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*To*: stevev@efn.org*Subject*: Re: your starship-design email of 8/20*From*: DotarSojat@aol.com*Date*: Mon, 26 Aug 1996 00:39:20 -0400*cc*: T.L.G.vanderLinden@student.utwente.nl

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

**Follow-Ups**:**Re: your starship-design email of 8/20***From:*Steve VanDevender <stevev@efn.org>

**The dreaded derivation, Re: your starship-design email of 8/20***From:*Steve VanDevender <stevev@efn.org>

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