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Irrational
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I'm doing a night course in General Relativity and we're currently finishing off Special Relativity... We're working mainly off of D'Inverno.
We've just covered the relativistic doppler effect and some associated things like aberration.
When it came to talking about the transverse doppler effect,
[tex]\frac{\lambda}{\lambda_{0}}= \frac{1}{(1-v^{2}/c^{2})^{\frac{1}{2}}}[/tex]
we were given a handout from C.J Davidson's letter "The Theory of the Transverse Doppler Effect" and the lecturer ran through the details of it.
The following bit I am having trouble working through though.
From the conservation of energy:
[tex]\frac{E_{1}}{(1 - \beta_{1}^{2})^{\frac{1}{2}}} = h\nu + \frac{E_{2}}{(1 - \beta_{2}^{2})^{\frac{1}{2}}}[/tex]
From the conservation of momentum...
- along the X axis, we have
[tex]\frac{E_{1}\beta_{1}/c}{(1 - \beta_{1}^{2})^{\frac{1}{2}}} = \frac{E_{2}\beta_{2}/c}{(1 - \beta_{2}^{2})^{\frac{1}{2}}}cos{\gamma} + \frac{h\nu}{c}cos{\alpha}[/tex]
- along the Y axis, we have
[tex]0 = \frac{E_{2}\beta_{2}/c}{(1 - \beta_{2}^{2})^{\frac{1}{2}}}sin{\gamma} - \frac{h\nu}{c}sin{\alpha}[/tex]
where [tex]\beta_{1} = v_{1}/c[/tex], [tex]\beta_{2} = v_{2}/c[/tex]From here, we eliminate [tex]\beta_{2}[/tex] and [tex]\theta[/tex] to get
[tex]\nu = \nu_{0}\frac{(1 - \beta_{1}^{2})^\frac{1}{2}}{(1 - \beta_{1} cos{\alpha})}[/tex]
where:
[tex] \nu_{0} = \frac{(E_{1} + E_{2})}{2E_{1}}\frac{(E_{1} - E_{2})}{h}[/tex]
and after this, it's plain sailing...
My problem is eliminating [tex]\beta_{2}[/tex] and [tex]\theta[/tex]. I just end up in a mess everytime I try to do it.
It's not homework but I wasn't sure whether to post it here or there. It's more filling in the missing gap. I know this is probably basic algebra and/or trigonometry but any help would be appreciated.
Also, for some reason my LaTex tags are bit weird in preview mode so let me know if something is wrong.
Thanks
Dave
We've just covered the relativistic doppler effect and some associated things like aberration.
When it came to talking about the transverse doppler effect,
[tex]\frac{\lambda}{\lambda_{0}}= \frac{1}{(1-v^{2}/c^{2})^{\frac{1}{2}}}[/tex]
we were given a handout from C.J Davidson's letter "The Theory of the Transverse Doppler Effect" and the lecturer ran through the details of it.
The following bit I am having trouble working through though.
From the conservation of energy:
[tex]\frac{E_{1}}{(1 - \beta_{1}^{2})^{\frac{1}{2}}} = h\nu + \frac{E_{2}}{(1 - \beta_{2}^{2})^{\frac{1}{2}}}[/tex]
From the conservation of momentum...
- along the X axis, we have
[tex]\frac{E_{1}\beta_{1}/c}{(1 - \beta_{1}^{2})^{\frac{1}{2}}} = \frac{E_{2}\beta_{2}/c}{(1 - \beta_{2}^{2})^{\frac{1}{2}}}cos{\gamma} + \frac{h\nu}{c}cos{\alpha}[/tex]
- along the Y axis, we have
[tex]0 = \frac{E_{2}\beta_{2}/c}{(1 - \beta_{2}^{2})^{\frac{1}{2}}}sin{\gamma} - \frac{h\nu}{c}sin{\alpha}[/tex]
where [tex]\beta_{1} = v_{1}/c[/tex], [tex]\beta_{2} = v_{2}/c[/tex]From here, we eliminate [tex]\beta_{2}[/tex] and [tex]\theta[/tex] to get
[tex]\nu = \nu_{0}\frac{(1 - \beta_{1}^{2})^\frac{1}{2}}{(1 - \beta_{1} cos{\alpha})}[/tex]
where:
[tex] \nu_{0} = \frac{(E_{1} + E_{2})}{2E_{1}}\frac{(E_{1} - E_{2})}{h}[/tex]
and after this, it's plain sailing...
My problem is eliminating [tex]\beta_{2}[/tex] and [tex]\theta[/tex]. I just end up in a mess everytime I try to do it.
It's not homework but I wasn't sure whether to post it here or there. It's more filling in the missing gap. I know this is probably basic algebra and/or trigonometry but any help would be appreciated.
Also, for some reason my LaTex tags are bit weird in preview mode so let me know if something is wrong.
Thanks
Dave
Last edited: