Relativistic velocity and frequecny effects

[SonOfOle]

Homework Statement

Consider a blob of luminous matter moving along a direction $$\hat{x}$$ at an angle $$\theta$$ to your line of sight at a relativistic speed v.

(a) According to you, the stationary observer at a distance d, what is the apparent transverse velocity of the blob? Hint: you must take into account light travel time effects.

(b) If the blob emits light monochromatically at frequency $$\omega_0$$ in its rest frame, what frequency will you measure in the lab frame?

Homework Equations

The Attempt at a Solution

The main difficulty I'm having is dealing with the blob moving off axis from the line of sight of the observer (also, I'm a bit confused on the wording of the problem). I know the traditional relativistic motion and frequency effects, but they assume motion directly on axis with the observer. Anyone have some insights to help me in solving this?

 

[Hootenanny]

Homework Statement

Consider a blob of luminous matter moving along a direction $$\hat{x}$$ at an angle $$\theta$$ to your line of sight at a relativistic speed v.

(a) According to you, the stationary observer at a distance d, what is the apparent transverse velocity of the blob? Hint: you must take into account light travel time effects.

(b) If the blob emits light monochromatically at frequency $$\omega_0$$ in its rest frame, what frequency will you measure in the lab frame?

Homework Equations

The Attempt at a Solution

The main difficulty I'm having is dealing with the blob moving off axis from the line of sight of the observer (also, I'm a bit confused on the wording of the problem). I know the traditional relativistic motion and frequency effects, but they assume motion directly on axis with the observer. Anyone have some insights to help me in solving this?

Try resolving the velocity into two orthogonal components, i.e. motion towards the observer and motion directly upwards in the diagram. You should then be able to apply the appropriate transformations to these two components separately, as if the blob was traveling rectilinearly.

 

[dynamicsolo]

For part (b), keep in mind that there is both a radial relativistic Doppler shift (relativistic modification of the classical effect) and a transverse Doppler shift (which is a non-classical effect).