Relativistic Doppler Shift Theory Question?

[MissSilvy]

Homework Statement

What sort of frequency shift does a rocket ship traveling along the x-axis as some speed close to c observe when it passes directly over a light source emitting photon on the y-axis (so when the angle between them is 90 degrees). At this angle, the rocket is neither approaching the source nor receding from it... is the photon redshifted, blueshifted, or unshifted in this case?

Homework Equations

f'/f = 1-\betacos(\theta)/\sqrt{1-\beta}²

The Attempt at a Solution

Intuitively, I guess that there would be no shift, since the photon is not moving on the same axis as the spaceship but the equation gives f'/f= 5/3 when I assume beta is 0.8, so the math suggests a redshift. I have a nagging feeling that this has something to do with the time-dilation experienced by the spaceship but I'm not exactly certain why. If anyone could explain why this phenomenon occurs, I'd be very grateful.

[This is a freshman honors mechanics course, so we've done Lorentz transformations, time dilation, and other stuff at a relatively low level.]

 

[Andrew Mason]

Homework Statement

What sort of frequency shift does a rocket ship traveling along the x-axis as some speed close to c observe when it passes directly over a light source emitting photon on the y-axis (so when the angle between them is 90 degrees). At this angle, the rocket is neither approaching the source nor receding from it... is the photon redshifted, blueshifted, or unshifted in this case?

Homework Equations

f'/f = 1-\betacos(\theta)/\sqrt{1-\beta}²

The Attempt at a Solution

Intuitively, I guess that there would be no shift, since the photon is not moving on the same axis as the spaceship but the equation gives f'/f= 5/3 when I assume beta is 0.8, so the math suggests a redshift. I have a nagging feeling that this has something to do with the time-dilation experienced by the spaceship but I'm not exactly certain why. If anyone could explain why this phenomenon occurs, I'd be very grateful.

[This is a freshman honors mechanics course, so we've done Lorentz transformations, time dilation, and other stuff at a relatively low level.]

If you understand time dilation then you will understand that relative to the source of the light, the moving observer measures time more slowly. So if there is a time interval T = 1/f between wavelengths of light at the source, the moving observer will measure that time as T' = \gamma T. Consequently, the frequency f' = 1/\gamma T measured by the moving observer will be less than f = 1/T, so it is red shifted.

AM

 

[MissSilvy]

Oh that makes perfect sense! I forgot that frequency depends on time. Thank you very much!

 

[Delphi51]

Most interesting!
My thinking on the problem was that looking at light coming from ahead of the ship, you would see it blue shifted. Looking behind, light from back home would appear red shifted. So wouldn't it make sense that light coming from the sides would be half way between - not shifted either way?

Relativity never seems to make sense to me, so not surprising if it doesn't this time.