Deriving Doppler Shift Using Spacetime Diagram

[Mindscrape]

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:

$$ct=cTcos(\theta)$$
and so we know that $$sin(\theta) = \frac{v}{c}$$ and $$cos(\theta) = \sqrt{1 - \frac{v^2}{c^2}}$$
so if
$$t=Tcos(\theta)$$
and using kinematics we know the distace between pulses is
$$x=vTcos(\theta)$$
and the intervals will then be
$$t = T + \frac{x}{c}$$
then with some algebra
$$t=T\gamma(1 + \frac{v}{c} \sqrt{1- \frac{v^2}{c^2}})$$
When we take the ratio between the two time, which should be the doppler shift then we get
$$\frac{t}{T} = \frac{1 + \frac{v}{c} \sqrt{1 - \frac{v^2}{c^2}}}{ \sqrt{1- \frac{v^2}{c^2}}}$$

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

 

[anathema]

Hello,

Thank you for sharing your approach to deriving the doppler shift using a spacetime diagram. Your method seems to be on the right track, but I think there may be a slight error in your calculations.

Firstly, in your equation ct=cTcos(\theta), I believe the correct form should be ct=cTcos(\theta) - x, as the light pulses are traveling at an angle and not directly along the x-axis. This means that the time interval between the pulses should be t=Tcos(\theta) - \frac{x}{c}, not t=Tcos(\theta).

Next, when you use kinematics to find the distance between the pulses, the correct equation should be x=v(Tcos(\theta)-\frac{x}{c}), as the pulses are moving at an angle and not directly along the x-axis. This means that the time interval between the pulses should be t=Tcos(\theta)-\frac{x}{c}, not t=Tcos(\theta).

Finally, when you take the ratio between the two time intervals, the correct equation should be \frac{t}{T}=\frac{1+\frac{v}{c}\sqrt{1-\frac{v^2}{c^2}}}{\sqrt{1-\frac{v^2}{c^2}}}, as you have correctly written. However, this can be simplified to \frac{t}{T}=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}, which is the correct form for the doppler shift.

I hope this helps to clarify any confusion and leads you to the correct solution. Keep up the good work!

 

[kyle8921]

I appreciate your attempt to derive the Doppler shift using a spacetime diagram. However, I believe there may be some errors in your calculations.

Firstly, your equation ct=cTcos(\theta) assumes that the speed of light is constant, which is not the case in the presence of a moving observer. The correct equation is ct' = cTcos(\theta'), where t' and \theta' are measured by the moving observer.

Secondly, in your calculation of the distance between pulses, you have used the velocity v instead of the relative velocity between the observer and the source of light. This could be a source of error in your final result.

Additionally, your equation for the interval between pulses, t = T + \frac{x}{c}, is not accurate for a moving observer. The correct equation is t = T\gamma(1 + \frac{v}{c} \sqrt{1- \frac{v^2}{c^2}}), where \gamma is the Lorentz factor.

I would suggest revisiting your calculations and making sure you are using the correct equations for a moving observer. Also, be mindful of the assumptions you are making and make sure they are valid in the context of the problem.

Overall, the concept of using a spacetime diagram to derive the Doppler shift is valid and can provide valuable insights. However, it is important to use the correct equations and be aware of potential sources of error. I hope this helps in your understanding and further exploration of this topic.