Appendix A: The 4-vector perspective

from ``A one-map two-clock approach to teaching relativity in introductory physics'',
arXiv:physics/9611011 (xxx.lanl.gov archive, Los Alamos, NM, 1996).

Abstract

  • See also our earlier one-map three-clock note on four-vectors.
  • by Phil Fraundorf, Dept. of Physics & Astronomy, University of Missouri-StL
  • Related papers: modernizing Newton, anyspeed modeling, and anticipation.
  • See also: aps1996nov07_001.
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  • Symbols: This is a paper has a v-w-u shift in notation from earlier non-coordinate kinematic papers, particularly those which involve the Galilean kinematic.
  • Version release date: 22 Dec 1996.
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    I. Introduction

    II. A traveler, one map, and two clocks

    III. Acceleration and force with one map and two clocks

    IV. Problems involving more than one map

    V. Conclusions

    VI. Acknowledgments

    VII. Appendix A: The 4-vector perspective

    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/dtau, where m is the object's rest mass and tau 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:

    (A1)
    Here the usual Delta 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,

    (A2)
    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:

    (A3)

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

    (A4)
    Here we've also taken the liberty to use a velocity 4-vector U = dX/dtau. The equality in Eqn. 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:

    (A5)
    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 Eqn. 2 for gamma 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.:

    (A6)
    Here we've taken the liberty to define acceleration 4-vector A = d^2X/dtau ^2 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:

    (A7)
    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 dgamma/dtau = gamma^4 a v||/c, that dw||/dtau = a gamma^4/gammaperp^2, and that dwperp/dtau = a gamma^3 vperp v||/c^2. Putting these results into the dot-product expression for the fourth term in Eqn. A6 and simplifying yields alpha^2 = a^2 gamma^6/gammaperp^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 tau 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

    (A8)
    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 Eqn. A8 shows for tau, vperp, and etao set to zero, this is nothing more than F = 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.

    VIII. References

    IX. Tables

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