-float frames (like earth orbit) and act on every ounce of our being (causing the same acceleration regardless of mass) will be treated as observed facts or curiosities, even though they are expected properties of geometric (connection-coefficient) forces in Einstein's "theory of invariants in all frames". 

1.2. ...to this approach to modernizing content

An emergent cross-disciplinary objective of Bayesian model selection is the search for models that are least surprised by incoming data, because this goal at once focuses on both goodness-of-fit and algorithmic-simplicity (Burnham2002, Gregory2005). The task of content-modernization in physics-education-research (PER) might, in theory, also be inspired by this objective. In that context this is a paper on kinematics content-modernization. The immediate goal is only to suggest a minor shift in our introduction to the subject, with hopes that future experimental physics education research can explore its impact on the physics experience of future physics-majors as well as the much larger population of future non-majors.

Because introductory physics sequences are often taught by multiple instructors at the same school, it is not easy to adopt texts which involve radically different approaches. Hence progress in modernizing the approach has been slow, even though the introductory series comprises the last course contact that physics departments have with the lion's share of the future consumers and voters that they encounter. Since modern physics is increasingly relevant to everyday life, this is an increasingly important opportunity missed.

The problem has been addressed variously in recent decades for example by Jonathan Reichert's seven-chapter particle-physics introduction to unidirectional motion (Reichert1991), by Tom Moore's Six Ideas text (Moore2003) which has e.g. at Washington University in St. Louis spawned a parallel introductory-physics sequence, and by Louis Bloomfield's How Things Work text (Bloomfield2013) designed to look like a collection of everyday topics to the students while faculty can choose to see it as a standard-format two-semester sequence, with modern physics insights, squeezed into a one-semester course with few math requirements. Although these are important initiatives, the standard-format approach retains a significant market-share.

Given the standard-format approach and limited time, individual teachers and text authors may nonetheless work to minimize the cognitive-dissonance between traditional approaches and modern insights by integrating those historical approaches into a modern context from the start. In statistical physics, for example, this might involve starting from the statistical roots of concepts like temperature and entropy (Moore1997, Fraundorf2003) i.e. the approximations underlying the ideal gas law and equipartition, as has been done in senior undergrad texts for many decades (Kittel1980p, Stowe1984p, Garrod1995, Schroeder2000). In quantum applications it might involve mention of ``explore all paths" logic (Taylor98b) and subsystem-correlations before moving to energy-eigenvalue stationary states. In modern physics generally it might involve illustrations of how the value and application of a given concept (like position in the subatomic world, or time on cosmic scales) can change significantly from one size-scale to the next (Primack2006).

It is the more modest objective of this paper to suggest a few paragraphs to offer at the beginning of the section on unidirectional kinematics. These paragraphs will introduce Galilean kinematics and Newtonian gravity as the approximations that they are, while placing a weak spot in everyone's implicit assumption of global time. Thanks to synergy between relativistic insights at all levels, this opens the door to a more integrative perspective on everyday topics like the perspective from accelerated frames, while offering to ambitious students a range of other types of problems with which they can play from the vantage point of a single bookkeeper or map frame (Bell1987, Scherr2001, Scherr2007, Levrini2008).

This paper therefore provides a ``kinematics teaser" as one possible format for those opening paragraphs, followed by a survey of complementary problem-types that may pique student interest but which will be largely ignored. That's because with or without a more modernized context, classes from most of today's standardized texts have much business-as-usual to cover.

In particular here we argue that intro-physics students in an engineering-physics class might augment their introduction to unidirectional-kinematics with a few paragraphs in harmony with the general trend toward metric-first approaches (Pais1982, Taylor2001, Hartle2002). This might help nip the implicit Newtonian-assumption of universal time in the bud. It would also introduce: (i) momentum-proportional proper-velocities (SearsBrehme1968, Shurcliff1996) that can be added vectorially at any speed, and (ii) proper-acceleration (Taylor1963) in harmony with the modern (equivalence-principle based) understanding of geometric-accelerations i.e. accelerations that arise from choice of a non free-float-frame coordinate-system.

In the process of putting into context the Newtonian-concepts of: reference-frame, elapsed-time t, position x, velocity v, acceleration a, constant-acceleration-integral and gravitational-acceleration g, this intro is designed to give students a taste of the more robust technical-concepts highlighted in bold below. If these engender critical-discussion (rather than a focus on intuition-conflicts), all the better as such concepts might help inspire the empirical-scientist inside students even if they never take another physics course.

Hence you might simply consider these notes an alternate, but fun, introduction to the Galilean-kinematic relationships for tracking motion and its causes (including the assumption of global-time) that are often just presented or assumed fait accompli. It remains to be seen if a few paragraphs, like those in the kinematics teaser below, will better prepare majors and non-majors for the challenges that they face downstream.

Some related references