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Old Sep 18th 2017, 01:53 AM   #1
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Derivation of lorentz factor

If you look at attachment I have a minkowski diagram showing a world line and corresponding time line (shown in orange). I decided to try and calculate length of line segment BC assuming that the ratio of AB to BC will give me gamma (the lorentz factor). I figure segment $\displaystyle \overline{AB} = x_b - vct_b$. I tried calculating CB by first finding the eqn of the line going through both points as $\displaystyle c(t - t_b) = v(x - x_b)$. Solving this simultaneously with ct=x/v.

This yields a formula for point C:

$\displaystyle t= \frac{- c t_{b} + v x_{b}}{c \left(v^{2} - 1\right)}, x=\quad \frac{v}{v^{2} - 1} \left(- c t_{b} + v x_{b}\right)$

Now when I try to compute the length of the segment BC I get...

$\displaystyle \sqrt{\frac{\left(v^{2} + 1\right) \left(c t_{b} v - x_{b}\right)^{2}}{\left(v - 1\right)^{2} \left(v + 1\right)^{2}}}$

At this point I get a bit lost as I can't see how this reduces to $\displaystyle \sqrt{1-v^2}$ or have I missed something?
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Derivation of lorentz factor-simultaneity3-1.png  
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Old Sep 18th 2017, 02:35 PM   #2
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Originally Posted by kiwiheretic View Post
...or have I missed something?
You made a lot of assumptions which are invalid. Please walk me through your thinkin process in the steps that led you to that last expression and I'll explain why. E.g.

This statement - I decided to try and calculate length of line segment BC assuming that the ratio of AB to BC will give me gamma (the lorentz factor). is not true and this one

I figure segment $\displaystyle \overline{AB} = x_b - vct_b$.

is also not true. Why did you think they were true in the first place?

Note: In spacetime diagrams the notation is (x, ct), not (ct, x). I.e. when (a, b) marks a point in a diagram a is always the abscissa and and b is always the ordinate.

If you're not familiar with those terms then see
https://en.wikipedia.org/wiki/Abscissa_and_ordinate
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Old Sep 18th 2017, 03:17 PM   #3
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Complete working is on https://gist.github.com/kiwiheretic/...b072379454967c (for some reason the image doesn't show on there but its the same as the attachment here).

My thinking is essentially using the lines ct=vx and ct=x/v as new basis vectors (as in linear algebra). That's the reason why the segment BC is parallel to ct=vx. If this is not so then how exactly does space transform for an observer travelling ct=v/x? Are you implying this does not happen as a linear transformation of space?
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Old Sep 18th 2017, 03:28 PM   #4
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Originally Posted by Pmb View Post
I figure segment $\displaystyle \overline{AB} = x_b - vct_b$.

is also not true. Why did you think they were true in the first place?
This line is horizontal so its just the difference between the x values of points A and B. This is for the stationary observer.

I was doing this late last night so not sure now why I thought BC was a length "contraction". Yeah I can see now something is flawed, just not sure still how to fix it.

Also it was because I was reading Peter Collier's book on relativity where he implied you can read off an event by drawing parallels to the travellers axis.

See attachment
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Derivation of lorentz factor-collier1.png  
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Last edited by kiwiheretic; Sep 18th 2017 at 04:13 PM.
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Old Sep 18th 2017, 04:58 PM   #5
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Originally Posted by kiwiheretic View Post
This line is horizontal so its just the difference between the x values of points A and B.
I know why x_b is the rightmost value of x, Why do believe that the leftmost value of x is vct_b?

In the page you linked to all you did is make statements of what you're doing with nearly no justification for them.

Frankly I give up at this point since it makes no sense whatsoever. You can't merely post a series of statements of what you're doing and make various claims. You have to explain why you're doing it and prove the claims you make before you can use them.

I'm sure you've seen this derivation before it SR texts. Take a close look at how arguments are made: Lorentz Contraction

Note: In SR one doesn't "derive" the Lorentz factor. That factor merely appears in a derivation of Lorentz contraction, time dilation and Lorentz transformations. Its merely a defined quantity, i.e. given a name.

I assume you were trying to use some sort of length contraction derivation.

Please note that I'm dropping out of this, not because I'm impatient, but because I'm neither in the mood or have much concentration nowadays. Too much pain and misery in my life now to concentrate very well.
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