Why does the traveling twin age less in the twin paradox?

[PeroK]

It seems the misconception is people get the idea that the time dilation effect and Doppler effect are competing with each other for approaching clocks. But the equation says what it says fairly unambiguously.

To be precise, that's the relativistic Doppler effect, which involves an element of time dilation. There is, for example, the transverse Doppler effect, which is essentially all due to time dilation.

In general, whether there is a redshift or blueshift also depends on the angle between the source and receiver. When the angle is zero there is always a blueshift (increased frequency).

 

[Ibix]

It seems the misconception is people get the idea that the time dilation effect and Doppler effect are competing with each other for approaching clocks. But the equation says what it says fairly unambiguously.

You can see the relativistic Doppler effect as the product of the Lorentz gamma factor and the naive distance-change-only Doppler factor of non-relativistic physics. In that sense, yes, the naive Doppler effect dominates the relativistic time dilation effect to get a net speed-up for sources approaching you.

But I'd say it's more sensible to call relativistic Doppler "Doppler" and refer to the distance-change-only effect by another term (distance-change-only, for example). Relativistic Doppler is the measurable thing - the other is a frame-dependent effect, so less important.