Re: Androcles asks for Derivation of LT

From: Daryl McCullough (stevendaryl3016_at_yahoo.com)
Date: 02/14/05 

Date: 14 Feb 2005 15:18:46 -0800

Androcles says...

>"Daryl McCullough" <stevendaryl3016@yahoo.com> wrote in message
>news:cuqj2f0i6e@drn.newsguy.com...
>> Androcles says...
>>
>>>"Daryl McCullough" <stevendaryl3016@yahoo.com> wrote
>>
>>>As for deriving the Lorentz tranformations, here's a derivation, going
>>>back to Sam and Joe. Let's introduce two new characters, Sally and
>>>Jane.
>>>
>>>Sam and Joe are at rest relative to frame B (for Boys).
>>>
>>>Sally and Jane are at rest relative to frame G (for Girls).
>>>
>>>Let the speed of frame B as measured in frame G be v.
>>>
>>>Androcles:
>>>_______________________________ --> v
>>>_|_|_|_|_|_|_|B|_|_|_|_|_|_|_|_
>>>
>>>
>>>_______________________________
>>>_|_|_|_|_|_|_|G|_|_|_|_|_|_|_|_
>>>
>>>
>>>McCullough:
>>>Also, by symmetry let the speed of frame G as measured in frame B also
>>>be v.
>>>
>>>Androcles:
>>>_______________________________
>>>_|_|_|_|_|_|_|B|_|_|_|_|_|_|_|_
>>>
>>>
>>>_______________________________ --> v
>>>_|_|_|_|_|_|_|G|_|_|_|_|_|_|_|_
>>>
>>>
>>>v = 0
>>
>> Obviously, I should have been more explicit.
>
>Yes, it does help to be obviously explicit instead of obviously
>misleading.
>
>
>In frame G,
>> Sam and Joe are travelling at speed v to the right. In
>> frame B, Sally and Jane are travelling at speed v to the
>> *left*. So
>>
>> In frame G
>>
>> _______________________________ --> v
>> _|_|_|_|_|_|_|B|_|_|_|_|_|_|_|_
>>
>>
>> _______________________________
>> _|_|_|_|_|_|_|G|_|_|_|_|_|_|_|_
>>
>> In frame B
>>
>> _______________________________
>> _|_|_|_|_|_|_|B|_|_|_|_|_|_|_|_
>>
>>
>> <-- v _______________________________
>> _|_|_|_|_|_|_|G|_|_|_|_|_|_|_|_
>>
>
>So in the ground frame, then, the B frame is moving at v/2
>and the G frame is moving at -v/2. That IS good news. Perhaps
>we can now cease being concerned about A and B being points
>in space in some god-forsaken rest frame.
>
>> Let e(i,j) be the event at which rung number i of the G-ladder passes
>> rung number j of the B-ladder. Let x(i,j) be the location of this
>> event, in G-coordinates, and let t(i,j) be the time of this event,
>> in G-coordinates. Let x'(i,j) and t'(i,j) be the location and time
>> of this event in B-coordinates. Let's assume that we pick our origin
>> so that x(0,0) = t(0,0) = x'(0,0) = t'(0,0) = 0. Let's figure out
>> the coordinates of e(i,j) for arbitrary i and j.
>>
>> Obviously, rung i of the G ladder is always at location x=iL, as
>> measured in frame G, so we have
>>
>> 1. x(i,j) = iL
>>
>> To compute t(i,j), note that in the G-frame, at time t=0, rung i of
>> the G-ladder is at location x = i L and rung j of the B-ladder is
>> at location x = j l. Therefore, the distance between these rungs
>> is (iL - jl). Since the B-ladder is travelling at speed v, these
>> two rungs will pass each other at time
>>
>> 2. t(i,j) = (iL - jl)/v
>>
>> The computation for frame B is similar, except that in frame B,
>> it is the B-rung number j that is stationary, so we have
>>
>> 3. x'(i,j) = jL
>>
>> In frame B, rung i of the G-ladder at time t=0 is at location
>> x' = il, while the location of rung j of the B-ladder is always
>> x' = jL. So the distance between these rungs is (il - jL), and so
>> they will pass at time
>>
>> 4. t'(i,j) = (il - jL)/v
>>
>> Since x(i,j) = iL, and x'(i,j) = jL, we can rewrite t
>> (for fixed i and j) as follows:
>>
>> 5. t = (x - x' l/L)/v (from equation 2)
>>
>> which gives us x' in terms of x and t:
>>
>> 6. x' = (x - vt)L/l
>>
>> Similarly, equation 4 can be rewritten (for fixed i and j) as
>>
>> 7. t' = (x l/L - x')/v
>>
>> Substituting form x' from equation 6 gives us
>>
>> 8. t' = (x l/L - (x-vt) L/l)/v
>> = L/l t + x/v (l/L - L/l)
>>
>> Let's let g be the ratio L/l. Then our transformation equations are:
>>
>> 9. x' = g (x-vt)
>> 10. t' = g t + x/v (1/g - g)
>> = g (t + x/v (1/g^2 - 1))
>>
>> Notice that so far, we haven't made any physical assumptions about
>> the speed of light or the length of moving objects, or the slowing
>> of moving clocks. We've only invoked the relativity principle, that
>> Sally and Jane have just as much right to consider themselves at rest
>> as Sam and Joe.
>> So our equations so far work for Galilean relativity,
>> as well as Einstein's relativity.
>
>Err... well, g = L/l, right? and l = L, right? so g = 1, right?

Not necessarily. Here were the definitions:

  L is the distance between rungs of the G-ladder
  as measured in the G-frame.

  l is the distance between rungs
  of the B-ladder as measured in the G-frame.

  v is the speed of the B-ladder as measured in the G-frame.

  L' is the distance between rungs of the G-ladder
  as measured in the B-frame.

  l' is the distance between rungs
  of the B-ladder as measured in the B-frame.

  u' is the speed of the G-ladder as measured in the B-frame.

I said that by symmetry,

   L = l'
   l = L'
   v = u'

I did not assume that L = l.
    
>> We haven't said anything
>> at all about the factor g = L/l.
>
>Oh, I thought we did. Isn't g = 1, then?

No.

>> To get Einstein's relativity, we now impose another assumption:
>> the speed of light is c in all inertial reference frames, regardless
>> of the motion of the source.
>
>Now I have to stop you dead in your tracks right there.
>Why one Earth would you expect me (or any other sane person)
>to make such an absurd assumption?

Because Maxwell's equations predict that light has speed c. If
Maxwell's equations are valid in every rest frame, then it follows
that light has speed c in every rest frame.

The principle of relativity (there is no standard for rest) plus
the validity of Maxwell's equations in vacuum implies that light
has the same speed in every reference frame. 

--
Daryl McCullough
Ithaca, NY

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