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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.
Hmm, the inadequacies of English for discussing relativistic physics
bite again. No, the accelerating object sees light from an inertial
(non-accelerating) object behind it forever; there is no proper time of
the accelerating object where the inertial object cannot be perceived
(however faint and doppler-shifted). However, as the accelerating
object's proper time goes to infinity, the light from the inertial
object is seen to come from proper times of the inertial object that
approach a finite limit. In the case of an accelerating object
initially coincident and non-moving with respect to the inertial object,
that finite time limit of intertial time is 1/a -- the accelerating
object sees the inertial object forever, but when the accelerating
object reads a clock on the inertial object, the reading on that clock
approaches a time 1/a after its departure.
> [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.
Unfortunately it appears that the archive of mailing list letters at
http://sunsite.unc.edu/lunar/ssdbbs.html is no longer accessible. I
would have to do some hunting to see if I have any of the material saved
in my personal archives. I intuitively worked out the notion of the
relationsihp between proper time of an emitter and proper time of an
accelerating receiver some time ago, but recently made progress in the
solution of a more complicated problem of which this was a special case
(tracing light rays from an arbitrary point in spacetime to the
worldline of an accelerated object).
I've been working on learning various aspects of relativistic physics
for some time, as Timothy can tell you. My particular interest is in
developing a spaceflight simulator that handles special relativity
effects, and so my research has been concentrated in deriving various
things I need to accomplish that, like modeling the spatial locations of
multiple relativistic objects and determining their apparent visual
locations; in other words, to concretely calculate things like "what do
stars in the solar neighborhood and a passing starship look like when
you are travelling towards Tau Ceti at 0.9c?"
> >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.)
These notational problems seem to keep coming up, since we all seem to
have learned or developed various idiosyncratic notations for handling
problems in relativistic physics.
In my own work I've learned to distinguish between "proper time" (time
experienced along the worldline of an object) and "frame position" (the
location of that object in some frame of reference, whose coordinates
include time in that frame). I generally describe frame positions as a
vector function of proper time. The vectors I use are modeled after the
ones used in _Gravitation_, which put the time coordinate first. For
example, a non-accelerating object has a frame position function of the
form S = V * t' + S0 where S is the frame position vector, S0 is the
frame position at t' == 0, V is what you below call "proper velocity" in
vector form, and t' is the proper time of the object.
At least we haven't run up against the most idiosyncratic nature of my
notation, which includes the use of what I dubbed the "Lorentz dot
product": the Lorentz dot product [ u0 u1 u2 u3 ] dot [ v0 v1 v2 v3 ]
is u0*v0 - u1*v1 - u2*v2 - u3*v3 (other people flip the sign of the
result). For example, if in a frame an observer sees an object move
through displacement [ v1 v2 v3 ] in a unit of time, then we can
describe the frame velocity as V = [ 1 v1 v2 v3 ], and the factor gamma
between that frame velocity and the proper velocity U as 1 / sqrt(V dot V),
where "dot" is the above-mentioned Lorentz dot product. Because I'm
interested in the computational aspects of how to solve these problems
and looking for efficient ways to do these computations, I've found that
recasting a lot of the math in terms of the Lorentz dot product is very
useful, if a bit obfuscating.
The derivation in _Gravitation_, chapter 6, gives the equations:
t = 1/a * sinh(a * t')
x = 1/a * cosh(a * t')
All that's different is that they choose a different origin that makes
things a little simpler for their solution; their curve is shifted right
along the x-axis by 1/a, and the asymptotes of the hyberbolic curve are
simply t = x and t = -x.
> 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.
The _proper time_ of arrival of a photon on the sail relative to the
_proper time_ of emission of the photon is asymptotic, in that as the
proper time of emission approaches a finite limit, the proper time of
reception goes to infinity. Note that "proper time of emission" and
"proper time of reception" are certainly not in the same frame. Since
the distance between emitter and receiver depends on the acceleration of
the receiver, clearly the time of reception is influenced by the
> 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?
You haven't really missed anything, and one can look at the relationship
of received power to emitted power just as well in terms of Doppler
shifting as by using the relationship between emitter time and receiver
time (and in a fundamental sense, they are the same thing expressed in
different terms; Doppler shifting is merely the result of different
rates of elapsed proper time between an emitter and a receiver).
However, your analysis doesn't consider at what time in the emitter's
frame the photons must be emitted to reach the receiver at a given
proper time of the receiver. I think the easiest way to describe this
is with a space-time diagram, but it would be rather hard to include the
diagram in this letter. I'll try to describe the diagram in words so
you can draw it yourself and hopefully see what's going on.
The parametric equation [ t, x ] = 1/a * [ sinh(a * t'), (cosh(a * t')-1) ]
describes a hyperbola with asymptotes t = x + 1/a and t = x - 1/a
approached as t' goes to infinity. So draw this hyperbola and its
asymptotes on paper, putting t on the y-axis and x on the x-axis; the
hyperbola represents the worldine of the receiver. Also draw a heavy
line up the y-axis representing the worldline of the emitter. Now you
can draw lines with slope 1 (or parallel to the upper asymptote) between
the worldline of the emitter and the worldline of the receiver
representing light rays sent from the emitter in the direction of of the
receiver. Note that because of this asymptote rays that leave the
emitter after time 1/a can never reach the receiver (they all travel
above the asymptote), and that as the emitter time approaches 1/a
photons emitted quite close together in emitter proper time are received
with a much larger difference in receiver proper time.
Or I could do this up as a GIF and arrange to mail it to you or give you
a web page URL to pick it up from.
> 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.
Indeed; the inverse square law will make things even worse than either
of us have talked about so far.
> 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.
I'm happy to answer the questions; of the people in this forum, you and
Timothy seem to have the best grasp of relativistic physics, and I'm
very appreciative of opportunities to test my understanding against
other people's knowledge.
> Regards, Rex