Let's say you have rain hitting you evenly on all sides (not very realistic, I know). If you were to move forwards at a constant speed, there would be more droplets of rain hitting you per second on your front, since the relative speed of droplets in front of you has increased.
Now, if you were to have photons 'hitting' you evenly on all sides and you move forwards at a constant speed, surely the relative speed of the photons in front of you WON'T increase (since light travels at the same speed to all observers), and therefore photons will still be hitting you evenly on all sides (per second).
However the searchlight effect seems to disagree with my conclusion. What have I done wrong?