Proper veloity and frame-invariant aeleration in speial relativity P. Fraundorf Department of Physis & Astronomy University of Missouri-StL, St. Louis MO (November, 99) We examine here a possible endpoint of the trends, in the teahing literature, away from use of relativisti masses (suh as m m in the momentum = massveloity expression) and toward use of proper veloity dx= v (e.g. in that same expression). We show that proper time & proper veloity, taken as omponents of a non-oordinate time/veloity pair, allow one to introdue time dilation and frame-invariant aeleration/fore -vetors in the ontext of one inertial frame, before subjets involving multiple frames (like Lorentz transforms, length ontration, and frame-dependent simultaneity) need be onsidered. We further show that many post-transform equations (like the veloity-addition rule) aquire elegane and/or utility not found in the absene of these variables. From physis/9...+p,..gm,.55.+b I. INTRODUCTION Eorts to onnet lassial and relativisti onepts will be with us as long as lassial kinematis is taught to introdutory students. For example, the observation that relativisti objets behave at high speed as though their inertial mass inreases in the! p = m! v expression, led to the denition (used in many early textbooks )of relativisti mass m m. Suh eorts might help to: (A) get the most from rst-taught relationships, and (B) nd what is true and fundamental in both lassial and relativisti approahes. The onepts of transverse (m ) and longitudinal (m m ) masses have similarly been used to preserve relations of the form F x = ma x for fores perpendiular and parallel, respetively, to the veloity diretion. Unfortunately for these relativisti masses, no deep sense in whih mass either hanges or has diretional dependene has emerged. They put familiar relationships to use in keeping trak of non-lassial behaviors (item A above), but do not (item B above) provide frameinvariant insights or make other relationships simpler as well. As a result, majority aeptane of their use seems further away now than it did a several deades ago. A more subtle trend in the pedagogial literature has been toward the denition of a quantity alled proper veloity ;5, whih an be written as! w! v. We use the symbol w here beause it is not in ommon use elsewhere in relativity texts, and beause w looks like v from a distane. This quantity also lets us ast the momentum expression above in lassial form as a mass times a veloity, ie. as! p = m! w. Hene it serves at least one of the \type A" goals served by m above. We show here that, when introdued as part of a nonoordinate time/veloity pair in pre-transform speial relativity, proper veloity allows us to introdue relativisti momentum, time-dilation, and frame-invariant relativisti aeleration in ontext of a single inertial frame. Moreover, through use of proper veloity many relationships (inluding post-transform relationships like veloity addition whih require onsideration of multiple frames) are made simpler and sometimes more useful. Hene it appears to serve goals of \type B" mentioned above as well. II. ONE OBJECT, ONE FRAME, BUT TWO TIMES: THE CONSEQUENCES. One may argue that a fundamental break between lassial and relativisti kinematis involves the observation that time passes dierently for moving observers, than it does for stationary ones. To quantify this, we dene two time variables when desribing the motion of a single objet (traveler) with respet to a single inertial oordinate frame. These are the oordinate time t and the proper (or traveler) time t o. Note that the proper time may be dierent for dierent travelers. It follows from above that we might also onsider two veloities, namely oordinate veloity! v d! x =, and proper (or traveler-kinemati) veloity! w d! x =. It is helpful to distinguish the units used to measure these veloities, by saying that the rst measures distane traveled per unit oordinate time, while the latter measures distane traveled per unit traveler time. Convenient units are [lightyears per oordinate year] and [lightyears per traveler year], respetively. Eah of these veloities an be alulated from the other by knowing the veloitydependene of the \traveler's speed of time" =, via the equation! w =! v () Note that all displaements dx are dened with respet to a single inertial (unprimed) frame. Thus proper veloity is not simply a oordinate veloity measured with respet to a dierent frame. It is rather one of an innite number of non-oordinate veloities, denable in ontext of time/veloity pairs experiened by one of the innite
number of observers who may hoose to trak the motion of our given objet on a given map using their own lok. The ardinal rule here is: measure all displaements from the vantage point of the hosen inertial (or \map") frame. Thus proper veloity! w is simply the rate at whih the oordinates of the traveler hange per unit traveler time on a map of the universe (e.g. in the traveler's \glove ompartment") whih was drawn from the point of view of the hosen inertial referene frame, and not from the point of view of whatever inertial frame the traveler happens to be in at a given time. A number of useful relationships for this \traveler's speed of time", inluding the familiar relationship to oordinate veloity, follow simply from the (+)D nature of the at spaetime metri (speially from the frameinvariant dot-produt rule applied to the displaement and veloity -vetors). Their derivation therefrom is outlined in the Appendix. For introdutory students, we an simply quote Einstein's predition that spaetime is p tied together so that instead of =, one has == (v=) = E=m, where E is Einstein's \relativisti energy" and is the speed of light. It then follows from above that: = q v r w = = + E m =+ K m () Here of ourse K is the kineti energy of motion, equal lassially to mv. One of the simplest exerises a student might do at this point is to show that sine v = w= = p w= +(w=), as proper veloity w goes to innity, the oordinate veloity v never gets larger than! All of lassial kinematis also follows from these things if and only if ', whih as one an see above is true only for speeds v. Given these tools to desribe the motion of an objet with respet to single inertial referene frame, perhaps the easiest type of relativisti problem to solve is that of time dilation. From the very denition of as a \traveler's speed of time", and the veloity relations whih show that, it is easy to see that the traveler's lok will always run slower than oordinate time. Hene if the traveler is going at a onstant speed, one has from equation that traveler time is dilated (spread out over a larger interval) relative to oordinate time, by the relation t = t o t o () In this way time-dilation problems an be addressed, without rst introduing the deeper ompliations (like frame-dependent simultaneity) assoiated with multiple inertial frames. This is the rst of several skills that this strategy an oer to students taking only introdutory physis, a \type A" benet aording to the introdution. A pratial awareness of the non-global nature of time thus need not require a readiness for the abstration of Lorentz transforms. Equations above also allow one to relate these veloities to energy. Hene an important part of relativisti dynamis is in hand as well. Another important part of relativisti dynamis, as mentioned in the introdution, takes on a familiar form sine momentum at any speed is! p = m! w = m! v () This relation has important sienti onsequenes as well. It shows that momentum like proper veloity has no upper limit, and that oordinate veloity beomes irrelevant to traking momentum at high speeds. One might already imagine that proper speed w is the important speed to a relativisti traveler trying to get somewhere with minimum traveler time. Equation shows that it is also a more interesting speed from the point of view of law enforement oials wishing to minimize fatalities on a highway in whih relativisti speeds are an option. This a \type B" benet of proper veloity. In this ontext, it is not surprising that the press doesn't report \land speed reord" for the fastest aelerated partile. New progress only hanges the value of v in the th or 8th deimal plae. The story of inreasing proper veloity, in the meantime, goes untold to a publi whose imagination might be aptured thereby. After all, the eduated lay publi (omprised of those who have had only one physis ourse) is by and large under the impression that the lightspeed limit rules out major progress along these lines. III. THE FRAME-INVARIANT ACCELERATION -VECTOR. The foregoing relations introdue, in ontext of a single inertial frame and without Lorentz transforms, many of the kinematial and dynamial relations of speial relativity taught inintrodutory ourses, in modern physis ourses, and perhaps even in some relativity ourses. In this setion, we over less familiar territory, namely the equations of relativisti aeleration. Fores if dened simply as rates of momentum hange in speial relativity have no frame-invariant formulation, and hene Newton's nd Law retains it's elegane only if written in oordinate-independent -vetor form. It is less ommonly taught, however, that a frame-invariant -vetor aeleration an be dened (again also in ontext of a single inertial frame). We show that, in terms of proper veloity and proper time, this aeleration has three simple integrals when held onstant. Moreover, it bears a familiar relationship to the speial frame-independent rate of momentum hange felt by an aelerated traveler. By again examining the frame-invariant salar produt of a -vetor (this time of the aeleration -vetor), one an show (as we do in the Appendix) that a \proper
aeleration" for a given objet, whih is the same to all inertial observers and thus \frame-invariant", an be written in terms of omponents for the lassial aeleration vetor (i.e. the seond oordinate-time t derivative of displaement! x )by:! =! a;where! a d! x. (5) This is quite remarkable, given that! a is so strongly frame-dependent! Here the \transverse time-speed\ is dened as p = (v =), where v is the omponent of oordinate veloity! v perpendiular to the diretion of oordinate aeleration! a. In this setion generally, in fat, subsripts k and refer to parallel and perpendiular omponent diretions relative to the diretion of this frame-invariant aeleration -vetor!, and not (for example) relative to oordinate veloity! v. Before onsidering integrals of the motion for onstant proper aeleration!, let's review the lassial integrals of motion for onstant aeleration! a. These an be written as a = v k =t = (v )=x k. The rst of these is assoiated with onservation of momentum in the absene of aeleration, and the seond with the workenergy theorem. These may look more familiar in the form v kf = v ki +at, and v = kf v +ax ki k. Given that oordinate veloity has an upper limit at the speed of light, it is easy to imagine why holding oordinate aeleration onstant in relativisti situations requires fores whih hange even from the traveler's point of view, and is not possible at all for t >( v ki )=a. Provided that proper time t o, proper veloity w, and time-speed an be used as variables, three simple integrals of the proper aeleration an be obtained by a proedure whih works for integrating other non-oordinate veloity/time expressions as well. The resulting integrals are summarized in ompat form, like those above, as = w k t = k t o =. () x k Here the integral with respet to proper time t o has been simplied by further dening the hyperboli veloity angle or rapidity k sinh [w k =] = tanh [v k =]. Note that both v and the \transverse time-speed" are onstants, and hene both proper veloity, and longitudinal momentum p k mw k, hange at a uniform rate when proper aeleration is held onstant. If motion is only in the diretion of aeleration, is, and p=t = m in the lassial tradition. In order to visualize the relationships dened by equation, it is helpful to plot for the (+)D or = ase all veloities and times versus x in dimensionless form from a ommon origin on a single graph (i.e. as v=, t o =, w= = t=, and versus x= ). As shown in Fig., v= is asymptoti to, t o = is exponential for large arguments, w= = t= are hyperboli, and also tangent to a linear for large arguments. The equations underlying this plot, from for = and oordinates sharing a ommon origin, an be written simply as: s x t += to + = osh r w. = = q + () v = This universal aeleration plot, adapted to the relevant range of variables, an be used to illustrate the solution of, and possibly to graphially solve, any onstant aeleration problem. Similar plots an be onstruted for more ompliated trips (e.g. aelerated twin-paradox adventures) and for the (+)D ase as well. veloities (e.g. v/) & times (e.g. at/) - - Fig : Variables in onstant (+)D aeleration speed of time oordinate veloity - 5 dimensionless position (ax/^) p-veloity & -time proper time FIG.. The variables involved in (+)D onstant aeleration In lassial kinematis, the rate at whih traveler energy E inreases with time is frame-dependent, but the rate at whih momentum p inreases is invariant. In speial relativity, these rates (when gured with respet to proper time) relate to eah other as time and spae omponents, respetively, of the aeleration -vetor. Both are frame-dependent at high speed. However, we an de- ne proper fore separately as the fore felt by an aelerated objet. We show in the Appendix that this is simply! F o m!. That is, all aelerated objets feel a frame-invariant -vetor fore! F o in the diretion of their aeleration. The magnitude of this fore an be alulated from any inertial frame, by multiplying the rate of momentum hange in the aeleration diretion times, or by multiplying mass times the proper aeleration. The lassial relation F = dp= = mdv= = md x= = ma then beomes:
dp k F o = = m dw k = m dv k = m d d(v k ) = m x k = m a = m (8) Even though the rate of momentum hange joins the rate of energy hange in beoming frame-dependent at high speed, Newton's nd Law for -vetors thus retains a frame-invariant form. Although they depend on the observer's inertial frame, it is instrutive to write out the omponents of momentum and energy rate-of-hange in terms of proper fore magnitude F o. The lassial equation relating rates of momentum hange to fore is d! p = =! F = ma! i k, where! i k is the unit vetor in the diretion of aeleration. This beomes d! p = F o!ik v + w k!i. (9) Note that if there are non-zero omponents of veloity in diretions both parallel and perpendiular to the diretion of aeleration, then momentum hanges are seen to have a omponent perpendiular to the aeleration diretion, as well as parallel to it. These transverse momentum hanges result beause transverse proper veloity w = v (and hene momentum p )hanges when traveler hanges, even though v is staying onstant. As mentioned above, the rate at whih traveler energy inreases with time lassially depends on traveler veloity through the relation de= = Fv k = m(! a! v ). Relativistially, this beomes de = F o w k = m (!! w ). () Hene the rate of traveler energy inrease is in form very similar to that in the lassial ase. Similarly, the lassial relationship between work, fore, and impulse an be summarized with the relation de=dx k = F = dp k =. Relativistially, this beomes de dx k = F o = dp k. () One again, save for some hanges in saling assoiated with the \transverse time-speed" onstant, the form of the lassial relationship between work, fore, and impulse is preserved in the relativisti ase. Sine these simple onnetions are a result, and not the reason, for our introdution of proper time/veloity in ontext of a single inertial frame, we suspet that they provide insight into relations that are true both lassially and relativistially, and thus are benets of \type B" disussed in the introdution. IV. PROPER-VELOCITY EQUATIONS IN \POST-TRANSFORM" RELATIVITY. The foregoing treats alulations made possible, and analogies with lassial forms whih result, if one introdues the proper time/veloity variables in ontext of a single inertial frame, prior to disussion of multiple inertial frames and hene prior to the introdution of Lorentz transforms. Are the Lorentz transform, and other post-transform relations, similarly simplied or extended The answer is yes, although our insights in this area are limited by the fats that: (i) we have taken only a ursory look at post-transform material, and (ii) one key expansion area, the treatment of aeleration, is already taken are of by the pre-transform material above. The Lorentz transform itself is simplied, in that it an be written using proper veloity in the symmetri matrix form: t x y z 5 = B @ w w C A t x 5. () y z This seems to us an improvement over the asymmetri equations normally used, but of ourse requires a bit of matrix and -vetor notation that your students may not be ready to exploit. The expression for length ontration, namely L = L o =, is not hanged at all. The developments above do suggest that the onept of proper length L o, as the length of a yardstik in the frame in whih it is at rest, may have broader use as well. The relativisti p Doppler eet expression, given as f = f o f+(v=)g=f (v=)g in terms of oordinate veloity, also simplies to f = f o =f (w=)g. The most notieable eet of proper veloity, in the post-transform relativity we've onsidered so far, involves simpliation and symmetrization of the veloity addition rule. The rule for adding oordinate veloities! v and! v to get relative oordinate veloity! v, namely v k =(v +v k )=( + v k v = ) and v = v with subsripts referring to omponent orientation with respet to the diretion of! v, is inherently ompliated. Moreover, for high speed alulations, the answer is usually uninteresting sine large oordinate veloities always add up to something very near to. By omparison, if one adds proper veloities! w =! v and! w =! v to get relative proper veloity! w, one nds simply that the oordinate veloity fators add while the -fators multiply, i.e. w k = v + v k,with w = w. () Note that the omponents transverse to the diretion of! v are unhanged. Physially more interesting questions an be answered with equation than with the oordinate veloity addition rule. For example, one might ask what relative proper veloity (and hene momentum) is attainable with olliding beams from an aelerator able to
produe partiles of proper speed w for impat onto a stationary target. If one is using 5GeV eletrons in the LEP aelerator at CERN, and are E=m ' 5GeV=5keV ' 5, v and v are essentially, and w and w are hene 5. Upon ollision, equation tells us that in a ollider the relative proper speed w is ( 5 ) ( + ) =. Investment in a ollider thus buys a fator of = 5 inrease in the momentum (and energy) of ollision. Compared to the ost of building a PeV aelerator for the equivalent eet on a stationary target, the ollider is a bargain indeed! V. CONCLUSIONS. We show inthis paper that introdution of two variables in the ontext of a single inertial frame, speially the non-oordinate proper (or traveler-kinemati) time/veloity pair, lets students takle time dilation and relativisti aeleration problems, prior to onsideration of issues involving multiple inertial frames (like Lorentz transforms, length ontration, and frame-dependent simultaneity). The ardinal rule to follow when doing this is simple: All distanes must be dened with respet to \maps" drawn from the vantage point of a single inertial referene frame. We show further that a frame-invariant proper aeleration -vetor has three simple integrals of the motion, in terms of these variables. Hene students an speak of the proper aeleration and fore -vetors for an objet in frame-independent terms, and solve relativisti onstant aeleration problems muh as they now dofor non-relativisti problems in introdutory ourses. We further show that the use of these variables does more than \superially preserve lassial forms". Not only are more than one lassial equation preserved with minor hange with these variables. In addition, more interesting physis is aessible to students more quikly with the equations that result. The relativisti addition rule for proper veloities is a speial ase of the latter in point. Hene we argue that the trend in the pedagogial literature, away from relativisti masses and toward use of proper time and veloity in ombination, may be a robust one whih provides: (B) deeper insight, as well as (A) more value from lessons rst-taught. ACKNOWLEDGMENTS This work has beneted indiretly from support by the U.S. Department of Energy, the Missouri Researh Board, as well as Monsanto and MEMC Eletroni Materials Companies. It has beneted most, however, from the interest and support of students at UM-St. Louis. APPENDIX A: THE -VECTOR PERSPECTIVE This appendix provides a more elegant view of matters disussed in the body of this paper by using spaetime -vetors not used there, along with some promised derivations. We postulate rst that: (i) displaements between events in spae and time may be desribed by a displaement -vetor X for whih the time{omponent may be put into distane-units by multiplying by the speed of light ; (ii) subtrating the sum of squares of spae-related omponents of any -vetor from the time omponent squared yields a salar \dot-produt" whih is frame-invariant, i.e. whih has a value whih is the same for all inertial observers; and (iii) translational momentum and energy, two physial quantities whih are onserved in the absene of external intervention, are omponents of the momentum-energy -vetor P m dx, where m is the objet's rest mass and t o is the frame-invariant displaement in time-units along its trajetory. From above, the -vetor displaement between two events in spae-time is desribed in terms of the position and time oordinate values for those two events, and an be written as: X t x 5. (A) y z Here the usual -notation is used to represent the value of nal minus initial. The dot-produt of the displaement -vetor is dened as the square of the frameinvariant proper-time interval between those two events. In other words, (t o ) X X =(t) (x +y +z ). (A) Sine this dot-produt an be positive or negative, proper time intervals an be real (time-like) or imaginary (spaelike). It is easy to rearrange this equation for the ase when the displaement is innitesimal, to onrm the rst two equalities in equation via: s = dx + = q dx. (A) The momentum-energy -vetor, as mentioned above, is then written using and the omponents of proper veloity! w d! x as: E w P mu = x m 5 = p x w y p y 5. (A) w z p z Here we've also taken the liberty to use a veloity - vetor U dx=. The equality in equation between 5
and E=m follows immediately. The frame-invariant dot-produt of this -vetor, times squared, yields the familiar relativisti relation between total energy E, momentum p, and frame-invariant rest mass-energy m : P P = m = E (p). (A5) If we dene kineti energy as the dierene between rest mass-energy and total energy using K E m, then the last equality inequation follows as well. Another useful relation whih follows is the relation between in- nitesimal unertainties, namely de = dx. dp Lastly, the fore-power -vetor may be dened as the proper time derivative of the momentum-energy -vetor, i.e.: F dp ma = m d dwx dwy dwz 5 = de dpx dpy dpz 5. (A) Here we've taken the liberty to dene aeleration - vetor A d X= o as well. The dot-produt of the fore-power -vetor is always negative. It may therefore be used to dene the frameinvariant proper aeleration, by writing: above, is that seen by the aelerated objet itself. As equation A8 shows for t o ;v and o set to zero, this is nothing more than F! o m!. Thus some utility for the rapidity/proper time integral of the equations of onstant proper aeleration (rd term in eqn. ) is illustrated as well. e.g. A. P. Frenh, Speial Relativity (W. W. Norton, NY, 98), p.. e.g. F. J. Blatt, Modern Physis (MGraw-Hill, NY, 99). H. Goldstein, Classial Mehanis, th printing (Addison- Wesley, Reading MA, 95), p. 5. Sears and Brehme, Introdution to the Theory of Relativity (Addison-Wesley, NY, 98). 5 W. A. Shurli, Speial Relativity: The Central Ideas (9 Appleton St., Cambridge MA 8, 99). P.Fraundorf, Non-oordinate time/veloity pairs in speial relativity, gr-q/98 (xxx.lanl.gov arhive, NM, 99). E. Taylor and J. A. Wheeler, Spaetime Physis, st edition (W. H. Freeman, San Franiso, 9). F F (m) de = dp. (A) We still must show that this frame-invariant proper aeleration has the magnitude speied in the text (eqn. 5). To relate proper aeleration to oordinate aeleration! a d! v dw k d! x, note rst that d = v k a, that a. Putting these re- = a, and that dw = v sults into the dot-produt expression for the fourth term in A and simplifying yields = a as required. As mentioned in the text, power is lassially framedependent, but frame-dependene for the omponents of momentum hange only asserts itself at high speed. This is best illustrated by writing out the fore -vetor omponents for a trajetory with onstant proper aeleration, in terms of frame-invariant proper time/aeleration variables t o and. If we onsider separately the momentum-hange omponents parallel and perpendiular to the unhanging and frame-independent aeleration -vetor!, one gets F = de dp k dp 5 = m v k sinh to v osh to sinh to + o + o + o 5, (A8) where o is simply the initial value for k sinh [ w k ]. The fore responsible for motion, as distint from the frame-dependent rates of momentum hange desribed