# (Re:)^3 Doppler effect

```Hi Steve

I'm glad to hear that a second edition of "Spacetime Physics"
has been published; I'll try to find a copy.  All I have of the
1966 edition are Xerox copies of some pages that I found inter-
esting in 1989.

To quote selectively my Xeroxed page 97:
...
"(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)"

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.)

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.

Regards, Rex

```