Proper velocity and frame-invariant acceleration in special relativity


We examine here a possible endpoint of the trends, in the teaching literature, away from use of relativistic masses (such as m' = gamma m in the momentum = mass times velocity expression) and toward use of proper velocity dx/dto = gamma v (e.g. in that same expression). We show that proper time & proper velocity, taken as components of a non-coordinate time/velocity pair, allow one to introduce time dilation and frame-invariant acceleration/force 3-vectors in the context of one inertial frame, before subjects involving multiple frames (like Lorentz transforms, length contraction, and frame-dependent simultaneity) need be considered. We further show that many post-transform equations (like the velocity-addition rule) acquire elegance and/or utility not found in the absence of these variables.

PACS: 03.30.+p, 01.40.Gm, 01.55.+b

  • by Phil Fraundorf, Dept. of Physics & Astronomy, University of Missouri-StL,
    St. Louis MO 63121-4499, Phone: (314)516-5044, Fax:(314)516-6152
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  • Cite: author, title, physics/9611011 ( archive, Los Alamos, NM, 1996). See also aps1996nov07_001. This is a paper for teachers (with a v-w-u notation shift from earlier non-coordinate kinematic papers which involve the Galilean kinematic).
  • Version release date: 08 Nov 1996.
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    Efforts to connect classical and relativistic concepts will be with us as long as classical kinematics is taught to introductory students. For example, the observation that relativistic objects behave at high speed as though their inertial mass increases in the p=mv expression, led to the definition (used in many early textbooks {e.g. French}) of relativistic mass m' = m gamma. Such efforts might help to: (A) get the most from first-taught relationships, and (B) find what is fundamentally true in both classical and relativistic approaches. The concepts of transverse (m') and longitudinal (m'' = m gamma^3) masses have similarly been used {e.g. Blatt} to preserve relations of the form Fx = max for forces perpendicular and parallel, respectively, to the velocity direction.

    Unfortunately for these relativistic masses, no deep sense in which mass either changes or has directional dependence has emerged. They put familiar relationships to use in keeping track of non-classical behaviors (item A above), but do not (item B above) provide frame-invariant insights or make other relationships simpler as well. As a result, majority acceptance of their use seems further away now than it did several decades ago {Goldstein}.

    A more subtle trend in the pedagogical literature has been toward the definition of a quantity called proper velocity {cf. SearsBrehme,Shurcliff}, which can be written as w = gamma v. We use the symbol w here because it is not in common use elsewhere in relativity texts, and because w looks like gamma v from a distance. This quantity also lets us cast the momentum expression above in classical form as a mass times a velocity, ie. as p = mw. Hence it serves at least one of the ``type A'' goals served by m' above.

    We show here that, when introduced as part of a non-coordinate time/velocity pair in pre-transform special relativity, proper velocity allows us to introduce relativistic momentum, time-dilation, and frame-invariant relativistic acceleration in context of a single inertial frame. Moreover, through use of proper velocity many relationships (including post-transform relationships like velocity addition which require consideration of multiple frames) are made simpler and sometimes more useful. Hence it appears to serve goals of ``type B'' mentioned above as well.

    One object, one frame, but two times: the consequences

    One may argue that a fundamental break between classical and relativistic kinematics involves the observation that time passes differently for moving observers, than for stationary ones. To quantify this, we define two time variables when describing the motion of a single object (traveler) with respect to a single inertial coordinate frame. These are the coordinate time t and the proper (or traveler) time to. Note that the proper time may be different for different travelers.

    It follows from above that we might also consider two velocities, namely coordinate velocity v = dx/dt, and proper (or traveler-kinematic) velocity w = dx/dto. It is helpful to distinguish the units used to measure these velocities, by saying that the first measures distance traveled per unit coordinate time, while the latter measures distance traveled per unit traveler time. Convenient units are [lightyears per coordinate year] and [lightyears per traveler year], respectively. Each of these velocities can be calculated from the other by knowing the velocity-dependence of the ``traveler's speed of time'' gamma=dt/dto, via the equation

    Note that all displacements dx are defined with respect to a single inertial (unprimed) frame. Thus proper velocity is not simply a coordinate velocity measured with respect to a different frame. It is rather one of an infinite number of non-coordinate velocities, definable in context of time/velocity pairs experienced by one of the infinite number of observers who may choose to track the motion of our given object on a given map using their own clock{Noncoord}. The cardinal rule here is: measure all displacements from the vantage point of the chosen inertial (or ``map'') frame. Thus proper velocity w is simply the rate at which the coordinates of the traveler change per unit traveler time on a map of the universe (e.g. in the traveler's ``glove compartment'') which was drawn from the point of view of the chosen inertial reference 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'', including the familiar relationship to coordinate velocity, follow simply from the (3+1)D nature of the flat spacetime metric (specifically from the frame-invariant dot-product rule applied to the displacement and velocity 4-vectors). Their derivation therefrom is outlined in the Appendix. For introductory students, we can simply quote Einstein's prediction that spacetime is tied together so that instead of gamma =1, one has gamma =1/Sqrt[1-(v/c)^2}=E/mc^2, where E is Einstein's ``relativistic energy'' and c is the speed of light. It then follows from above that:

    Here of course K is the kinetic energy of motion, equal classically to (1/2)mv^2. One of the simplest exercises a student might do at this point is to show that since v = w/gamma = w/Sqrt[1+(w/c)^2], as proper speed w goes to infinity, the coordinate speed v never gets larger than c! All of classical kinematics also follows from these things if and only if gamma is near 1, which as one can see above is true only for speeds v much less than c.

    Given these tools to describe the motion of an object with respect to single inertial reference frame, perhaps the easiest type of relativistic problem to solve is that of time dilation. From the very definition of gamma as a ``traveler's speed of time'', and the velocity relations which show that gamma is greater than or equal to 1, it is easy to see that the traveler's clock will always run slower than coordinate time. Hence if the traveler is going at a constant speed, one has from equation (2) that traveler time is dilated (spread out over a larger interval) relative to coordinate time, by the relation

    In this way time-dilation problems can be addressed, without first introducing the deeper complications (like frame-dependent simultaneity) associated with multiple inertial frames. This is the first of several skills that this strategy can offer to students taking only introductory physics, a ``type A'' benefit according to the introduction. A practical awareness of the non-global nature of time thus need not require a readiness for the abstraction of Lorentz transforms.

    Equations (2) above also allow one to relate these velocities to energy. Hence an important part of relativistic dynamics is in hand as well. Another important part of relativistic dynamics, as mentioned in the introduction, takes on a familiar form since momentum at any speed is

    This relation has important scientific consequences as well. It shows that momentum like proper velocity has no upper limit, and that coordinate velocity becomes irrelevant to tracking momentum at high speeds.

    One might already imagine that proper speed w is the important speed to a relativistic traveler trying to get somewhere with minimum traveler time. Equation (4) shows that it is also a more interesting speed from the point of view of law enforcement officials wishing to minimize fatalities on a highway in which relativistic speeds are an option. This a ``type B'' benefit of proper velocity. In this context, it is not surprising that the press doesn't report ``land speed record'' for the fastest accelerated particle. New progress only changes the value of v in the 7th or 8th decimal place. The story of increasing proper velocity, in the meantime, goes untold to a public whose imagination might be captured thereby. After all, the educated lay public (comprised of those who have had only one physics course) is by and large under the impression that the lightspeed limit rules out major progress along these lines.

    The frame-invariant acceleration 3-vector

    The foregoing relations introduce, in context of a single inertial frame and without Lorentz transforms, many of the kinematical and dynamical relations of special relativity taught in introductory courses, in modern physics courses, and perhaps even in some relativity courses. In this section, we cover less familiar territory, namely the equations of relativistic acceleration. Forces if defined simply as rates of momentum change in special relativity have no frame-invariant formulation, and hence Newton's 2nd Law retains it's elegance only if written in coordinate-independent 4-vector form. It is less commonly taught, however, that a frame-invariant 3-vector acceleration can be defined (again also in context of a single inertial frame). We show that, in terms of proper velocity and proper time, this acceleration has three simple integrals when held constant. Moreover, it bears a familiar relationship to the special frame-independent rate of momentum change felt by an accelerated traveler.

    By again examining the frame-invariant scalar product of a 4-vector (this time of the acceleration 4-vector), one can show (as we do in the Appendix) that a ``proper acceleration'' for a given object, which is the same to all inertial observers and thus ``frame-invariant'', can be written in terms of components for the classical acceleration vector (i.e. the second coordinate-time t derivative of displacement x) by:

    This is quite remarkable, given that a is so strongly frame-dependent! Here the ``transverse time-speed" gammaperp is defined as 1/Sqrt[1-(vperp/c)^2], where vperp is the component of coordinate velocity v perpendicular to the direction of coordinate accceleration a. In this section generally, in fact, subscripts || and perp refer to parallel and perpendicular component directions relative to the direction of coordinate acceleration a, and not (for example) relative to coordinate velocity v.

    Before considering integrals of the motion for constant proper acceleration alpha, let's review the classical integrals of motion for constant acceleration a. These can be written as a = Delta v||/Delta t = (1/2) Delta(v^2)/Delta x||. The first of these is associated with conservation of momentum in the absence of acceleration, and the second with the work-energy theorem. These may look more familiar in the form v||f = v||i + a Delta t, and v||f^2 = v||i^2 + 2 a Delta x||. Given that coordinate velocity has an upper limit at the speed of light, it is easy to imagine why holding coordinate acceleration constant in relativistic situations requires forces which change even from the traveler's point of view, and is not possible at all for Delta t greater than (c-v||i)/a.

    Provided that proper time to, proper velocity w, and time-speed gamma can be used as variables, three simple integrals of the proper acceleration can be obtained by a procedure which works for integrating other non-coordinate velocity/time expressions as well{Noncoord}. The resulting integrals are summarized in compact form, like those above, as

    Here the integral with respect to proper time to has been simplified by further defining the hyperbolic velocity angle or rapidity{TaylorWheeler} eta|| = ArcSinh[w||/c] = ArcTanh[v||/c]. Note that both vperp and the ``transverse time-speed'' gammaperp are constants, and hence both proper velocity, and longitudinal momentum p|| = mw||, change at a uniform rate when proper acceleration is held constant. If motion is only in the direction of acceleration, gammaperp is 1, and Delta p/Delta t = m alpha in the classical tradition.

    In order to visualize the relationships defined by equation 6, it is helpful to plot for the (1+1)D or gammaperp=1 case all velocities and times versus x in dimensionless form from a common origin on a single graph (i.e. as v/c, alpha to/c, w/c=alpha t/c, and gamma versus alpha x/c^2). As shown in Fig. 1, v/c is asymptotic to 1, alpha to/c is exponential for large arguments, w/c=alpha t/c are hyperbolic, and also tangent to a linear gamma for large arguments. The equations underlying this plot, from eqn 6 for gammaperp=1 and coordinates sharing a common origin, can be written simply as:

    This universal acceleration plot, adapted to the relevant range of variables, can be used to illustrate the solution of, and possibly to graphically solve, any constant acceleration problem. Similar plots can be constructed for more complicated trips (e.g. accelerated twin-paradox adventures) and for the (3+1)D case as well{Noncoord}.

    FIG. 1. The variables involved in (1+1)D constant acceleration

    In classical kinematics, the rate at which traveler energy E increases with time is not frame-independent, but the rate at which momentum p increases is invariant. In special relativity, these rates (when figured with respect to proper time) relate to each other as time and space components, respectively, of the acceleration 4-vector. Both are frame-dependent at high speed. However, we can define proper force separately as the force felt by an accelerated object. We show in the Appendix that this is simply Fo = m alpha. That is, all accelerated objects feel a frame-invariant 3-vector force Fo in the direction of their acceleration. The magnitude of this force can be calculated from any inertial frame, by multiplying the rate of momentum change in the acceleration direction times gammaperp, or by multiplying mass times the proper acceleration alpha. The classical relation F = dp/dt = mdv/dt = md^2x/dt^2 = ma then becomes:

    Even though the rate of momentum change joins the rate of energy change in becoming frame-dependent at high speed, Newton's 2nd Law for 3-vectors thus retains a frame-invariant form.

    Although they depend on the observer's inertial frame, it is instructive to write out the rates of momentum and energy change in terms of the proper force magnitude Fo. The classical equation relating rates of momentum change to force is dp/dt = F = ma i||, where i|| is the unit vector in the direction of acceleration. This becomes

    Note that if there are non-zero components of velocity in directions both parallel and perpendicular to the direction of acceleration, then momentum changes are seen to have a component perpendicular to the acceleration direction, as well as parallel to it. These transverse momentum changes result because transverse proper velocity wperp = gamma vperp (and hence momentum pperp) changes when traveler gamma changes, even though vperp is staying constant.

    As mentioned above, the rate at which traveler energy increases with time classically depends on traveler velocity through the relation dE/dt = Fv|| = ma*v. Relativistically, this becomes

    Hence the rate of traveler energy increase is in form very similar to that in the classical case.

    Similarly, the classical relationship between work, force, and impulse can be summarized with the relation dE/dx|| = F = dp||/dt. Relativistically, this becomes

    Once again, save for some changes in scaling associated with the ``transverse time-speed'' constant gammaperp, the form of the classical relationship between work, force, and impulse is preserved in the relativistic case. Since these simple connections are a result, and not the reason, for our introduction of proper time/velocity in context of a single inertial frame, we suspect that they provide insight into relations that are true both classically and relativistically, and thus are benefits of ``type B'' discussed in the introduction.

    For an example...

    Proper-velocity equations in ``post-transform'' relativity

    The foregoing treats calculations made possible, and analogies with classical forms which result, if one introduces the proper time/velocity variables in context of a single inertial frame, prior to discussion of multiple inertial frames and hence prior to the introduction of Lorentz transforms. Are the Lorentz transform, and other post-transform relations, similarly simplified or extended? The answer is yes, although our insights in this area are limited by the facts that: (i) we have taken only a cursory look at post-transform material, and (ii) one key expansion area, the treatment of acceleration, is already taken care of by the pre-transform material above.

    The Lorentz transform itself is simplified, in that it can be written using proper velocity in the symmetric matrix form:

    This seems to us an improvement over the asymmetric equations normally used, but of course requires a bit of matrix and 4-vector notation that your students may not be ready to exploit.

    The expression for length contraction, namely L = Lo/gamma, is not changed at all. The developments above do suggest that the concept of proper length Lo, as the length of a yardstick in the frame in which it is at rest, may have broader use as well. The relativistic Doppler effect expression, given as frequency f = fo Sqrt[{1+(v/c)}/{1-(v/c)}] in terms of coordinate velocity, also simplifies to f = fo/{gamma-(w/c)}.

    The most noticeable effect of proper velocity, in the post-transform relativity we've considered so far, involves simplification and symmetrization of the velocity addition rule. The rule for adding coordinate velocities v' and v to get relative coordinate velocity v'' in inherently complicated, namely v||'' = (v'+v||)/(1+v||v'/c^2) with vperp'' NOT equal to vperp. Here subscripts refer to component orientation with respect to the direction of v'. Moreover, for high speed calculations, the answer is usually un-interesting since large coordinate velocities always add up to something very near to c. By comparison, if one adds proper velocities w' = gamma' v' and w = gamma v to get relative proper velocity w'', one finds simply that the coordinate velocity factors add while the gamma-factors multiply, i.e.

    Note that the components transverse to the direction of v' are unchanged.

    Physically more interesting questions can be answered with equation (12) than with the coordinate velocity addition rule. For example, one might ask what relative proper velocity (and hence momentum) is attainable with colliding beams from an accelerator able to produce particles of proper speed w for impact onto a stationary target. If one is using 50GeV electrons in the LEP2 accelerator at CERN, gamma and gamma' are E/mc^2 = 50GeV/511keV or 10^5, v and v' are essentially c, and w and w' are hence 10^5c. Upon collision, equation (12) tells us that in a collider the relative proper speed w'' is (10^5)^2(c+c) = 2 x 10^10 c. Investment in a collider thus buys a factor of 2 gamma = 2 x 10^5 increase in the momentum (and energy) of collision. Compared to the cost of building a 10PeV accelerator for the equivalent effect on a stationary target, the collider is a bargain indeed!


    We show in this paper that introduction of two variables in the context of a single inertial frame, specifically the non-coordinate proper (or traveler-kinematic) time/velocity pair, lets students tackle time dilation and relativistic acceleration problems, prior to consideration of issues involving multiple inertial frames (like Lorentz transforms, length contraction, and frame-dependent simultaneity). The cardinal rule to follow when doing this is simple: All distances must be defined with respect to ``maps'' drawn from the vantage point of a single inertial reference frame.

    We show further that a frame-invariant proper acceleration 3-vector has three simple integrals of the motion, in terms of these variables. Hence students can speak of the proper acceleration and force 3-vectors for an object in frame-independent terms, and solve relativistic constant acceleration problems much as they now do for non-relativistic problems in introductory courses.

    We further show that the use of these variables does more than ``superficially preserve classical forms''. Not only are more than one classical equation preserved with minor change with these variables. In addition, more interesting physics is accessible to students more quickly with the equations that result. The relativistic addition rule for proper velocities is a special case of the latter in point. Hence we argue that the trend in the pedagogical literature, away from relativistic masses and toward use of proper time and velocity in combination, may be a robust one which provides: (B) deeper insight, as well as (A) more value from lessons first-taught.



    This work has benefited indirectly from support by the U.S. Department of Energy, the Missouri Research Board, as well as Monsanto and MEMC Electronic Materials Companies. It has benefited most, however, from the interest and support of students at UM-St. Louis.


  • {French} A. P. French, Special Relativity (W. W. Norton, NY, 1968), p. 22.
  • {Blatt} F. J. Blatt, Modern Physics (McGraw-Hill, NY, 1992).
  • {Goldstein} H. Goldstein, Classical Mechanics, 7th printing (Addison-Wesley, Reading MA, 1965), p. 205.
  • {SearsBrehme} Sears and Brehme, Introduction to the Theory of Relativity (Addison Wesley, 1968).
  • {Shurcliff} W. A. Shurcliff, Special Relativity: The Central Ideas (19 Appleton St., Cambridge MA 02138, 1996).
  • {Noncoord} P. Fraundorf, "Non-coordinate time/velocity pairs in special relativity", General Relativity & Quantum Cosmology gr-qc/9607038, ( e-Print archive, Los Alamos NM, 1996).
  • {TaylorWheeler} F. Taylor and J. A. Wheeler, Spacetime Physics (W. H. Freeman, San Francisco, 1963 and 1992).
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