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Most diagrams regarding relativistic doppler effect always show in terms of an outside observer that is looking at both the light and the moving source. In this context, yes, the source moves towards the observer too so the observer does not see the source at the center of a sphere. The wavefront is still a sphere of course, just one with concentric spheres with different centers.

Now, let's remove this other obsever and just consider the source. No matter how fast the source moves, he always sees light moving at $c$. Not only that, but if we say that there is nothing else except for the source and the light that is being emitted, the source should just see himself as completely stationary. Therefore the observation should simply be that they are in the center of a sphere of light that has different frequencies. Is this correct?

EDIT: Now that I think about it, if we remove the other observer and anything else to use as a reference, it does not make sense to think of the source as having a velocity. So the situation is exactly the same as if the source was stationary - is this correct?

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In SR, if you are at a point where light is emitted, it will always expand around you as a sphere, assuming you are moving inertially. So if two people, A and B, pass each other at some speed, and light is emitted from the point where they meet, each individual will continue to see the same light expanding as a sphere centred on their position. A will think B is not at the centre of the sphere, while B will think that A is not at the centre of the sphere. The cause of the effect is the relativity of simultaneity, which causes each observer to be viewing different points on the expanding wavefront at different times. If the light was originally emitted at the observer's location, the observer will not see any Doppler effect.

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"So the situation is exactly the same as if the source was stationary - is this correct?" If this is your question, answer is yes, but it has nothing to do with the Doppler effect until you Lorentz transform to the observer's rest frame.

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