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Proper velocity and frame-invariant acceleration in special relativity

Abstract

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

Main Paper

  • 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 (xxx.lanl.gov 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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    Appendix

    This appendix provides a more elegant view of matters discussed in the body of this paper by using space-time 4-vectors not used there, along with some promised derivations. We postulate first that: (i) displacements between events in space and time may be described by a displacement 4-vector X for which the time--component may be put into distance-units by multiplying by the speed of light c; (ii) subtracting the sum of squares of space-related components of any 4-vector from the time component squared yields a scalar ``dot-product'' which is frame-invariant, i.e. which has a value which is the same for all inertial observers; and (iii) translational momentum and energy, two physical quantities which are conserved in the absence of external intervention, are components of the momentum-energy 4-vector P = m dX/dto, where m is the object's rest mass and to is the frame-invariant displacement, in time-units, along its trajectory.

    From above, the 4-vector displacement between two events in space-time is described in terms of the position and time coordinate values for those two events, and can be written as:

    (13)
    Here the usual ``Delta''-notation is used to represent the value of final minus initial. The dot-product of the displacement 4-vector is defined as the square of the frame-invariant proper-time interval between those two events. In other words,
    (14)
    Since this dot-product can be positive or negative, proper time intervals can be real (time-like) or imaginary (space-like). It is easy to rearrange this equation for the case when the displacement is infinitesimal, to confirm the first two equalities in equation (2) via:
    (15)

    The momentum-energy 4-vector, as mentioned above, is then written using gamma and the components of proper velocity w = dx/dto as:

    (16)
    Here we've also taken the liberty to define a velocity 4-vector U. The equality in equation (2) between gamma and E/mc^2 follows immediately. The frame-invariant dot-product of this 4-vector, times c squared, yields the familiar relativistic relation between total energy E, momentum p, and frame-invariant rest mass-energy mc^2:
    (17)
    If we define kinetic energy as the difference between rest mass-energy and total energy using K = E-mc^2, then the last equality in equation (2) follows as well. Another useful relation which follows is the relation between infinitesimal uncertainties, namely dE/dp = dx/dt.

    Lastly, the force-power 4-vector may be defined as the proper time derivative of the momentum-energy 4-vector, i.e.:

    (18)
    Here we've taken the liberty to define an acceleration 4-vector A as well.

    The dot-product of the force-power 4-vector is always negative. It may therefore be used to define the frame-invariant proper acceleration alpha, by writing:

    (19)
    We still must show that this frame-invariant proper acceleration has the magnitude specified in the text (eqn. 5). To relate proper acceleration alpha to coordinate acceleration a = dv/dt = d^2x/dt^2, note first that c d(gamma)dto = gamma^4 v||/c a, that dw||/dto = gamma^4/gammaperp^2 a, and that dwperp/dto = gamma^3 vperp/c v||/c a. Putting these results into the dot-product expression for the sixth term in (19) and simplifying yields alpha^2 = gamma^2/gammaperp^2 a^2 as required.

    As mentioned in the text, power is classically frame-dependent, but frame-dependence for the components of momentum change only asserts itself at high speed. This is best illustrated by writing out the force 4-vector components for a trajectory with constant proper acceleration, in terms of frame-invariant proper time/acceleration variables to and alpha. If we consider separately the momentum-change components parallel and perpendicular to the unchanging and frame-independent acceleration 3-vector alpha, one gets

    (20)
    where etao is simply the initial value for eta|| = ArcSinh[w||/c].

    The force responsible for motion, as distinct from the frame-dependent rates of momentum change described above, is that seen by the accelerated object itself. As equation 20 shows for to,vperp and etao set to zero, this is nothing more than Fo = m alpha. Thus some utility for the rapidity/proper time integral of the equations of constant proper acceleration (3rd term in eqn. 6) is illustrated as well.

    For more on this subject, see our table of contents. Please share your thoughts via our review template, or send comments, answers to problems posed, and/or complaints, to philf@newton.umsl.edu. This page contains original material, so if you choose to echo in your work, in print, or on the web, a citation would be cool. (Thanks. /philf :)