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Traveler-Point Dynamics: a 3-Vector Bridge to Curved Spacetimes Phil Fraundorf

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Traveler-Point Dynamics: a 3-Vector Bridge to Curved Spacetimes Phil Fraundorf Traveler-point dynamics: A 3-vector bridge to curved spacetimes Phil Fraundorf To cite this version: Phil Fraundorf. Traveler-point dynamics: A 3-vector bridge to curved spacetimes. 2021. hal- 01503971v3 HAL Id: hal-01503971 https://hal.archives-ouvertes.fr/hal-01503971v3 Preprint submitted on 24 May 2021 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. Traveler-point dynamics: A 3-vector bridge to curved spacetimes P. Fraundorf∗ Physics & Astronomy/Center for Nanoscience, U. Missouri-StL (63121), St. Louis, MO, USA† (Dated: May 24, 2021) Locally-defined parameters chosen to be minimally frame-variant can be useful for describing mo- tion in accelerated frames and in curved spacetimes. In particular the metric-equation’s synchrony- free “traveler-point parameters”, namely scalar proper-time and the 3-vectors: proper-velocity and proper-force, are useful in curved spacetime because extended arrays of synchronized clocks (e.g. for local measurement of the denominator in ∆x/∆t) may be hard to find. Combined with a recognition of improper (geometric) forces, these same parameters can better prepare intro-physics students for a world where Newton’s approach is a only useful approximation. This strategy also suggests an “accelerated-traveler perspective” on high speed travels that are exact at any speed in flat spacetime, and approximate in curved settings e.g. for computing gravitational trajectories at high speed. CONTENTS proper velocity4, and 3-vector proper acceleration5 are inherently local to one traveler’s perspective, hence we I. introduction 1 refer to them as ”traveler-point” quantities. The advan- tage is that they are either frame-invariant (in the case II. proper time 1 of proper time and the magnitude of proper-acceleration) or synchrony-free6 (i.e. do not require an extended net- III. proper velocity 2 work of synchronized clocks which of course is not always available in curved space time and accelerated frames). IV.properacceleration 3 In a larger context, general-relativity revealed a cen- tury ago why we can get by with using Newton’s laws V.causesofmotion 4 in spacetime so curvy that it’s tough to jump higher than a meter. It’s because those laws work locally in all VI. discussion 6 frames of reference7,8, provided we recognize that motion is generally affected by both proper and “geometric” (i.e. Acknowledgments 6 accelerometer-invisible connection-coefficient) forces. In that context, it’s probably time to give introductory References 6 students the good news. Here we discuss a way to do this without telling them to measure time and distance in e.g. A.possiblecoursenotes 7 kilograms, and without asking them to juggle more than 1. spacetime version of Pythagoras’ theorem 7 one concurrent defintion of simultaneity. 2. traveler-point kinematics 7 The metric equation of course avoids these things (cf. 3. dynamics in flat spacetime 8 Fig. 1) by specifying locally-defined frame invariants like 4. dynamics in curved spacetime 8 proper-time, and contenting itself with a single defini- tion of simultaneity i.e. that associated with the set of book-keeper (or map) coordinates that one chooses to de- I. INTRODUCTION scribe the spacetime metric. Hence the focus here is on a metric-first9, as distinct from transform-first10, strategy In an excellent 1968 article1, Robert W. Brehme dis- for describing motion in accelerated frames, in curved cussed the advantage of teaching relativity with four- spacetime, as well as at high speeds. vectors. This is not done for intro-physics courses. One reason may be that spacetime’s 4-vector symmetry is bro- ken locally into (3+1)D, so that e.g. engineers working II. PROPER TIME in curved spacetimes2 (including that here on earth) will naturally want to describe time-intervals in seconds and Proper-time is the time elapsed locally along the world- space-intervals e.g. in meters. In this note, we therefore line of any physical clock (broadly defined). It is also the explore in depth an idea about use of minimally frame- frame-invariant time-interval δτ that shows up in metric 2 variant scalars and 3-vectors, perhaps first hinted at in equations of the form (cδτ) = ΣµΣν gµν δXµδXν , which Percy W. Bridgman’s 1928 ruminations3 about the wis- in the flat-space Cartesian case takes the Pythagorean dom of defining 3-vector velocity using an ”extended” form (cδτ)2 = (cδt)2 − (δx)2 − (δy)2 − (δz)2, where book- time system rather than defining 3-vector velocity using keeper or map coordinates are found on the right-hand traveler proper time instead. side of the equation, and c is the “lightspeed” spacetime The problem with the latter, of course, is that mini- constant for converting e.g. seconds into meters. mally frame-variant quantities like proper time, 3-vector When simultaneity is defined by a flat-spacetime book- 2 FIG. 2. Proper-velocity addition for a sci-fi puzzler, in which FIG. 1. Even one extended-frame with synchronized clocks your starfleet battle-cruiser drops in from hyperspace heading may be more than we have in curved spacetime. away from a nearby star, only to find the enemy ship heading off in another direction. keeper coordinate frame, time-elapsed on traveling clocks is always “dilated” (e.g. spread out), since from the ordinate velocity ~v ≡ d~x/dt it has no upper limit. Also metric equation the differential-aging or Lorentz factor unlike coordinate velocity, its measurement does not de- γ ≡ dt/dτ ≥ 1. Expressions for this in terms of velocity pend on map-clock readings (synchronized or not) along will be discussed below. that world line. Proper-velocity is also represented by We won’t discuss the utility of proper-time τ in detail the space-like components of a traveling object’s velocity here because it is widely used now even by introductory 4-vector. textbook authors11. However it’s worth pointing out that An interesting thought problem for future engineers early treatments of special relativity were so focused on (as well as teachers) might be to ask if speed-limit signs, the equivalence between frames, each with their own ex- in a “Mr. Tompkins” world14 with a much slower light- tended set of synchronized clocks, that proper-time (if speed spacetime constant, would use coordinate-speed or discussed12) was generally a late-comer to the discussion, proper-speed. Four criteria to consider might involve the and of course used only for “rest frame” (i.e. unacceler- measure’s connection to momentum, kinetic-energy, and ated) travelers10. to reaction-times (after the “warning photons” arrive) for Moving yardsticks also have a “proper-length” associ- both the driver, and a pedestrian who is tempted to cross ated with them, which can be useful too. However this the street. is a “non-local” quantity not defined by the metric equa- The time-like component of that 4-vector is cγ, where tion, the concept of “rigid body” is only an approxima- γ ≡ dt/dτ is the also-useful (but not synchrony free) tion in spacetime, and length contraction itself involves Lorentz differential-aging or “speed of map-time” fac- three separate events as distinct from the two involved in tor mentioned above. In flat spacetime, from the met- time-dilation. Hence we don’t include proper-length in a ric equation it is easy to show that γ = w/v = collection of those traveler-point parameters which show p1 + (w/c)2 ≥ 1. promise of wide-ranging usefulness in accelerated frames In gravitationally-curved Schwarzschild spacetime and curved spacetime. with “far-coordinate” map distances and times, one gets instead from the metric for non-radial proper-velocity wφ 2 that γ ≡ dt/dτ = γrp1 + (wφ/c) ≥ 1, where the grav- III. PROPER VELOCITY itational time-dilation factor is γr ≡ 1/p1 − (rs/r) ≥ 1 and r ≥ rs is distance from the center of a mass M 13 Proper-velocity ~w ≡ d~x/dτ is the rate at which book- object whose Schwarzschild (event-horizon) radius rs = keeper or map 3-vector position ~x is changing per unit- 2GM/c2. Global positioning system satellites have to time elapsed locally on traveler clocks. Because it is consider both of these factors in compensating for the proportional to momentum (i.e. ~p = m ~w), unlike co- rate of time’s passage in their orbits around earth. 3 For unidirectional velocity addition, the advantage of a collider over an accelerator is easily seen because wAC = γABγBC (vAB +vBC ). In other words, the Lorentz-factors multiply even though the coordinate-velocites (which add) never exceed c. Proper velocity’s direct connection to momentum makes it more relevant to speed-limits e.g. in worlds with a reduced value for lightspeed, and its di- rect connection to traveler time makes it more relevant to passengers and crew when planning long high-speed trips. More generally, unlike coordinate-velocities, proper ve- locities add vectorially provided that one rescales (in magnitude only) the “out of frame” component. In other words ~wAC = γAC~vAC = ( ~wAB )C + ~wBC . Here C’s view of the out-of-frame proper-velocity ( ~wAB)C is in the same FIG. 3. Accelerometer data from a phone dropped and caught direction as ~wAB but rescaled in magnitude by a factor three times.
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