The proper-force 3-vector P Fraundorf To cite this version: P Fraundorf. The proper-force 3-vector. 2016. hal-01344268v2 HAL Id: hal-01344268 https://hal.archives-ouvertes.fr/hal-01344268v2 Preprint submitted on 16 Sep 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The proper-force 3-vector P. Fraundorf1, a) Physics & Astronomy/Center for Nanoscience, U. Missouri-StL (63121), St. Louis, MO, USA (Dated: 16 September 2016) The distinction between proper (i.e. cell-phone detectable) and geometric (i.e. connection-coefficient) forces allows one to use Newton’s 3-vector laws in accelerated frames and curved spacetime. Here we show how this is assisted by use of quantities that are either (i) frame-invariant or (ii) synchrony-free i.e. do not rely on extended-networks of synchronized-clocks. The acceleration four-vector’s invariant magnitude, and quantities that build on the metric-equation’s book-keeper frame to define simultaneity, point the way to more robust student understanding at both low and high speeds. In the process, we gain a simple (3+1)D flat-space work- energy theorem using the proper-acceleration 3-vector ~α (net proper-force per unit mass), whose integrals of the motion simplify with a hyperbolic velocity angle (rapidity) written as 2/(γo + 1)ατ/c, where c is lightspeed and τ is traveler-time from “turnaround” when the Lorentz-factor ispγo. CONTENTS These latter might be difficult to come by on accelerated platforms and in curved spacetime. I. introduction 1 The first proper-time derivative of an accelerated trav- eler’s 4-vector position has lightspeed c as its invari- II. frame dependence & synchrony 1 ant magnitude. Here we simply define simultaneity us- ing bookkeeper coordinates and then examine the sec- III. low speed applications 2 ond proper-time derivative of position, as seen from the proper reference-frame3 of that accelerated traveler. IV. bringing in the metric 3 In the process we show: (a) that the distinction be- tween proper and geometric forces is already quite useful V. any speed applications 4 for introductory physics, (b) that via the metric equa- tion a lot can be done with only a single extended map- VI. discussion 5 frame of yardsticks and synchronized clocks, and (c) that Acknowledgments 6 the traveler’s view of anyspeed-acceleration is less frame- variant than the map perspective. We also exploit the A. 3-vector relativity 6 frame-invariance of proper-force in an empirical observa- tion exercise on the electrostatic origin of magnetism, which provides some visceral experience with length- I. INTRODUCTION contraction at the same time. Relativists have long expressed unhappiness with coordinate-acceleration and coordinate-force (for good II. FRAME DEPENDENCE & SYNCHRONY reason1,2), but have also pointed out that general- relativity makes a case for the local-validity of Newton’s The value of frame-independence in the modeling of laws in all frames3–5 provided that we consider geometric relativistic-motion and curved-spacetime goes without (curved-frame or “connection-coefficient”) forces as well saying. The frame-invariance of lightspeed c (the mag- as proper-forces whenever we find ourselves in a non- nitude of the velocity 4-vector U λ ≡ dXλ/dτ) has “free-float” trajectory6. The distinction has everday rel- been central to our understanding of spacetime from evance to intro-physics students, because texts often in- the beginning7. Proper-time (the magnitude of the dis- troduce gravity as a real force but inertial forces as fake placement 4-vector Xλ) is finding increasing use by even though their cell-phones detect neither for the same introductory text authors as we speak. reason: Accelerometers only detect proper-forces, while The Lorentz-transform view of proper-time, of course, gravity and inertial (e.g. centrifugal) forces are geomet- is that it is time-passing on the synchronized clocks of a ric. tangent but co-moving free-float-frame in flat spacetime. In this paper we explore an approach to accelerated The metric equation’s view of proper-time is simpler but motion designed to be: (i) the most frame-independent, more general, i.e. as a quantity measured on a single and (ii) the least in need of synchronized-clock arrays. clock under any conditions i.e. accelerated or not, in curved space-time or not. Proper-time is frame-invariant in the sense that its value may be agreed upon using any general-relativistic a)[email protected] book-keeper coordinates that we choose. These book- 2 TABLE I. Accelerated-motion definitions in flat (3+1)D spacetime. Note that acceleration/force magnitudes are spacelike, while the others are timelike along a traveler’s worldline, and that we’ve defined x and y as spatial coordinates and to the direction of proper-acceleration 3-vector ~α. || ⊥ 4-vector magnitude time-components to spatial 3-vector ~α to spatial 3-vector ~α || dp ⊥ dpo P 1 dE || dp⊥ power/force ΣFo dτ c =( c ) dτ F|| dτ F⊥ dτ ≡ dw ≡F f ≡ ΣFo dγ P P || || || dw⊥ F⊥ f⊥ acceleration α = m c dτ = mc = γ mc dτ = m = γ m dτ = m = γ m E energy/momentum mc c = γmc p|| = mw|| p⊥ = mw⊥ p dt E dx || dx p⊥ velocity c cγ c = w = = γv w⊥ = = γv⊥ ≡ dτ mc || ≡ dτ dτ || ≡ dτ dτ coordinate cτ ct x y TABLE II. Relationship between variables: Here τ is traveler-time elapsed from “turnaround” (when γ γo) for as long as ≡ proper acceleration ~α doesn’t change, and γ± (γo 1)/2. The right arrow denotes the non-relativistic limit. ≡ p ± → 4-vector invariant time-components/c to spatial 3-vector ~α to spatial 3-vector ~α dw|| ⊥ dγ α ατ α 2 || ατ dw⊥ ατ accel. α = γ+ sinh τ = α cosh α = αγ− sinh 0 dτ c h cγ+ i c dτ h cγ+ i dτ h cγ+ i → → 2→ 2 2 ατ ατ ατ velocity c γ = γ + γ cosh 1 w|| = cγ+ sinh ατ w⊥ = c√2γ− cosh v⊥ − + h cγ+ i h cγ+ i h 2cγ+ i → 2 → → 2 3/2 c ατ 2 2c2 ατ 1 2 c2 ατ coord. τ t = γ−τ + γ sinh τ x = γ+ sinh ατ y = γ+γ−cτ + γ+ sinh v⊥τ + α h cγ+ i → α h 2cγ+ i → 2 2α h cγ+ i → keeper coordinates are alone used to define extended si- approach to kinematics here by choosing a single “book- multaneity (i.e. the global place and time of events), keeper” coordinate-system in terms of which both “map- while the frame-invariance of proper-time drastically im- time” t and “map-position” ~x are measured. Simultane- proves the transformation-properties of quantities differ- ity will be defined in terms of synchronized (but not al- entiated with respect to it. The proper-velocity 3-vector ways local e.g. in the case of Schwarzschild “far-time”) ~w ≡ d~x/dτ (which unlike coordinate velocity ~v ≡ d~x/dt clocks in this book-keeper frame. adds vectorially with appropriate rescaling of the “out-of- In addition purely space-like vectors, along with frame- frame” component) and the proper-acceleration 3-vector invariants, may be described as “synchrony-free” to use (discussed here) are cases in point. a word employed by William Shurcliff when discussing The topic of this paper is in particular the frame- proper-velocity12,13 ~w ≡ d~x/dτ = ~p/m. These are quan- invariant magnitude of the acceleration 4-vector, in tities whose operational-definition does not require an ex- standard notation3: tended network of synchronized-clocks, something of lim- ited availability around gravitational-objects (like earth), λ λ λ DU dU λ µ ν and impossible to find on platforms (like spaceships) un- A := = +Γ µν U U (1) dτ dτ dergoing accelerated motion. The time-like energy of a moving object via its dependence on the Lorentz-factor and uses for this vector’s components (as power/force) γ ≡ dt/dτ is (like “mixed objects” such as coordinate- when they are multiplied by frame-invariant rest-mass λ velocity ~v ≡ d~x/dt) not synchrony-free, because it re- m. Here free-float or geodesic trajectories have A = 0, quires map-time t data from clocks at multiple locations. so that we can think of coordinate acceleration dU λ/dτ The “traveler’s point of view” that we argue offers the as a sum of proper and geometric terms, the latter most direct way to communicate about an accelerated depending on local space-time curvature through the traveler is the frame that Misner, Thorne and Wheeler3 64-component affine-connection Γλ which gives rise to µν refer to as “the proper reference frame of an accelerated “apparent” forces in accelerated coordinate-systems and traveler”. One can always convert these to expressions curved space-time. As usual greek indices run from written in terms of bookkeeper variables like map-time 0 (time-component) to 3 (space-components) and obey t and coordinate-velocity ~v, but we show here that the the Einstein summation convention when repeated in a algorithmically-simplest way to describe the effects of the product.