Re: VRD--Kip S. Thorne book

Hi Carlos,
     I must start with a couple of apologies. Firstly I must apologize
that it appeared that i have not answered your question sooner. In fact
I have been on a 1g interstellar journey; I answered your question in 1
day but you received it after 14 days. Hopefully after reading my answer
you will be able to calculate how fast I was traveling!
    Secondly I must apologize for having not read Kip Thorne's  book and
so i don't know the exact situation he was describing. What I shall do
then is to explain to you how to calculate the times and speeds for
yourself for a situation that, to me at least, seems quite general. I
shall explain to you the method behind the calculations and then give
you the equations. If you don't understand the explanation too well
don't worry. You can either contact me again or just apply the equations
without worrying about the theory.
    In the first quarter of this century Albert Einstein revolutionized
our notions of the space and time; in fact he changed there fundamental
meanings. He did this with the development of two theories; Special
Relativity (SR) and General Relativity (GR). SR describes how the laws
of physics behave in what are called "inertial frames of reference". A
fancy name for the XYZ and T coordinate systems of the collection of all
observers that are NOT accelerating relative to each other. For example
two rockets, piloted by Ben and Catherine, in space float past each
other at constant velocity (i.e. without there engines on) are inertial
observers and if they both carry a ruler and a clock to define a
coordinate system then they can be said to exist in inertial frames of
reference.
    The relative velocity between Ben and Catherine's rockets can take on
any value it likes so long as that velocity is less than the speed of
light and it never changes. If a third rocket were to come along,
piloted by Diane say, that had its motor burning brightly then the
relative velocity between Diane and her friends would be changing
constantly; i.e. she will be accelerating and as such she is NOT in an
inertial frame. SR deals ONLY with inertial frames. If Ben and Catherine
perform measurements of whatever, each other size etc., SR can tell us
how there results will compare. SR can not directly tell us what Diane
will observe.
    Despite Diane's impatience  we can still use SR to describe what Diane
will observe in her accelerating frame. We can do this by considering what
will be observed not by Diane herself but by an inertial observer
traveling at the same velocity as Diane at a certain time. For example
say at 1:30 pm (according to Ben's clock) Diane has accelerated to a
velocity of 0.5 times the speed of light (which we write as 0.5 x c or
simply 0.5c where c is the speed of light). SR can not tell us directly
what Diane will observe. If the relative velocity between Ben and
Catherine is also 0.5c then SR can tell us what Catherine will observe.
Since Catherine's velocity relative to Ben is constant (remember they
are inertial observers) then both Catherine and Diane will be traveling
at 0.5c relative to Ben at 1:30 pm; i.e. for an instant Catherine and
Diane will be identical. SR tells us what Catherine will observe in this
instant and so we can infer what Diane will observe in this instant. Of
course an instant later feisty Diane would have accelerated a little
more and Catherine and Diane will no longer agree. But all we have to do is
bring in another friend traveling slightly faster, Catherine's sister
perhaps,(but still with constant velocity relative to both Ben and
Catherine) who matches Diane's speed at say 1:31 pm. This approach is valid
based on an assumption. That assumption is that acceleration itself has no
effect on Diane's clock only her instantaneous velocity. This assumption
has been tested successfully experimentally and so I shall use it with
out further justification.
    If we do this repeatedly we can know what Diane will observe at each
instant by considering an inertial observer who has the same velocity as
Diane at that instant. The only snag is that we need a different
inertial observer for each instant and in order to accurately describe
what Diane will see we need a lot of them!
    The approach I have used is explained in "Discovering Relativity for
yourself"- Sam Lilley, Cambridge University Press, p213-266. Read this,
it gives an excellent account of the method.
    The method basically uses the approach of Ben, Catherine and Diane
except that Catherine has many many sisters. 
     Because we are always comparing Diane's clock to an equivalent
inertial observer we need to repeat a calculation over and over again; once
for each equivalent frame that we are comparing Diane's clock to or
equivalently once for each instant in Diane's path. The number of times
we have to repeat the calculation or rather the difference between each
equivalent frame is defined by how accurately we want the answer to be.
    The most accurate we can get is if we use an infinite number of
Catherines sisters(inertial observers) which differ from each other in only
an infinitesimal way. i.e. use the calculus (you wont find this step in the
reference).
    Even if you understood little of what's above  here are the
equations that can give you your answers.

Consider Ben and Diane and forget about Catherine and all her sisters
(They were used to calculate Dianes observations and so are sort of
behind the scenes

  -----\                   -----\
  |  B  \          ------  |  D  \ -----> acceleration = a
  |     /           ------ |     /
  -----/                   -----/

D starts with zero velocity relative to B and accelerates away with
acceleration a. After a while she reverse here motor to decelerate back
to zero velocity relative to B at a distance (according to B) of D_B. D
then continues to accelerate towards B and then reverses her motor again
to decelerate back to a velocity of zero relative to Ben back at the
starting position. So to summarize, Diane and Ben start off at the same
point. Diane accelerates away to a distance of D_B and then accelerates
back coming to a halt back at Ben's unchanged position (according to Ben).


Time taken according to Ben (in years), T_B

(1)                          T_B = (2/a)(k - 1/k)

Time taken according to Diane (in years),, T_D

(2)                          T_D = (4/a)ln(k)

Maximum distance reached  according to Ben (in light years),D_B

(3)                          D_B = (1/a)[(k + 1/k) -2]

In these equations a is the acceleration in g's i.e for a 1g
acceleration a = 1, for a 20g a = 20. in equation (2) "ln(k)" means take
the natural logarithm of the number k. Some people call this the log to
base e. It should be on your calculator. K in these equations relates to
the maximum velocity obtained by the following equations. k has an
important significance in SR, it is known as the Doppler Factor. You can
just consider it a way of reducing the number of terms in equations (1),
(2), (3).

(4)
        1 + (v/c)
k =   -------------
    sqrt(1 - (v/c)^2)

(5)
      k^2 - 1
v =   ------
      k^2 + 1

Where v is the maximum velocity reached and c is the speed of light. You
can calculate k by inserting the maximum velocity or by rearranging
equations (1), (2), or (3) depending on what you know about your
journey.
e.g. 1) For instance if you know that T_D is 10 years then you can work
out k by rearranging (2);


k = exp(aT_D/4)

where exp(aT_D/4) means raise e to the power of (aT_D/4); i.e its the
inverse of ln (should be written as e^x on your calculator and is
usually accessed by SHIFT + ln)

If T_D is 10 years and a = 1 then k = 12.18. Consequently equation (1)
gives the time according to Ben as;


                  T_D = (2/1)(12.18 - 1/12.18) = 24.2 years

So for a 1g trip that lasts 10 years according to Diane would have
lasted 24 years according to Ben.

e.g. 2) If we know the the maximum velocity is 0.75c and a = 20 (your
example) then we can calculate k from (4)

       1 + (0.75)
k =   -------------    =  2.65
    sqrt(1 - (0.75)^2)

and so we can calculate;

                  T_B = (2/20)( 2.65- 1/2.65) = 0.23 years

Time taken according to Diane (in years),, T_D

                  T_D = (4/20)ln(2.65) = 0.19 years

Maximum distance reached  according to Ben (in light years),D_B

                  D_B = (1/20)[(2.65 + 1/2.65) -2] = 0.05 light-years


Have fun!

Matt