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Re: your starship-design email of 8/20

Hi Steve

>I've recently been working on the physics of light signals
>between a "stationary" object and an object undergoing rela-
>tivistic acceleration relative to itself.

I think I have seen two previous remarks by you to the effect
that an object accelerating at one g can not receive light
from behind after an acceleration time of about one year.
[Note: 1 g = 1.0324 lt-yr/yr^2.  Incidentally, I prefer units
of yr for time, lt-yr for distance, lt-yr/yr for velocity and
lt-yr/yr^2 for acceleration.  With these units, values for
interstellar flight are all O(1).]  I'm not organized well
enough to retrieve your previous remarks to enable me to com-
pare them with my recollection stated above.  Please correct
me if I've misremembered.  The existence of previous remarks
indicates that 1) you have been working on this before "recent-
ly", and/or 2) you may possibly have run across some earlier
work relating to this.  I admit that your previous remarks
triggered a feeling of familiarity with the idea, but I can't
determine what that feeling might be based on.

>Consider an object undergoing uniform acceleration relative to
>itself; its frame position at its proper time t1 is:
>[ t x ] = [ 1/a * sinh(a * t1)
>            1/a * cosh(a * t1) ]

I interpret "acceleration relative to itself" as "proper accel-
eration" (i.e., acceleration measured by an inertial platform
mounted on the object), but I'm unfamiliar with the meanings of
"frame position" and "[ t x ]".  ("Coordinates"?  "(t,x)," with
the ordinate first?  Incidentally, I'm of the opinion that the
Minkowski (x,t) coordinate system was a mistake.  It should be
replaced by the "Epstein" (x,t') coordinate system, in which
the "interval equation" conforms geometrically with the Pythag-
orean theorem.  I have adopted the Lorentz-transformation nota-
tion of t for apparent time and t' for proper time.)  I'm also
uncertain about the equation stated above.  Similar equations
that I'm familiar with are

     t = 1/a * sinh(a * t')          and
     x = 1/a * [cosh(a * t') - 1]   .

I can't quite rationalize these with your stated equation.  (I
remember that you left out the [- 1] in the x equation at first
once before.)

Regarding the rest of your discussion, I am locked up by the
idea, supported by the "asymptotic relationship," that the time
of arrival of a beam photon at the accelerated sail depends on
the sail's acceleration.  My intuitive preference, unsupported
by any analysis, is that the time of reception of light by an
accelerating object depends only on the distance from the
source, not on the acceleration of the object.

Let me try a derivation based on distance and velocity only.
(I'm not totally comfortable with it, but it may be usable as
a starting point for discussion.)

Let's consider a beam of photons that is catching up with an
accelerating object receding at an apparent velocity v and a
proper velocity u, where u = gamma * v.  (Incidentally, u = at,
where t is the apparent, not proper, time.)  The energy of
each photon is Doppler-shifted downward by the factor
sqrt[(1 - v)/(1 + v)].  The arrival rate of the photons at the
object is also reduced by the Doppler factor, so the power re-
ceived by the sail is reduced from the power radiated by the
source by the factor (1 - v)/(1 + v).  [If v were given by at'
(which it isn't), one could deduce that the power received by
the sail after an acceleration time of 1/a would be reduced to
zero, consistent with my recollection of your earlier remarks.]
>From the actual relations, as stated above, the received power
would be reduced by the factor (1 - at/gamma)/(1 + at/gamma),
where gamma is sqrt(1 + u^2), which is sqrt[1 + (at)^2].  Then,
if the object were to accelerate at 1 lt-yr/yr^2 (0.9686 g) for
1 yr proper time, the apparent time would be sinh(1) yr, and
the received power would be 13.5 percent of the radiated power.
For acceleration times of 1 and 2 yr and for acceleration
levels of 1 and 0.5 lt-yr/yr^2, the calculated ratio of receiv-
ed power to radiated power is given in the table below:

    Acceleration time     ratio of received to radiated power
         (yr)                   (a = 1)      (a = 0.5)
          1                     0.1353        0.3275
          2                     0.0183        0.0663

What have I missed?

Of course all the considerations above ignore the spillage of
beam power around the sail.  The combined effects of Doppler-
shifting (or whatever) and beam spillage will certainly raise
the ratio of radiated power to captured power to even more
exorbitant levels.

This letter is intended to be more exploratory than analytic.
I need more information to verify your "asymptotic relation-
ship."  I just haven't yet had the time to invest the required
concentration, and I hope that further comments by you would
give some clues to make the verification less time-consuming.
Also, I'm sure that it's a lighter load on you to answer spec-
ific questions than to try to address all plausible questions
in a write-up.

Regards, Rex