The proper-force 3-vector 1, a)

Transcription

1 The proper-fore 3-vetor 1, a) P. Fraundorf Physis & Astronomy/Center for Nanosiene, U. Missouri-StL (63121), St. USA (Dated: 18 Otober 2017) Louis, MO, The distintion between proper (i.e. ell-phone detetable) and geometri (i.e. onnetion-oeffiient) fores allows one to use Newton s 3-vetor laws in aelerated frames and urved spaetime. Here we show how this is assisted by use of quantities that are either (i) frame-invariant or (ii) synhrony-free i.e. do not rely on extended-networks of synhronized-loks. The aeleration four-vetor s invariant magnitude, and quantities that build on the metri-equation s book-keeper frame to define simultaneity, point the way to more robust student understanding at both low and high speeds. In the proess, we gain a simple (3+1)D flat-spae workenergy theorem using the proper-aeleration 3-vetor α (net proper-fore per unit mass), whose integrals of the motion simplify with a hyperboli veloity angle (rapidity) written as 2/(γ o + 1)/, where is lightspeed and τ is traveler-time from turnaround when the Lorentz-fator is γ o. CONTENTS I. introdution 1 II. frame dependene & synhrony 1 III. low speed appliations 2 IV. bringing in the metri 3 V. any speed appliations 4 VI. disussion 5 Aknowledgments 6 A. 3-vetor relativity 6 I. INTRODUCTION Relativists have long expressed unhappiness with oordinate-aeleration and oordinate-fore (for good reason 1,2 ), but have also pointed out that generalrelativity makes a ase for the loal-validity of Newton s laws in all frames 3 5 provided that we onsider geometri (urved-frame or onnetion-oeffiient ) fores as well as proper-fores whenever we find ourselves in a non- free-float trajetory 6. The distintion has everday relevane to intro-physis students, beause texts often introdue gravity as a real fore but inertial fores as fake even though their ell-phones detet neither for the same reason: Aelerometers only detet proper-fores, while gravity and inertial (e.g. entrifugal) fores are geometri. In this paper we explore an approah to aelerated motion designed to be: (i) the most frame-independent, and (ii) the least in need of synhronized-lok arrays. a) pfraundorf@umsl.edu These latter might be diffiult to ome by on aelerated platforms and in urved spaetime. The first proper-time derivative of an aelerated traveler s 4-vetor position has lightspeed as its invariant magnitude. Here we simply define simultaneity using bookkeeper oordinates and then examine the seond proper-time derivative of position, as seen from the proper referene-frame 3 of that aelerated traveler. In the proess we show: (a) that the distintion between proper and geometri fores is already quite useful for introdutory physis, (b) that via the metri equation a lot an be done with only a single extended mapframe of yardstiks and synhronized loks, and () that the traveler s view of anyspeed-aeleration is less framevariant than the map perspetive. We also exploit the frame-invariane of proper-fore in an empirial observation exerise on the eletrostati origin of magnetism, whih provides some viseral experiene with lengthontration at the same time. II. FRAME DEPENDENCE & SYNCHRONY The value of frame-independene in the modeling of relativisti-motion and urved-spaetime goes without saying. The frame-invariane of lightspeed (the magnitude of the veloity 4-vetor U λ dx λ /) has been entral to our understanding of spaetime from the beginning 7. Proper-time (the magnitude of the displaement 4-vetor X λ ) is finding inreasing use by introdutory text authors as we speak. The Lorentz-transform view of proper-time, of ourse, is that it is time-passing on the synhronized loks of a tangent but o-moving free-float-frame in flat spaetime. The metri equation s view of proper-time is simpler but more general, i.e. as a quantity measured on a single lok under any onditions i.e. aelerated or not, in urved spae-time or not. Proper-time is frame-invariant in the sense that its value may be agreed upon using any general-relativisti book-keeper oordinates that we hoose. These book-

2 2 TABLE I. Aelerated-motion definitions in flat (3+1)D spaetime. Note that aeleration/fore magnitudes are spaelike, while the others are timelike along a traveler s worldline, and that we ve defined x and y as spatial oordinates and to the diretion of proper-aeleration 3-vetor α. 4-vetor magnitude time-omponents to spatial 3-vetor α to spatial 3-vetor α power/fore ΣF o dpo aeleration α = ΣFo m energy/momentum m dγ P = ( 1 ) de F dp F dp = P = γ P dw = F = γ f dw = F m m m m m = γ f m E = γm p = mw p = mw veloity γ dt = E w m dx = p = γv w dx = p oordinate τ t x y = γv TABLE II. Relationship between variables: Here τ is traveler-time elapsed from turnaround (when γ γ o) for as long as proper aeleration α doesn t hange, and γ ± (γ o ± 1)/2. The right arrow denotes the non-relativisti limit. 4-vetor invariant time-omponents/ [ ] to spatial 3-vetor α to spatial 3-vetor α dγ ael. α = α γ+ sinh γ + ( ) [ ] [ ] α 2 dw τ dw = α osh γ + α = αγ sinh γ + 0 [ ] [ ] [ ] veloity γ = γ 2 + γ+ 2 osh γ + 1 w = γ + sinh γ + w = γ +γ (osh γ + + 1) v [ ] [ ] 2 [ oord. τ t = γ τ 2 + γ+ 3 sinh α γ + τ x = γ sinh α 2γ y = γ 2 +γ (τ + sinh α γ + ]) v τ keeper oordinates are alone used to define extended simultaneity (i.e. the global plae and time of events), while the frame-invariane of proper-time drastially improves the transformation-properties of quantities differentiated with respet to it. The proper-veloity 3-vetor w d x/ (whih unlike oordinate veloity v d x/dt adds vetorially with appropriate resaling of the out-offrame omponent) and the proper-aeleration 3-vetor (disussed here) are ases in point. The topi of this paper is in partiular the frameinvariant magnitude of the aeleration 4-vetor, in standard notation 3 : A λ := DU λ = du λ + Γλ µνu µ U ν (1) and uses for this vetor s omponents (as power/fore) when they are multiplied by frame-invariant rest-mass m. Here free-float or geodesi trajetories have A λ = 0, so that we an think of oordinate aeleration du λ / as a sum of proper and geometri terms, the latter depending on loal spae-time urvature through the 64-omponent affine-onnetion Γ λ µν whih gives rise to apparent fores in aelerated oordinate-systems and urved spae-time. As usual greek indies run from 0 (time-omponent) to 3 (spae-omponents) and obey the Einstein summation onvention when repeated in a produt. Beause this proper-aeleration four-vetor beomes purely spae-like in a frame instantaneouslyomoving with our traveler, its physial interpretation is simply the proper-fore/mass felt to be pressing on our traveler, as well as the 3-vetor proper-aeleration 8 10 α seen by free-float observers in the o-moving frame. In addition to a preferene here for frame-invariane, the onept of simultaneity is a messy one in aelerated frames (e.g. using radar-time methods 11 ) as well as in urved spaetime. Hene we take a metri-first approah to kinematis here by hoosing a single bookkeeper oordinate-system in terms of whih both maptime t and map-position x are measured. Simultaneity will be defined in terms of synhronized (but not always loal e.g. in the ase of Shwarzshild far-time ) loks in this book-keeper frame. In addition purely spae-like vetors, along with frameinvariants, may be desribed as synhrony-free to use a word employed by William Shurliff when disussing proper-veloity 12,13 w d x/ = p/m. These are quantities whose operational-definition does not require an extended network of synhronized-loks, something of limited availability around gravitational-objets (like earth), and impossible to find on platforms (like spaeships) undergoing aelerated motion. The time-like energy of a moving objet via its dependene on the Lorentz-fator γ dt/ is (like mixed objets suh as oordinateveloity v d x/dt) not synhrony-free, beause it requires map-time t data from loks at multiple loations. The traveler s point of view that we argue offers the most diret way to ommuniate about an aelerated traveler is the frame that Misner, Thorne and Wheeler 3 refer to as the proper referene frame of an aelerated traveler. One an always onvert these to expressions written in terms of bookkeeper variables like map-time t and oordinate-veloity v, but we show here that the algorithmially-simplest way to desribe the effets of the loal spae-time metri on motion (following the ritera above) involves the parameterization desribed here. III. LOW SPEED APPLICATIONS For appliations at low speed, telling students about proper-fores as distint from geometri-fores (that at on every oune of a objet s being) is a good start in

3 3 FIG. 1. Two views of proper (red) and geometri (dark blue or brown) fores in some everyday settings. preparing them for the value of Newton s laws in both free-float and aelerated frames. The simple example (a) of a ar leaving a stop-sign is illustrated by the sreen apture from a wikimedia ommons animation in the topleft of Fig. 1, whih shows the red proper-fore seen by observers in both frames as pressing on the driver s bak while the ar aelerates. This of ourse is aneled only in the ar frame by a geometri fore whih (like gravity) ats on every oune of the driver s being. Animation sreen aptures are also provided in that figure whih show proper and geometri fores from two perspetives in the ases of (b) arousel motion, () the proess of rolling off a liff, and (d) the later stages of a bungie jump. The sreen apture in Fig. 2 from the real-time animation of a 50[m] diameter rotating-wheel spae-habitat with 1 gee of artifiial gravity at its perimeter is similarly instrutive. We also reommend telling intro-physis students that time itself is dependent on a given lok s loation and state of motion, with the speed of map-time relative to a traveler s lok (i.e. dt/) an important lue to the traveling-lok s energy (potential and/or kineti). These things may be done at the outset, followed by the assertion that introdutory physis texts by default refer to map-time (t) sine traveler-time (τ) differenes at low speed are negligible, and they traditionally treat gravity as another proper-fore even though we now know that it too is a geometri-fore, aused not by a traveler s motion but by gravity s urvature of spae-time around massive objets. Traditional treatments often further fous only on appliation of Newton s laws from inertial-frame perspetives, in whih ase geometrifores (other than gravity) an be ignored. With these minor metri-first hanges to the introdution, traditional introdutory physis treatments remain perfetly self-onsistent and intat. IV. BRINGING IN THE METRIC In order for teahers to feel grounded when addressing introdutory issues in ontext of an intimidating Riemann-geometry framework, it is ruial that the onsequenes of their assumptions be easy to for them to verify. Thankfully the metri-equation, unlike Lorentz transforms, requires only one bookkeeper frame whose time-variable may (or may not) be possible to assoiate with time s passage on loks synhronized aross a meaningful region of spaetime. Our first step, namely hoosing the metri parameterization to desribe a speifi problem, is espeially important beause it defines both the meaning of measurments and our (perhaps impliit) definition of simultaneity. This is good news for introdutory teahers, sine

4 4 its bad enough to be talking about different times on different loks, without having to also be juggling multiple definitions of simultaneity. For general relativity appliations in a world where time is measured on wathes, and distanes are measured with yardstiks, whenever possible we will seek metri parameterizations whose time-variable orresponds to loks that an be synhronized. We therefore follow Newton in flat-spae settings by hoosing a set of freefloat (e.g. inertial or un-aelerated) frame variables like oordinate-time t and oordinate-position x to desribe aelerated motion. As teahers, one we have a metri and a orresponding definition of what simultaneity means, we are bak on familiar territory. The aveat is that frameindependene may be attributed only to four-vetor magnitudes, and no longer to time-intervals, distanes, or rates of momentum-hange. For the flat-spae (1+1)D ase, for instane, the proper time-interval δτ and derivatives with respet to τ yield the following frame-invariant magnitudes: with the lightspeed onstant (δτ) 2 = (δt) 2 (δx) 2, (2) 2 = ( δt ) 2 δτ and proper-aeleration α: α 2 = ( ) 2 δ2 t δτ 2 ( ) 2 δx, (3) δτ ( δ 2 ) 2 x δτ 2. (4) Given this, the hallenge of finding the integrals of the motion e.g. for onstant aeleration is muh like that hallenge of showing that x = 1 2 at2 via the same derivative relations, but using Newton s assumptions that oordinate-intervals and oordinate-aeleration a δ2 x δt 2 are frame-invariant. Simple-form versions of the metribased integrals are tabulated in ontext of the disussions to follow. V. ANY SPEED APPLICATIONS Table I defines notation for desribing aelerated motion in (3+1)D flat spaetime. Table II shows the instantaneous relationship between these varables (also at low speed), as parameterized by the traveler-time τ and Lorentz-fator γ o from turnaround were the instantaneous proper aeleration to remain onstant (f. Appendix A). In both tables, only values in the timeomponents olumn rely on synhrony between mapframe loks at more that one loation. Values in the spatial-oordinate olumns to the right are synhronyfree, while values in the olumn to the left are frameinvariant as well. FIG. 2. Free-float and ship frame views of a pentagonal dropped-ball trajetory in a rotating-wheel spae habitat, to illustrate the non-ontat nature of the ell-phone undetetable entrifugal fore. Of ourse a map-frame observer s measurements of map-position as a funtion of map-time (along with deliverables like inferred oordinate-fores) will be parameterized in terms of synhrony-dependent map-time instead of frame-invariant traveler-time. Although map-frame observers an alulate synhrony-free quantities like momentum and proper-veloity in terms of synhronydependent parameters, it will take extra steps going to there from what they measure, and perhaps also going from there to what they want to infer. If on the other hand the traveler measures their felt proper-aeleration, as well as the rates at whih they pass map-landmarks on their route, the equations to everything else are simpler and organially related as shown in Table II. Plus, everything that the traveler measures and reports on (exept for elements in the timeomponent olumn of the table) will either be synhronyfree or frame-invariant. The onnetion between the traveler ontrolparameters and Table II is reinfored if we imagine long-distane travel in a spaeraft with traveler ontrol over thust (i.e. proper-fore) magnitude and diretion. The table onnets proper-aeleration s magnitude and diretion to instantaneous values of proper-time from turnaround and v, whih in turn are related via the same table to navigational objetives (like the x and y values for the turnaround-point itself). Although variable-rearrangement is ompliated relative to the low-speed ase via gamma-fator oupling between diretions, a wide range of puzzles involving high-speed navigation in free-spae may be addressed with this table. Of most interest perhaps to beginning students are of ourse the possibilities that relativity opens up for onstant proper-aeleration (e.g. 1 gee ) round-trips between distant loations. Not only are these equations even simpler than the (3+1)D ase, but the real limiting fator (namely the payload to launh-mass ratio) is quite simple to alulate as well. A pratial lassroom appliation of the frameindependene of proper-fore in this ontext involves an

5 5 FIG. 3. Two views of proper fore on a moving harge from a neutral urrent-arrying wire, with 40 milliseond time-steps between after-images. The shorter light-arrow in the wire-frame is the oordinate-fore f dp/dt = F o/γ. Effets of the depited fores on the harge-motion are ignored, as is the B-field in the moving-harge frame whih has no effet. TABLE III. Relationships between variables for aeleration in (1+1)D flat-spaetime: Here τ is traveler-time elapsed from turnaround for as long as proper aeleration α doesn t hange. The right arrow shows simplifiation when. 4-vetor invariants time-omponents/ spae-omponents aeleration α ΣFo dγ = P = γp = α sinh [ ] ( m m 2 m 2 α ) 2 dw τ = ΣF = γσf = α osh [ ] m m α veloity γ dt = E = 1 + ( w m 2 )2 = osh [ ] 1 w dx = p = γv = sinh [ ] m oordinate τ t = sinh [ ] ( [ α τ x = 2 α osh ] ) empirial observation exerise for students interested in the eletrostati origins of the magneti fore between moving harges. In essene, students are asked to take data in real time from animations (f. Fig. 3) showing neutral-wire and moving-harge perspetives on the proper-fore felt by the moving harge. Simple ratios (in either spae or time) allow students to quantify the length-ontration, the urrents and harge densities from these two perspetives, and a variety of other physial quantities. In order to see signifiant differenes in these quantities from the two perspetives, of ourse, harge veloities have to be relativisti. Sine veloities are also perpendiular to observed fores, a signifiant differene between the oordinate-fore observed in the neutral wire frame, and the proper-fore felt by the moving harge, also shows up. VI. DISCUSSION As mentioned above, extended arrays of synhronized loks are diffiult to ome by in urved spaetime (f. the relativisti orretions needed to make global positioning estimates aurate). They are perhaps even more diffiult to ome by on aelerated platforms (f. disussions of aelerated-frame Rindler oordinates ). Lorentz-transform first analyses of any-speed motion of ourse require at least two relativistially o-moving frames of synhronized loks. No wonder aelerated motion is of little interest in that ontext. Metri-first approahes require only one suh mapframe, sine proper-time on traveler loks is a frameinvariant. The integrals of onstant proper-aeleration, espeially in (1+1)D e.g. as α = w/ t = η/ τ = 2 γ/ x where η sinh[ w ], are also quite manageable. As shown Table III, whih is a (1+1)D version of Tables I and II ombined, the general magnitude-inequality between oordinate-fore f d p/dt (where we are using the relativisti momentum p) and proper-aeleration α, namely Σf m α, also beomes the more familiarlooking signed-equality Σf = mα. The approah also works in urved-spaetime. Table IV illustrates for the radial-only Shwarzshild ase using the exat Lorentz-fator from Hartle 4, even though the integration (even in the Newtonian ase) is simplest if we an ignore variations of g with r. The ompetition between veloity-related, and gravitational, time-dilation e.g. for GPS-system orbits is nonetheless quite lear.

6 6 TABLE IV. Relationship between variables for aeleration in (1+1)D gravity: Here τ is traveler-time from turnaround for fixed proper aeleration, while as usual g GM and r r 2 s 2GM. Here neglets hanges in g and assumes that. 2 4-vetor invariants time-omponents/ [ ] spae-omponents aeleration α ΣFo dγ = P = γp α g sinh (α g)τ ( ) [ ] α g 2 m m 2 m 2 τ dw = ΣF = γσf (α g) osh (α g)τ (α g) m m veloity γ dt = E = γ m 2 r 1 + ( γ rw ) 2 γ r [ ] 1 1 rs w dr = p = γv sinh (α g)τ (α g)τ m [ ] r ( [ ] ) oordinate τ t sinh (α g)τ τ r 2 osh (α g)τ (α g)τ α g α g 2 Just as in flat-spaetime, the metri equation in general assoiates a set of {t, x, y, z} bookkeeper-oordinates with eah event. In the Shwarzshild ase, however, loks an only be synhronized at fixed-r. Hene a radartime model 11 (or some suh) of extended-simultaneity might be needed to answer the question What time is it now at radius r? The good news for the ase of Shwarzshild (and other steady-state metris) is that γ dt = E m an be defined 2 regardless of one s model for extended-simultaneity. Although in general momentum p d x remains synhronyfree, definitions of synhrony-dependent energy may enounter signifiant ompliation when the bookkeeper dt time-derivative beomes dependent on extendedsimultaneity. We further show that frame-invariane (where all frames agree) is quite valuable for illustrations. The synhrony-free nature of proper-veloity and momentum, as well as of fore-omponents desribed as derivatives using proper-time τ instead of map-time t, also lead to a simpler and more robust piture of aelerated motion when examined from the point of view of the aelerated traveler. ACKNOWLEDGMENTS Thanks are due to: Roger Hill for some lovely ourse notes, Bill Shurliff for his ounsel on minimally-variant approahes, as well as Eri Mandell and Edwin Taylor for their ideas and enthusiasm. 1 A. P. Frenh, Speial relativity, The M.I.T. Introdutory Physis Series (W. W. Norton, New York, 1968) page 154:...aeleration is a quantity of limited and questionable value in speial relativity. 2 T.-P. Cheng, Relativity, gravitation and osmology (Ox, 2005) page 6: in speial relativity... we are still restrited to... inertial frames of referene and hene no aeleration. 3 C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Franiso, 1973). 4 J. B. Hartle, Gravity (Addison Wesley Longman, 2002). 5 R. J. Cook, Physial time and physial spae in general relativity, Amerian Journal of Physis 72, (2004). 6 E. Taylor and J. A. Wheeler, Exploring blak holes, 1st ed. (Addison Wesley Longman, 2001). 7 A. Einstein, Relativity: The speial and the general theory, a popular exposition (Methuen and Company, 1920, 1961). 8 A. Einstein and N. Rosen, The partile problem in the general theory of relativity, Phys. Rev. 48, (1935). 9 E. Taylor and J. A. Wheeler, Spaetime physis, 2nd ed. (W. H. Freeman, San Franiso, 1992). 10 C. Lagoute and E. Davoust, The interstellar traveler, Amerian Journal of Physis 63, 221 (1995). 11 C. E. Dolby and S. F. Gull, On radar time and the twin paradox, Amerian Journal of Physis 69, (2001). 12 F. W. Sears and R. W. Brehme, Introdution to the theory of relativity (Addison-Wesley, NY, New York, 1968) setion W. A. Shurliff, Speial relativity: The entral ideas, (1996), 19 Appleton St, Cambridge MA J. D. Jakson, Classial Eletrodynamis, 3rd ed. (John Wiley and Sons, 1999). 15 D. G. Messershmitt, Speial relativity from the traveler s perspetive, (2015), private ommuniation. 16 J. M. Levy, A simple derivation of the Lorentz transform and of the related veloity and aeleration formulae, Amerian Journal of Physis 75, (2007). Appendix A: 3-vetor relativity By way of example, A. P. Frenh 1 examines oordinateaeleration omponents with respet to oordinateveloity, J. D. Jakson s relativity hapters 14 do an exellent job at showing both 3-vetor and Lorentz-ovariant (4-vetor) versions of the way that eletro-magneti proper fores work, and D. G. Messershmitt 15 has examined saling relations for proper-aeleration from a modern engineering perspetive. Eah of these has foused on omponents relative to the diretion of motion, rather than relative to the diretion of proper aeleration e.g. of a roketship whih has loal ontrol of its diretion of thrust. One must of ourse use aution in using relativisti 3- vetors (espeially in urved-spaetime and aeleratedframes) beause many impliit Newtonian assumptions are no longer valid. One may also enounter dissonane from uni-diretional simplifiations, like the breakdown of relativisti-momentum p into a produt of relativisti-mass γm and oordinate-veloity v d x/dt, rather than into a produt of frame-invariant restmass m and synhrony-free proper-veloity w d x/ = γ v. In flat spaetime, quantities demoted to the status of frame-variant or effetive for the indiated frame only (eifo 13 ) inlude: oordinate-time t in omparison to the frameinvariant proper-time τ elapsed e.g. on a traveling lok,

7 7 simultaneity whose frame invariane may be (only temporarily) put aside by using the metri equation to selet but one book-keeper definition of simultaneity, and fore ΣF d p/ whih we deal with by introduing the net felt or proper-fore ΣF o (from the traveler s perspetive) whose magnitude (like proper-time τ and lightspeed ) is frame-invariant. Note that rates of energy-hange de/ are framevariant even at low speeds: For instane, de/ is always zero in the rest-frame (as well as the tangent free-floatframe ) of an aelerated objet even if energy is rapidly hanging from the vantage point of other frames. In urved-spaetimes and aelerated-frames, key onepts inlude the added ideas of: bookkeeper oordinates in the metri-equation whih may be hosen for onveniene but might not permit extended networks of synhronized loks, vetors as loally-defined tangents 3 instead of as lines between points A and B, free-float (geodesi) trajetories and tangent freefloat-frames in partiular, and geometri (onnetion-oeffiient) fores as distint from proper-fores. As disussed in the artile, of ourse, the latter are also not new to students of low-speed physis if inertial fores like entrifugal and Coriolis have been studied. These aveats in mind, familiar relations in salar and/or 3-vetor form an often be written with an additive (γ 1) term for the eifo-orretion. For instane, the motion-related eifo-orretion to frame-invariant restenergy m 2 in flat spaetime is simply kineti-energy (γ 1)m 2, oordinate-time dilation for proper-time interval δτ may be written as δt = δτ + (γ 1)δτ, and vetor length-ontration for proper-length interval L o is L = L o + (γ 1) L o γ w, (A1) where the subsript w selets only that omponent of L o whih is parallel to proper-veloity w. More generally, for 4-vetor {X t, X} the Lorentz boost to a primed-frame moving at proper-veloity w has timeomponent X t = X t w X + (γ w 1)X t. The spaeomponent is X = X w X t + (γ w 1) X w. Sine proper-aeleration is purely spaelike in the frame of the aelerated traveler, we an say that the time-omponent yields the flat spaetime work-energy expression de/ = m α w = ΣF o w. The spaeomponent says that the frame-variant fore an be expressed in terms of proper-aeleration 16 and properveloity as: Σ F = d p = md w = m α + (γ 1)m α w, (A2) The seond term here is a orretion to the net properfore ΣF o = mα that (like the only term in the workenergy expression sine proper-power is always zero) allows one to determine the net-fore ΣF d p/ from the map-frame perspetive. Note that sine proper-veloity uses a time-variable loalized to the traveler (and hene does not require synhronized map-loks along the traveler s trajetory), these 3-vetor expressions may be useful loally in urved as well as flat spaetime settings, provided that we have a definition for γ dt/ (from the metri) and hene an effetive value for traveler total energy E = γm 2. In flat spaetime, where the metri tells us that γ dt/ = 1 + (w/) 2, one an obtain the energy-integral differential equation: 2 ( 1 + γ + ( ) α γ = α γ)2 α (A3) 1 + γ where the dot refers to differentiation with respet to proper-time τ, and w = ( 2 /α)dγ/. This integrates pretty quikly to the ontents of Table II. Table III entries then follow diretly for the (1+1)D ase by letting γ o 1. For the Shwarzshild potential, dt/ beomes γ r (1 + (γr w/) 2 ) if γ r 1/(1 r s /r) and r s is the event-horizon radius 2GM/ 2. Appliation of equation A2 then only qualitatively yields the approximate relationships in Table IV. Note also that the resaled veloity-term in equation A2 is reminesent of the fator (γ BC + (γ AB 1) w BC wab ) that resales (in magnitude only) the out-offrame proper-veloity w AB ( w AB ) C when alulating relative proper-veloity 3-vetors by 3-vetor addition: w AC γ AC v AC = ( w AB ) C + w BC. (A4) Thus a fous on frame-invariant and synhrony-free variables might help autiously open the door to a wider range of dimensioned 3-vetor relativisti explorations. Finally, let s examine the onnetion of these equations to the Lorentz-equation for eletromagneti proper-fore, whih underpins Fig. 3 as well as most proper-fores that we enounter in everyday life. It is also a prototype for the Maxwell-like equations that underpin field-mediated proper-fores in general. In terms of eletri E and magneti B fields in a frame with respet to whih a harge Q is moving at proper-veloity w = γ v, we an use the Lorentz-transform equations (SI-units version 14 ) for the eletri field in the primed frame of harge Q to write the proper-fore as: Σ F o = Q E = Q E w + γq( E w + v B) (A5) Although the fore on our moving harge (as a rate of momentum hange) is in general frame-variant, all observers (traveling at any speed even in urved spaetime) should be able to agree on the proper-fore and properaeleration that the harge is experiening. Putting this

8 8 general expression, for the net frame-invariant properfore on a moving harge, into the expression for net frame-variant fore above gives us: Σ F d p = γ d p dt = γq( E + v B) (A6) This Lorentz-fore expression, here obtained from the eletrostati definition of E and the field transformationrules, illustrates how a magneti field that yields a fore perpendiular to veloity may serve as a natural omplement to any stati proper-fore field.