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(Re:)^3 Doppler effect
> To quote selectively my Xeroxed page 97:
> "Exercise 51. Clock paradox III
> "(b) How much velocity does the spaceship have after a given
> time? (question in italics) This is the moment to object to
> the question and rephrase it. Velocity [beta] (the original
> has the greek letter) is not the simple quantity to analyze.
> The simple quantity is the velocity parameter [theta]. It is
> simple because it is additive in this sense: Let the velocity
> parameter of the spaceship in Figure 76 with respect to the
> imaginary instantaneously comoving inertial frame change from
> 0 to d[theta] in an astronaut time d[t']. (Let me use t' in
> place of the authors' [tau]). Then the velocity parameter
> of the spaceship with respect to the laboratory frame changes
> in the same astronaut time from the initial value [theta] to
> the subsequent value [theta] + d[theta]. Now relate d[theta]
> to the acceleration [g/c] in the instantaneously comoving
> inertial frame. In this frame
> [g/c] * d[t'] = d[beta] = tanh(d[theta]) ~ d[theta] "
> (Note: I never felt comfortable with this sequence.)
> "so that
> d[theta] = [g/c] * d[t'] (64)"
Since for small values of d[theta] the quantity tanh(d[theta]) is very
close to d[theta], it's reasonable for small accelerations (1 g is about
3.27 * 10^-8 1/s (or (m/s^2)/c); for such a small value the difference
between tanh(x) and x truly is negligible).
> I call a variant of this last equation (with g replaced by F/m),
> F = m * c * d[theta]/dt'
> the "velocity-parameter equation of motion." That is not the
> authors' term. I believe that my derivation of the VPEM (my
> term, abbreviated) on pp 12-13 of my paper is more straightfor-
> ward because it doesn't need the questionable approximations
> in the sequence above.
> (This would be a place for a correction in the second edition.)
I don't believe the second edition contains any form of this problem (I
assume it has to do with relativistic acceleration?). All of the useful
information I've found on special relativistic treatment of acceleration
is in chapter 6 of _Gravitation_ by Misner, Thorne, and Wheeler. They
do mention methods of analysis much like this, including use of the
quantities sinh(a * t'), cosh(a * t'), and tanh(a * t'). Most of the
rest of the book is well over my head and generally irrelevant to the
kinds of relativistic phenomena you'd experience in interstellar travel,
unless you intend to get near a black hole or neutron star, but the
treatment of special relativity in the first few chapters (particularly
3 and 6) has been an invaluable reference to me.
> My paper "An Engineering Review of Relativity for Interstellar
> Flight" is in 4 files in MSWORD6.0a for Windows. If you have
> WORD6.0a to open them in, I can attach the files, one at a time,
> to email notes (I successfully did this to send a copy to
> Timothy, although I backed it up with snail mail). I don't know
> what TeX, troff, or Postscript are.
Sigh. I don't have MS Word 6.0; my home computer runs Linux. Perhaps
you could induce MS Word 6.0 to print your paper as Postscript, so you
could send the Postscript to me? I would think given all the other
features of MS Word this should be possible. I do have a good
Postscript viewer for Linux. As long as the printed version is less
than 5 megabytes in size I will have no problem receiving it.
> Regards, Rex