Re: relativistic acceleration stuff

[Kevin C Houston <hous0042@maroon.tc.umn.edu>]

> Kevin C. Houston writes:
> 
>  > > 
>  > This can be kept to an order of two magnitude. Beamer stays constant.
> 

> What are you saying here?  That the beam only has to vary in
> magnitude by an "order of two", or that it stays constant?

Both.
The Beamer sends a constant wavelength.  The ship only sees the 
wavelength vary by a factor of ~130 near midpoint. (velocity is limited 
to .99997 c due to the ship reducing acceleration before midpoint)

> 
> To maintain constant acceleration _in the ship's rest frame_, the
> ship must receive a constant amount of energy _in its frame_.  As
> the ship accelerates away from the beamer and reaches higher
> velocities, _it will see the beamer output go down in energy_.
> If you claim anything else, you haven't been studying your
> relativity theory.

You keep making the mistake (or the mis-type) of equating energy and 
power (power = energy/time) so that as time slows down, power goes up for 
an equal amount of energy or stays constant for a decreasing amount of 
energy.  In order to maintain a constant acceleration, the ship must 
recieve a constant amount of _power_ in it's frame, and that can be 
affected by time dialation parameters.

>  > > If you really could maintain 1 g acceleration in the ship frame,
>  > > you run into an even worse problem with getting beamed power --
>  > > during the first year of acceleration, you watch the beam
>  > > decrease in power to effectively 0!  
>  > 
>  > But at the same (well I was gonna say time but that would lead to 
>  > confusion) in the same way, the rate of change of everything on the ship 
>  > decreases.  It's just like going into a kind of hibernation. Thus the 
>  > beamer sends out an energy pulse of about 2 L.Y. long.  
>  > The crew then use the first L.Y.'s worth of energy to approach c to four 
>  > or five decimals.  The ship then nearly "keeps pace" with it's energy 
>  > supply, and uses hardly any at all.  This is possible because with very 
>  > long time parameters, very little energy is needed to effect change, so 
>  > that the crew "feels" awake, the clocks and other (mechanical/electrical/ 
>  > chemical/bio-chemical) systems behave normally within the frame of the 
>  > ship itself.
> 
> I don't think you understand what goes on.  If the ship maintains
> constant acceleration at 1G, it can only see about 1 year's worth
> of power.  IT NEVER SEES LIGHT EMITTED FROM THE BEAMER AFTER
> THAT, UNTIL ITS ACCELERATION DECREASES.

please don't shout.  I understand full well what is going on here, I'm 
just having a little difficulting explaining what I mean to you.  first 
point, I'n not talking about open-ended acceleration, I'm talking about 
the trip we are planning to T.C.  the ship will be under acceleration for 
about 3/4 of a year.  that is, they will only accelerate with power from 
the first 9 months of the beam.  then they will turn around and begin 
decelerating.  

> 
> Draw a spacetime diagram with a hyperbola opening to the right
> whose asymptotes are the lines t = x/c and t = -x/c.  The
> worldline of the uniformly accelerated ship corresponds to the
> hyperbola.  In this diagram, the ship starts at a nonzero x = 1/a
> (a is the acceleration) at t = 0.  Light is constrained to move
> along worldlines of the form t = x/c + k and t = -x/c + k, for
> any arbitrary constant k.  What light lines can intersect the
> hyperbola?  At what times and locations can those light beams be
> emitted in the global frame?  Using the four regions bounded by
> the asymptotes, you'll see that only light from the region below
> t = x/c can ever reach the ship.  Light from above that region
> never reaches the ship.  Similarly, the ship can only send light
> to the region above t = -x/c.
> 
Imagine yourself to be floating in space watching the trip from a long 
ways off.  what would you see as a stationary observer.

1) Time=0 days Sol system begins beaming and "Asimov" begins accelerating.

2) Time~365 days.  Ship reaches near light speed (time=c/a) dist ~.5 L.Y.

3) Time~547 days.  Dist ~1 L.Y. Earth stops sending Beam.  Ship and beam 
travel nearly as one, that is to say that the tail end of the beam does 
not move apreciably nearer to the ship.  Time is very slow on ship by 
factor of ~130:1

4) dist=5.98 L.Y. ship begins decelerating.  Tail end of beam begins to 
move toward ship again.

5) midpoint+547 days dist=11.9 L.Y. ship arrives at T.C.  tail end of 
beam arrives at T.C. also

>  > No, I won't get much beamed _energy_ , I'll have plenty of power tho' 
>  > because each of my seconds becomes worth hundreds of beamer seconds.  So 
>  > when I absorb one seconds worth of energy, it suffices me for 
>  > nearly two minutes beamer time because my demand for energy (in any form) 
>  > has decreased.
> 
> No, you are mixing frames, and thereby ruining your argument.
> Draw everything out in a frame that is inertial (the ship's frame
> is accelerating, so it is invalid as an ongoing reference for
> analysis in special relativity).  When the beamer emits one
> second of energy in your global frame, it takes much more time
> than that along the ship's worldline for that second of energy to
> be received when the ship is at high velocity.  In the case of
> sustained acceleration, only a finite amount of energy from the
> beamer is ever visible to the ship; energy emitted after a time
> 1/a (a is the acceleration in 1/m units) can never reach the
> ship.

I agree that we can't accelerate forever, and for the reason you state.  
However, by the time you had accelerated to that point, you're time rate 
would be so slow, that for all practical purposes you'd be standing 
still. (i.e. you could cover light-years of distance in a single "hour" of 
your awareness)  the small amount of additional acceleration you might 
accomplish over these multi-light-years of space would feel like 10m/s^2
to you on the ship, but it wouldn't increase your energy or momentum very 
much (as measured by the outside observer)

Ok, I'm traveling at .99997 c (no accel), and getting my power from a beam  
coming from behind.  Power in the beam is 1000 KJ/s.  The beam is at a 
wavelength of 130 cm as measured by the stationary beamer.  So that the 
wavelength I "see" is 1 cm. (of course, I cooked the numbers).  So the 
energy in the beam that I can use is 130 times less than if I were stationary
or energy in one second of beam = 7.69 KJ.  But my time rate is 130 times 
slower than the beamer's.  so that I will recieve 130 seconds of beam (as 
measured by the beamer) in only 1 second (as measured by me) this is 7.69 
*130 =1000 KJ.  Time dialation conspires with lorentz contraction to 
provide me with the same amount of _power_.

> 
> You really should sit down and read _Spacetime Physics_.  You're
> making elementary errors in reasoning about relativistic physics.
> 
I have taken a modern physics course in college, although much of it was 
wave equations, we did have some relativity training.  I will be reading 
"Spacetime Physics" and the gravitation book you mentioned over the summer.

Since you have read these books recently, here's a project for you:
come up with equations for the ship that:
1) show ships velocity as a function of ship time (St) assuming constant accel.
   ex. v=f(St,a)

2) show ship's distance from earth in earth-sized meters D as a function of 
   ship time assuming constant accel. 
   ex D (in meters)=f(St,a) 

3) the amount of exhaust mass  Em moving at .9996 c required to accelerate a 
   ship at 10 m/s^2 (as felt by the crew) as a function of ship's total mass 
   Mt. 
   ex. Em=f(Mt,a)

4) the rest mass (as measured by the crew) of that exhaust.  
   ex. Rm=f(Em,.9996c)

5) The power (in units of energy) needed on board to accelerate that mass 
   flow to .9996c (assume stored energy on board to make it easier)
   ex. P=f(Rm,.9996c)

This will enable us to track time and energy requirements of the "Asimov" 
much better than in the past.  Hopefully I'll be able to dump these 
formulas into the spreadsheet, and model the trip much better.

Kevin