What is the relativistic doppler shift in a multi-dimensional scenario?

[Pengwuino]

I have a problem here. I stole me a grad student and he didn't really know what to do either. Without further adoo (or whatever), i give you... the problem!

There are 3 trucks transmitting on the same frequency. #2 is stationary. #1 moves to the left at velocity v. #3 moves up at velocity v as well. I need to determine the relativistic doppler shift of the signal signal between #3 and #1.

We tried to determine a function for the velocity vector seperating #3 and #2 but that got ugly fast and we couldn't do anything there. Any suggestions as to what to do? Mind you, I'm suppose to know just a bit over intro-series modern physics and the 3 semester calculus series.

 

[Galileo]

Let me see if I understand the question correctly. Truck #2 is attached to our inertial frame. #1 goes to the left (say -x direction) with speed v and #3 is going in the positive x-direction with speed v? Then you are asked to determine the doppler shift of a signal coming from #3 as seen from truck #1?
In that case you need to find the speed of #3 wrt #1 as seen from the frame of truck #1.

 

[Pengwuino]

No, truck #3 is going in the positive y direction with speed v.

It's a little weird because the book has said absolutely nothing about 2-dimensional relativistic transformations like this

 

[Galileo]

Ah, totally read past the 'up' word there.

Well, let's just take #1 to go in the positive x-direction then and assume the origins of the three frames coincided at t=0 (standard configuration).
The coordinates of the motion of #3 is described by x=z=0, y=vt.
Now you can just do a Lorentz-transformation to find what these coordinates are in the frame of truck #1.

 

[Pengwuino]

Well we still haven't been taught how to do anything more then 1-dimensional LT's so i don't know how to do that...

 

[Gokul43201]

Can you find the relative velocity of #3 w.r.t #1 ?

 

[Pengwuino]

Not with what we've learned since both are moving perpendicular and we've only studied reference frames moving in the x direction from a stationary frame (or something also moving in the x-direction)

 

[Galileo]

Ok, well let's take #2 to be in reference frame S and #1 in frame S'.
The Lorentz transformation take the form:
x'=\gamma(x-vt')
y'=y
z'=z
t'=\gamma(t-\frac{vx}{c^2})

In other words. Nothing happens to the coordinates perpendicular to the direction of motion. This and previous post are all you need to solve the problem.

 

[George Jones]

This and previous post are all you need to solve the problem.

This gives a good start, but there is more to this question than might be seen at first glance. For example,

\sqrt{\frac{1 - V}{1+V}},

where V is the relative speed between #3 and #1 gives the wrong answer for (frequencies) because this formula is correct only when the spatial direction of signal propagation is along the direction of relative motion.

A generalization of this formula must be derived from scratch.

Regards,
George

 

[Galileo]

Unless the velocity is in the radial direction, as it is in this case.
I did assume that at t=0 the trucks were all at the origin.

 

[George Jones]

Unless the velocity is in the radial direction, as it is in this case. I did assume that at t=0 the trucks were all at the origin.

Right - if the worllines of #1 and #3 intersect at any event, then the spatial direction of signal propagation is along the direction of relative motion, and the standard longitudinal formula works.

I was just picking a nit that is a bit of a pet peeve of mine.

Standard introductory treatments of special relativity use spactime diagrams that have one time dimension and one spatial dimension. On such diagrams, the worldlines of any two inertial observers intersect. Often, students never see a treatment that of special relativity that goes beyond this, and they are left with the impression that the worldlines of any 2 inertial observers intesect, but this is not the case.

When 2 or 3 spatial dimensions are considered, it is more likely the case that the worldlines of two inertial observers do not intersect. In this problem, which does have more than one spatial dimension, it must be part of the given that worldlines of #1 and #3 intersect, or else the longitudinal Doppler formula doesn't apply.

As you say, it seems reasonable to assume the intersection.

Regards,
George