Each rain drop is a little prism that refracts (bends) light. Each wavelength of light bends by a different amount, which creates a visible color spectrum ordered by wavelength.
As the rain falls, the apparent angle between it, the sun, and *you* changes, meaning that the portion of its refracted light spectrum that reaches your eye changes. At first you see the red portion. Then as the raindrop descends you see orange, yellow, green, etc.
The colors it emits don’t change. Only which portion of this spectrum that reaches your eye changes.
From afar, we wouldn’t notice such a small detail. But we certainly do notice when a whole field of billions of raindrops all refract sunlight from roughly that direction.
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