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A Traveler-Centered Intro to Kinematics

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A Traveler-Centered Intro to Kinematics A traveler-centered intro to kinematics P. Fraundorf Physics & Astronomy/Center for NanoScience, U. Missouri-StL (63121)∗ (Dated: September 4, 2018) Treating time as a local variable permits robust approaches to kinematics that forego questions of extended-simultaneity, which because of their abstract nature might not be addressed explicitly until a first relativity course and even then without considering the dependence of clock-rates on position in a gravitational field. For example we here use synchrony-free “traveler kinematic” relations to construct a brief story for beginning students about: (a) time as a local quantity like position that depends on “which clock”, (b) coordinate-acceleration as an approximation to the acceleration felt by a moving traveler, and (c) the geometric origin (hence mass-independence) of gravitational acceleration. The goal is to explicitly rule out global-time for all from the start, so that it can be returned as a local approximation, while tantalizing students interested in the subject with more widely-applicable equations in range of their math background. PACS numbers: 05.70.Ce, 02.50.Wp, 75.10.Hk, 01.55.+b Contents In the process of putting into context the Newtonian-concepts of: reference-frame, elapsed- I. introduction 1 time t, position x, velocity v, acceleration a, constant-acceleration-integral and gravitational- II. time as merely local 1 acceleration g, this intro is designed to give students a taste of the more robust technical-concepts highlighted in III. kinematics teaser 2 bold below. If these engender critical-discussion (rather than a focus on intution-conflicts), all the better as such IV. constant acceleration for robots 2 concepts might help inspire the empirical-scientist inside A. low-speed results 3 students even if they never take another physics course. B.any-speedresults 3 The few take-home equations from this introduction that students will likely be asked to master in an intro V. cool any-speed applications 4 course will be highlighted in red. Hence you might simply A.proper-velocityathome 4 consider these notes an alternate, but fun, intro to some B.proper-accelerationatwork 5 key relationships that are often just handed to students (explicitly or implicitly) as given. VI. curving space-time 5 A. acceleration-relatedcurvature 5 B. gravity’s acceleration 6 II. TIME AS MERELY LOCAL VII. discussion 7 In the first part of the 20th century it was discovered Acknowledgments 7 that time is a local variable, linked to each clock’s loca- tion through a space-time version of Pythagoras’ theo- References 7 rem i.e. the local metric equation. Both height in the earth’s gravitational field, and clock-motion, affect the rate at which time passes on a given clock. Both of these I. INTRODUCTION effects must, for example, be taken into account in the arXiv:1206.2877v3 [physics.pop-ph] 11 Mar 2014 algorithms used by handheld global positioning systems. Intro-physics students in an engineering-physics class The fact that time is local to the clock that’s measuring might augment their introduction to unidirectional- it means that we should probably address the question kinematics with a technical story that’s in harmony of extended-simultaneity (i.e. when an event happened with the general trend toward metric-first approaches1,2. from your perspective if you weren’t present at the event) This might help nip the implicit Newtonian-assumption only as needed, and with suitable caution. Care is espe- of universal time in the bud. It would also intro- cially needed for events separated by “space-like” inter- duce: (i) momentum-proportional proper-velocities3,4 vals i.e. for which ∆x>c∆t where c is the space/time that can be added vectorially at any speed5, and constant sometimes called “lightspeed”. (ii) proper-acceleration6 in harmony with the mod- Recognizing that traveling clocks behave differently ern (equivalence-principle based) understanding of also gives us a synchrony-free4 measure of speed with geometric-accelerations i.e. accelerations that arise from minimal frame-dependence, namely proper-velocity3 ~w ≡ choice of a non free-float-frame coordinate-system. d~x/dτ = γ~v, which lets us think of momentum as a 3- 2 FIG. 1: The synchrony-free traveler-kinematic in (1+1)D. FIG. 2: Galileo’s approximation to the traveler-kinematic. vector proportional to velocity regardless of speed. Here τ is the frame-invariant proper-time elapsed on the trav- eling object’s clock, Lorentz-factor γ ≡ dt/dτ, and as servers who are not themselves being accelerated. usual coordinate-velocity ~v = d~x/dt. Before we take this leap, however, we might spend Recognition of the height-dependence of time as a kine- a paragraph describing kinematics in terms of traveler- matic (i.e. metric-equation) effect, moreover, allows us to centered variables (Fig. 1) that allow one to describe mo- explain the fact that free-falling objects are accelerated tion locally regardless of speed and/or space-time curva- by gravity at the same rate regardless of mass. Hence ture. These variables are frame-invariant proper-time gravity is now seen as a geometric force instead of a τ on traveler clocks, synchrony-free proper-velocity proper one, which is only felt from the vantage point of w ≡ dx/dτ defined in the map-frame, and the frame- non “free-float-frame” coordinate-systems like the shell- invariant proper-acceleration α experienced by the frame normally inhabited by dwellers on planet earth. traveler, which for unidirectional motion in flat space- time equals (1/γ)dw/dτ. Acceleration from the traveler perspective is key, because as Galileo and Newton demon- III. KINEMATICS TEASER strated in the 17th century, the causes of motion are in- timately connected to this second-derivative of position The world is full of motion, but describing it (that’s as a function of time. what kinematics does) requires two perspectives: (i) the The relationships above allow us to write proper- perspective of that which is moving e.g. “the traveler”, acceleration as the proper-time derivative of hyperbolic and (ii) the reference perspective which ain’t moving e.g. velocity-angle or rapidity η, defined by setting c sinh[η] “the map”. Thus at bare minimum we imagine a map- equal to proper-velocity w in the acceleration direction. frame, defined by a coordinate-system of yardsticks say These relationships in turn simplify at low speeds (as measuring map-position x with synchronized-clocks fixed long as we can also treat space-time as flat) as shown to those yardsticks measuring map-time t, plus a traveler at right below, because one can then approximate the carrying her own clock that measures traveler (or proper) proper-values for velocity and acceleration (Fig. 2) with time τ. A definition of extended-simultaneity (i.e. not coordinate-values v ≡ dx/dt and a ≡ dv/dt. local to the traveler and her environs), where needed for dx dx problems addressed by this approach, is provided by that proper-vel. c sinh[η]= dτ v = dt coord.-vel. dη → dv synchronized array of map-clocks. proper-accel. α = c dτ a = dt coord.-accel. We will assume that map-clocks on earth can be syn- (1) chronized (ignoring the fact that time’s rate of passage As mentioned above, treating space-time as flat requires increases with altitude), but let’s initially treat traveler- that our map frame be seen as a free-float-frame (i.e. one time τ as a local quantity that may or may not agree experiencing no net forces). Introductory courses there- with map-time t. The space-time version of Pythago- fore concentrate on drawing out uses for the kinematic ras’ theorem says that in flat space-time, with lightspeed equations on the right hand side above. We provide the constant c, the Lorentz-factor or “speed of map-time” ones on the left, to show that only a bit of added com- is γ ≡ dt/dτ = 1 + (dx/dτ)2/c2. This indicates that plication will allow one to work in many other situations for many engineeringp problems on earth (except e.g. for as well. GPS and relativistic-accelerator engineering) we can ig- nore clock differences, provided we imagine further that gravity arises not from variations in time’s pas- IV. CONSTANT ACCELERATION FOR sage as a function of height (i.e. from kinematics) ROBOTS but from a dynamical force that acts on every ounce of an object’s being. In that case we can treat time as global, For a change of pace from most texts, let’s discuss the and imagine that accelerations all look the same to ob- assumptions needed for a computer to derive the equa- 3 tions of constant acceleration. For equations that work at any speed, we’ll also give you some practice treating time as a local instead of as a global variable i.e. as a value connected to readings on a specific clock. A. low-speed results If we define coordinate-acceleration a ≡ dv/dt and coordinate-velocity v ≡ dx/dt where x and t are map- position and map-time, respectively, then holding con- stant the coordinate-acceleration a (which is not the FIG. 3: The synchrony-free traveler-kinematic in (3+1)D. acceleration felt by our traveler at high speeds) allows one to derive the v ≪ c low-speed constant coordinate- acceleration equations familiar from intro-physics texts B. any-speed results for coordinate-velocity v[t] and map-position x[t]. Can you do it? For equations that work at any speed, we begin by The following is what Mathematica needs to pull it off: treating the “proper” time τ on the clocks of a trav- eler as a local-variable, whose value we’d like to figure FullSimplify[DSolve[{ out relative to the local value of the traveler’s position v[t] == x’[t], x and time t on the yardsticks and synchronized clocks a == v’[t], of a reference map-frame.
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