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Conclusions V. .Introduction I. eaiitcevrnetdelr nteerhsgravi- earth’s the in dwellers environment Relativistic References Acknowledgments .TeEuvlneFx4 Fix Equivalence The A. .TeSbetv-yaisFx3 Fix Subjective-Dynamics The A. .TeSeie-lc i 1 Fix Specified-Clock The A. .Cneune fEuvlne5 Equivalence of Consequences B. .Cneune fSbetv-yais3 Subjective-Dynamics of Consequences B. .Cneune fSeie-lcs2 Specified-Clocks of Consequences B. .Dabcs ieo hl 6 5 5 Shell a on Life Drawbacks: Gravity 3. Extreme Extension: Carousels 2. Anyspeed Extension: 1. .Dabcs ne-cl eso 4 4 4 3 Tension Inter-Scale Drawbacks: 4. Magnetism One-Frame Extension: Chase-Plane 3. Galileo’s Extension: Acceleration 2. Proper Extension: 1. .Dabcs reaiiy2 2 2 Irrelativity Drawbacks: Two-Clocks 3. One-Map Extension: 2. Airtracks Fast Extension: 1. ae etna ol htte pn omc ielann a learning time much course) of so (locally, even spend and they frames, r that non-inertial encounter speeds, they tools physi when introductory Newtonian dissonance to based cognitive advantages two less offer experience might context, in hsc n srnm/etrfrMlclrEetois U Electronics, Molecular for Astronomy/Center and Physics oicto ftreiesudryn etnsoiia wo original Newton’s underlying ideas three of Modification .INTRODUCTION I. 1 vni oriaefe insights coordinate-free if even , Contents oenzn etn owr tayspeed any at work to Newton, Modernizing 3 Dtd oebr1,2018) 17, November (Dated: . 2 Hence . .Fraundorf P. 6 6 6 6 4 3 1 1 omaueposition measure to time epfo h qiaec rnil)t rirr coordi- well. arbitrary as to systems principle) nate equivalence to the (with tools from and speeds, help further Newtonian high Their at It familiar acceleration involving of problems gravity. changes. extension of major the require views facilitates not modern does or either elimination considers motion later speed gener- one high when assumptions (iii) dissonance These cognitive and frames. ate dynamics, inertial frame-invariant to (ii) limitation time, worldview: classical global the (i) in implicit assumptions three ing nmpcok ie e fsnhoie lcscon- clocks synchronized of set a (i.e. map-clocks on supino lbltm,aantwihms spoken most protection. in which built against little have time, languages global of assumption urn adsedrcr eg o 0GVelectrons) GeV 50 10 for of order (e.g. the the on record year, being traveler speed per lightyears land of current speeds units could particle in time” if expressed traveler accelerators are unit in per interest acceleration distance public particle augment “map for describ- of record Likewise, terms speed birth. in land of world within one place the places their in ing to of travel travel years range light their 100 limiting long of for instead options relativ- lifetime, that their know enhances children ity few example, spacetime For extreme and/or curvature. in speeds tools high frame-based involving of application projects also wider might to pain changes cons door the the These and open minimize later. pros can tools understanding the that deeper the emphasis at of with in looking changes bets – some our motion of hedge spacetime. describe curved to to in ways used and here fundamen- in speeds discuss fail We relativistic motion at about conditions, ways thinking tal supersonic for tools under the apply so don’t students tory aeahbto pcfigtecokue omeasure to used clock the specifying of habit a Make cest mrvdtoscnb aiiae yavoid- by facilitated be can tools improved to Access Tm stesm o vroe”Ti steimplicit the is This everyone.” for same the is “Time uta oefli-o ol htw ffrt introduc- to offer we that tools fluid-flow some as Just ocurved-spacetime. to t swl stecodnt ytmyrsik used yardsticks system coordinate the as well as , ltvsi ffcs eody h map- the Secondly, effects. elativistic ssuet.Frt h tdnswill students the First, students. cs isuiS.Lus O63121 MO Louis, Missouri-St. . l iw ihol io changes minor only with view, rld I ICNETO ONE MISCONCEPTION II. .TeSeie-lc Fix Specified-Clock The A. otcnb xeddt high to extended be can bout x ihtm pcfida measured as specified time With . 5 ! 2 nected to the yardsticks with respect to which distance is (in their own terms) of phenomena underlying the metric measured), the usual kinematic equations of introductory equation as well as relativistic expressions for the kinetic physics need not be changed, even if their precise mean- energy, K = (γ 1)mc2, and momentum, p = mw, of a ing (at least in the case of equations describing relative traveler of mass−m.7 A particular student’s exploration motion) is different. might for example uncover, instead of the metric equa- tion, the mathematically equivalent relation that holds constant the sum of squares of their “coordinate speeds” B. Consequences of Specified-Clocks through space (dx/dt) and time (cdτ/dt).
When t represents time on “map-clocks”, the usual definitions of velocity and acceleration (as derivatives of map-position with respect to t) define what rela- tivists sometimes call “coordinate” velocity and acceler- 2. Extension: One-Map Two-Clocks ation, respectively2. These definitions are useful at any speed, although they become less global when spacetime The conceptual distinction between stationary and is curved. If the integrals of constant acceleration, like 2 moving clocks opens the door for students to discover, ∆v = a∆t and ∆(v )=2a∆x, represent relations be- and apply, Minkowski’s spacetime version of Pythago- tween map-time and map-position, they follow simply ras’ theorem8 (c∆τ)2 = (c∆t)2 (∆x)2, i.e. the metric from these definitions and hence are also good at any equation, as an equation for time− τ on traveler clocks in speed. The caveat, of course, is that holding coordinate context of coordinates referenced to a single map frame9. acceleration finite and constant is physically awkward at Lightspeed c here acts simply as a spacetime constant high speeds, and impossible to do forever. This dimin- that connects traditional units for distance and time. ishes the usefulness of these equations in the high speed 4 The metric equation connects (i) traveler and map time case , but not their correctness. Thus being specific intervals, (ii) coordinate velocity, v dx/dt, with proper about clocks yields classical definitions of velocity and velocity10, w dx/dτ, and (iii) special≡ to general rela- acceleration, plus familiar equations of constant coordi- tivity as well. ≡ nate acceleration, that work even at relativistic speeds. The kinematical equations for relative motion (e.g. One consequence of such an introduction to clock be- ′ havior at high speed, for example, is that an upper limit x ∼= x + vot and its time derivatives) now refer only to yardsticks and clocks in a single reference frame. Hence on coordinate-velocity v may appear more natural to stu- they only tell us about “map-frame” observations. While dents since it follows from the lack of any such limit on they are correct in these terms, the prediction of prime- proper-velocity w. One would hardly expect this latter frame perspectives is only valid at low speed. Since mul- quantity (map distance traveled per unit traveler time) tiple inertial frames are a challenge in and of themselves5, to exceed infinity for a real world traveler. The absence simply describing it as a low-speed approximation may of an upper bound also makes a proper-velocity of one be the best choice in an introductory course. Nonethe- [lightyear per traveler year] at v = c/√2 a natural “scale less much can be done from the perspective of a single speed” for the transition from sub-relativistic to relativis- inertial frame to give quantitative insight into motion at tic regimes. The distinction between traveler and map high speeds. Exact position, velocity, and acceleration time also yields a natural definition for Einstein’s gamma transforms from the metric equation (like relativistic ve- factor as a speed of map-time per unit traveler-time given simultaneity defined by the map, i.e. γ dt/dτ. The locity addition and the Lorentz transform) thus may not ≡ provide the most useful insights in limited time. In any metric equation then easily yields the familiar relations, γ =1/ 1 (v/c)2 = w/v. case with subtle changes like this, introductory physics − courses can become natural venues for amending the as- p sumption of “global-time” implicit in the wiring of our brains.
3. Drawbacks: Irrelativity 1. Extension: Fast Airtracks The downside of specifying clocks, of course, is that Rather than simply “being told” the metric equa- global time is more deeply-rooted and simpler than clock- tion and ways to use it, students might benefit from specific time. For example, we often treat magnetism and time spent first pondering how to explain data on trav- gravity as non-relativistic add-ons to a classical world eler/map time differences, in the spirit of Piaget and without giving it a second thought. On the other hand, modeling workshop6. In that case an airtrack thought our present understanding of these phenomena grows experiment, with adjustable kinetic-energy source (e.g. from a metric equation that looks like Pythagoras’ theo- a spring) and two gliders that stick on collision, is ad- rem with a minus sign. Thus only tradition tells us that equate to give students data for experimental discovery relativistic effects don’t color our lives everyday. 3
III. MISCONCEPTION TWO ordinate system” of an accelerated object2, while the cor- responding frame-variant forces differ and may be accom- “Coordinate acceleration and force are frame- panied by frame-variant energy changes. The potentials invariant.” Informal polls at one state11 and one that give rise to such frame-variant forces inevitably have national12 AAPT meeting suggest that physics teachers both time-like (e.g. electrostatic) and space-like (e.g. 8 often presume that the rate of change of conserved quan- magnetic) components . In curved space-time and/or tities (energy as well as momentum) is frame-invariant, rotating coordinate systems, the proper-acceleration and even though the expression for power as a product of proper-force retains it’s three-component form only from force and velocity makes this clearly incorrect at even instant to instant, within the traveling object’s “proper 2 the lowest speed. reference frame” .
A. The Subjective-Dynamics Fix B. Consequences of Subjective-Dynamics
Address the issue of frame invariance when dynami- To begin with, students are alerted to the fact that rel- cal quantities are first introduced. Point out that the ativity makes many quantities more dependent on one’s value of the conserved dynamical quantities momentum choice of reference frame. For some of these quanti- ties (time increments, object length, mass18, traveler and energy, along with their time derivatives force and 19 17 power, depends on the frame of reference (even though velocity , and acceleration ) there is a “minimally vari- force is nearly frame-independent at low speeds). Like- ant” or proper form which can simplify discussion across wise, note that coordinate acceleration a = dv/dt is not frames. necessarily the acceleration felt in the proper frame of a traveler. The former is impossible to hold finite and 1. Extension: Proper Acceleration constant indefinitely, while proper acceleration α is both frame-invariant and possible to hold constant13. This is what Newton had guessed (incorrectly) to be the case for Recognizing the distinction between coordinate and coordinate acceleration, by a perfectly reasonable appli- proper acceleration, students also gain (from the met- cation of Occam’s razor given the observational data that ric equation) simple 1D equations for describing constant he had to work with at the time14,15. proper-acceleration. In terms of coordinate integrals like Quantifying these accelerations at arbitrary speed is those for constant coordinate-acceleration above, there 20 simple in the unidirectional case16, where α = γ3a. An are three equations instead of two because the met- ric equation relates three coordinates: map-time t, map- invariant proper-force Fo = mα may also be defined in this context17. Quantifying the conserved dynamical position x, and traveler-time τ. The integrals are ∆w = α∆t, c2∆γ = α∆x, and c∆η = α∆τ where the “hyper- quantities further requires relativistic expressions for mo- −1 2 bolic velocity angle”17 or rapidity η tanh [v/c]. One mentum p = mw and energy E = γmc . Their deriva- ≡ tives with respect to map-time then become the famil- can also give students a set of map-based Newton-like iar net frame-variant force ΣF = mdw/dt and power laws good in (3+1)D flat spacetime, with only a modest 21 dE/dt =ΣF v. amount of added complication . These quantities• most directly represent the way that For example, Newton’s second law for flat spacetime momentum and energy associated with a traveling ob- looks like ← ject is changing, from the perspective of a map-frame α ← v⊥≪c ← v≪c ← observer. Proper-time derivatives of momentum and en- F = m i l = mα = ma (1) γt ergy are of course easier to transform from frame to frame (they form a 4-vector), but they also represent a per- where column three-vectors are denoted by left-pointing spective intermediate between that of the map-frame ob- arrows, and unit vectors are denoted by the letter i. As server and the traveling object. The focus here is on usual v and a denote coordinate-velocity and coordinate- local implementation of the insights in terms of map- acceleration, while α denotes proper-acceleration. Vec- frame observer and traveling-object experiences. Hence tor components transverse and longitudinal to the frame- references here to coordinate-free insights that underlie variant force direction are denoted by subscripts t and l, the conclusions (e.g. to the four-vector nature of certain respectively, while components perpendicular and par- quantities) are for instructor reference, but of limited use allel to the frame-invariant proper-acceleration three- to students wanting a sense of anyspeed motion only in vector are denoted by subscripts and . Also γt 2 ⊥ k ≡ concrete terms. 1 + (wt/c) , and the unit vector longitudinal to the Perhaps coincidentally in the unidirectional motion pframe-variant force is related to unit vectors perpendic- case, the net frame-variant force ΣF and the proper-force ular and parallel to the proper-acceleration by Fo are equal. In flat (3+1D) space-time, proper acceler- ation and proper force behave like a 3-component scalar ← 1 ← v⊥ vk ← i l = γtγ⊥ 2 i k + i ⊥ , (2) (i.e. no time component) within the “local space-time co- γ⊥ c c 4 where γ 1/ 1 (v /c)2. Hence the frame-variant IV. MISCONCEPTION THREE ⊥ ≡ − ⊥ force direction differsp from that of the invariant proper acceleration only when the velocity of the object being “Newton’s laws work only in unaccelerated frames.” effected is neither perpendicular to, nor parallel to, the This misconception is often echoed in classes whose first proper acceleration. example of a force is the affine-connection force2 gravity, which like centrifugal force arises only if one chooses a “locally non-inertial” coordinate system. Of course this 2. Extension: Galileo’s Chase-Plane would be fine for historical reasons, if the equivalence principle hadn’t elegantly shown that Newton’s laws are useful locally in any frame24. This is potentially motiva- Relativistic equations for constant proper accelera- tional news for those first struggling to understand how tion work poorly at low speeds because of roundoff er- Newton’s laws work, and perhaps worth sharing from the ror in calculating the difference between squares, while start. Galileo’s equations22 (which students spend so much time learning) are elegant in their simplicity. Perhaps it is good news then that by going from “one-map two-clock” A. The Equivalence Fix descriptions of motion ala the metric equation to one- map and three-clocks, Galileo’s equations can be shown to apply to constant proper-acceleration at any speed. Show examples of the way that Newton-like laws work locally in any frame, when one includes “geometric” Specifically, in terms of time T on the clocks of a (affine-connection) forces (e.g. centrifugal or gravity) suitably-motivated “chase-plane”, one can show23 that that act on every ounce of one’s being. Geometric forces the Galilean-kinematic velocity V dx/dT of a trav- also have the property that they may be made to van- eler undergoing constant proper-acceleration≡ α obeys ish at any point in space and time, by choosing a “lo- ∆(V 2)=2α∆x, ∆V = α∆T , etc. This familiar and cally inertial” coordinate system. Non-local effects, like simple time evolution seamlessly bridges the gap to low tides and coriolis forces, of course may not be possible to speeds since v
The question is not if these forces exist. One’s sense of perience a radial acceleration of the form being forced to the side of vehicles as they round curves is as real as one’s sense of being forced to the ground when d2r dφ 2 = r γΩ , (5) a chair leg breaks. Rather, we might better be asking: dτ 2 − dτ What is the range of positions and times over which such “geometric forces” can be seen to govern motion in their where γ is as usual dt/dτ. As any car passenger can at- non-inertial setting, while remaining consistent with a set test, accelerations like this are experienced as forces that of Newton-like rules. act on every ounce of one’s being. For observers at fixed φ, this “affine-connection force” is a relativistic version of the familiar centrifugal force felt as one goes around B. Consequences of Equivalence curves in a car. This treatment of the problem goes fur- ther even for low-speed application. For observers with Students are readied for simple equations that compare fixed r and changing φ e.g. for azimuthal track runners in centrifugal and centripetal perspectives on travel around a space station with artificial gravity, the above equation curves2, and electrostatic and gravitational point sources predicts a change in their “centrifugal weight” depending with regard to their relativistic (e.g. magnetic and cur- on how fast and in which direction they run. In particular vature) effects21. They also gain “geodesic frame” (e.g. by running quickly enough in a direction opposite to the satellite) and “shell-frame” (e.g. earth) based Newton- satellite’s rotation, they can make themselves weightless. like equations of motion at any speed, around objects of any mass. This is relevant to global positioning system applications which are forced to recognize the subjectiv- 2. Extension: Extreme Gravity ity of earth-based NIST clocks25, and extreme environ- ments like those discussed in Taylor and Wheeler’s most 0 1 2 recent text26. The metric tensor for x = ct, x = r, x = θ, and x3 = φ, here in “far coordinates”, becomes
2GM (1 c2r )0 0 0 1. Extension: Anyspeed Carousels − − 1 0 2GM 0 0 1− c2r gµν = 2 . (6) Consider a set of fiducial (map-frame) observers who 0 0 r 0 0 00 r2 sin[θ]2 find themselves rotating at angular velocity Ω along with a set of yardsticks arrayed around the circumference of a circle of radius r. In this case, the metric tensor for Although this space-time curvature only changes the metric coefficients by about a part per billion at the sur- x0 = ct = ct 1 ( Ωr )2, x1 = r = r , x2 = φ = φ +Ωt, o c o o face of the earth, it beautifully explains much of what we 3 q − and x = z = zo becomes experience about gravity today, and more.
2 Of course, the equation only applies exterior to a spher- 1 Ωr 0 Ωr r 0 ically symmetric mass M. For objects whose mass lies − − c c h i within the event horizon radius predicted by this metric, gµν = 0 100 . (3) Ωr 2 2GM r 0 r 0 at r = c2 , the metric also only applies exterior to the c 27 0 001 event horizon as well. Cook shows (strangely enough) that for both “shell frame” (r constant) and “rain frame” As shown by Cook27, the spatial metric defining local (free falling from infinity) observers, the local physical radar distance then assumes the decidedly non-Euclidean metric (i.e. Pythagoras’ theorem) remains Euclidean, at form least outside the event horizon. This in spite of the fact that the two are obviously in different states of accelera- 2 2 1 2 2 tion. (dℓ) = (dr) + 2 (rdφ) + (dz) (4) 1 Ωr Calculating far-coordinate affine-connection terms, − c and the resulting geodesic equation, predicts a radial ac- Thus distance is the same in the radial direction as in celeration for stationary objects of the form a stationary frame, while circumferential yardsticks are length-contracted in the azimuthal direction, increasing d2r GM 2GM = r . (7) local radar distance as appropriate for an azimuthal ve- dτ 2 r3 − c2 locity of Ωr and illustrating contraction-effects within one (azimuthal ring) frame. This acceleration in far-coordinates thus for example goes To find the geometric forces, we calculate the affine- away (because of the apparent slowing down of time) connection and resulting geodesic equation in terms of at the event horizon, but cleanly reduces to Newton’s coordinates in this rotating frame. Thus free objects ex- gravity law for r 2GM/c2. ≫ 6
3. Drawbacks: Life on a Shell between charges of given sign. On a chunk of wire of e length ds, this translates to a positive charge Q = ℓo ds and a negative charge of e ds and therefore a net charge Disadvantages of discussing the local validity of New- − ℓo ton’s laws, as distinct from their correctness only in un- of 0. If the positive charges are stationary in the wire, accelerated frames, are at least twofold. To begin with, but the negative charges are moving to the right with a e the concept of “local validity” is rather sophisticated. speed v, we say that the current in the wire is I = ℓo v, Admittedly we have many concrete examples, like the to the left. From Coulomb’s law, the electrostatic force local validity of the uniform gravitational field approxi- on a stationary test charge q a distance r above the wire mation at the earth’s surface, and it’s inability to deal is of course with less local phenomena such as orbits and lunar tides. qQ qQ A deeper problem is the difficulty of explaining what an Fup = F+ + F− = k k = 0 (A1) inertial (e.g. rain) frame is, if stationary frames (e.g. sit- r2 − r2 ting down on a stationary earth) can be non-inertial. It is simpler (or at least more traditional) to introduce iner- Since the test-charge is not moving, the flat-space ver- tial frames by referring to their uniform motion relative sion of Newton’s 2nd Law (here F = Fo/γ) predicts the to some inertial standard, rather than by referring to the same canceling proper-forces on our test charge as well. absence of a felt “geometric” acceleration acting on every However if we now move the test charge to the right at ounce of mass in one’s corner of the world. a speed v, the 2nd Law predicts that the proper-force Fo+ exerted on our test charge by the stationary posi- tive charges remains in the direction of their separation, V. CONCLUSIONS but increases in magnitude by a factor of γ. By sym- metry, the co-moving frame-variant force due to negative Measurement of time and mass in meters provides rel- charges (which now see the test charge as stopped) will ativistic insight into global symmetries, and the relation have its previous contribution to the proper-force Fo− between 4-vectors in coordinate-independent form. How- decreased by a factor of 1/γ. ever, frame-specific Newton-like laws with separate units This proper-force provides a frame-invariant platform for length, time and mass are perhaps still crucial to the for combining the two frame-variant but otherwise purely inhabitants of any particular world, as an interface to electrostatic forces, allowing us to conclude that the local physical processes. net proper-force experienced by our moving test charge We point out some advantages of the fact that New- equals the proper-force from the positive charges on the ton’s laws, written in context of a map-frame of choice, test charge before it began moving (kqQ/r2) times the have considerable potential beyond their classical ap- non-zero difference between γ and 1/γ. Finally the net plications. By avoiding the implicit assumptions of (i) frame-variant force on our moving test particle, reduced global time, (ii) frame-invariant dynamics, and (iii) lim- from the net proper-force again by that factor of γ, be- itation to inertial frames, introductory students can be comes... better prepared for an intuitive understanding of rela- 2 tivistic environments, as well as for getting the most out 1 1 qQ qQ v k Ids Fup = (γ )k = k = qv (A2) of the laws themselves. γ − γ r2 r2 c2 c2 r2
This force, due to a frame-dependence of forces in Acknowledgments spacetime that has nothing to do with electrostatic forces per se, is of course traditionally explained by saying Several decades of energetic work by E. F. Taylor, on that the current creates a magnetic field B according to pedagogically-informed content-modernization, has of- the Biot-Savart prescription in parentheses on the right, fered key inspiration. which in turn exerts a force according to the Lorentz Law F = qvB. Thus magnetic fields are a convenient tool for taking into account relativistic effects of the APPENDIX A: ONE-FRAME BIOT-SAVART Coulomb force (most noticable around neutral current- carrying wires), and the flat-space (one-map two-clock) By way of application, imagine a wire running from version of Newton’s 2nd law provides us with a derivation e left to right with a positive charge density of + ℓo and a that (except for the symmetry invocation) requires only e negative charge density of , where ℓo is the distance one map-frame with yardsticks and synchronized clocks. − ℓo
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