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Mindscrape

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Alright, I'm trying to derive the dopper shift using a spacetime diagram (see attached).

If we model light pulses then we can derive the distance between the pulses in time, and hence the doppler shift... right?

So, if we make some light pulses along some sort of time event in the x, cT frame and extend their perpindiculars then we can make some relations. Here is what I have done:

[tex]ct=cTcos(\theta)[/tex]

and so we know that [tex]sin(\theta) = \frac{v}{c} [/tex] and [tex]cos(\theta) = \sqrt{1 - \frac{v^2}{c^2}}[/tex]

so if

[tex]t=Tcos(\theta)[/tex]

and using kinematics we know the distace between pulses is

[tex]x=vTcos(\theta)[/tex]

and the intervals will then be

[tex]t = T + \frac{x}{c}[/tex]

then with some algebra

[tex]t=T\gamma(1 + \frac{v}{c} \sqrt{1- \frac{v^2}{c^2}})[/tex]

When we take the ratio between the two time, which should be the doppler shift then we get

[tex]\frac{t}{T} = \frac{1 + \frac{v}{c} \sqrt{1 - \frac{v^2}{c^2}}}{ \sqrt{1- \frac{v^2}{c^2}}}[/tex]

Which is really close, but off somehow. Anyone know what went wrong?

If we model light pulses then we can derive the distance between the pulses in time, and hence the doppler shift... right?

So, if we make some light pulses along some sort of time event in the x, cT frame and extend their perpindiculars then we can make some relations. Here is what I have done:

[tex]ct=cTcos(\theta)[/tex]

and so we know that [tex]sin(\theta) = \frac{v}{c} [/tex] and [tex]cos(\theta) = \sqrt{1 - \frac{v^2}{c^2}}[/tex]

so if

[tex]t=Tcos(\theta)[/tex]

and using kinematics we know the distace between pulses is

[tex]x=vTcos(\theta)[/tex]

and the intervals will then be

[tex]t = T + \frac{x}{c}[/tex]

then with some algebra

[tex]t=T\gamma(1 + \frac{v}{c} \sqrt{1- \frac{v^2}{c^2}})[/tex]

When we take the ratio between the two time, which should be the doppler shift then we get

[tex]\frac{t}{T} = \frac{1 + \frac{v}{c} \sqrt{1 - \frac{v^2}{c^2}}}{ \sqrt{1- \frac{v^2}{c^2}}}[/tex]

Which is really close, but off somehow. Anyone know what went wrong?

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