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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/dt_o = 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.

At UM-StLouis see also: a1toc, cme, i-fzx, progs, stei-lab, & wuzzlers.

For source, cite URL at http://www.umsl.edu/~run/trvxmp01.html

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For an example...

For an example, consider the 14.2 proper-year first half of a one ''gee'' uniformly-accelerated (alpha = 9.81m/s^2 = 1.03ly/yr^2) trip to Andromeda galaxy, discussed by Lagoute and Davoust. From equation (6), the maximum (final) rapidity is simply eta_|| = alpha t_o/c = 14.7. Hence the final proper velocity is w = sinh (alpha t_o/c) = 1.2 x 10^6ly/tyr. From equation (2) this means that gamma = Sqrt[1+(w/c)^2] = 1.2 x 10^6, and the coordinate velocity v = w/gamma = 0.99999999999963ly/yr. Going back to equation (6), this means that coordinate time elapsed is t = w/alpha c = 1.1 x 10^6years, and distance traveled x = gamma c^2/alpha = 1.1 x 10^6ly.

From equation (5), the coordinate acceleration falls from 1gee at the start of the leg to a = alpha/gamma^3 = 6 x 10^{-19}gee at maximum speed. The forces, energies, and momenta of course depend on the spacecraft's mass. At any given point along the trajectory from the equations above, F_o is of course just m alpha, dE/dx is gamma_perpF_o=F_o, dp/dt is F_o/gamma_perp = F_o, and dE/dt is gamma_perpF_ov_|| = F_ov. Note that all except the last of these are constant if mass is constant, albeit dependent on the reference frame chosen. However, the 4-vector components dp/dt_o and dE/dt_o are not constant at all, showing in another way the pervasive frame-dependences mentioned above.

The mass of course may not be constant. If the spacecraft is propelled by ejecting particles at velocity u opposite to the acceleration direction, the force felt in the frame of the traveler will be simply m alpha = -udm/dt_o. Hence in terms of traveler time the mass obeys m = m_o exp[- alpha t_o/u]. In terms of coordinate time, the differential equation becomes m alpha = -u gamma dm/dt, which solves to the equation derived with significantly more trouble in the reference above.

References

C. Lagoute and E. Davoust, The interstellar traveler, Am. J. Phys. 63 (1995) 221.

Postscript

The foregoing solution may seem routine, as well it should be. It is not. Note that the entire solution was implemented using distances measured (and concepts defined) in context of a single inertial coordinate frame. In other words, suffering through Lorentz transforms is not a prerequisite! Moreover, the 3-vector forces and accelerations used and calculated have frame-invariant components, i.e. those particular parameters are correct in context of any frame.

Compare these tools with those provided in the special relativity sections of your intro-physics, modern-physics, mechanics and relativity textbooks. If the author hasn't gotten the messages below, drop a line to the author and/or the publisher suggesting that they pay a visit here. If they have, let me know so I can put them on a list of places to look!

The messages to look for in those books are that: (i) special relativity deals perfectly well with accelerated objects, (ii) relativistic mass is out and proper velocity is in, (iii) Lorentz transforms can wait until multi-frame phenomena, like length contraction, velocity addition, and frame-dependent simultaneity are on the table, and (iv) the 3-vector form of Newton's second law is usable and component-wise frame invariant at relativistic speeds, if acceleration is "properly" defined. :)

For more on this subject, see our table of contents. Please share your thoughts through 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 :)