I understand that if I am standing on earth and a space ship takes off and travels at .6c, then I perceive the space traveler receding at .6c relative to me, and the space traveler perceive me as receding at .6c relative to him. If another traveler takes off in the 180-degree opposite direction, then likewise I perceive the other space traveler receding at .6c relative to me, and the other space traveler perceive me as receding at .6c relative to him.
So why don’t they perceive each other as traveling faster than c, the speed of light?
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Because you’re resting this on an incorrect assumption; that velocities are simply additive. In reality, they’re actually not.
For an example; say you have two people, one on a moving train, and one outside the moving train, standing on the ground. At the moment the person on the train passes the person on the ground, the person on the train fires a gun (in the same direction the train is traveling) at a certain speed relative to themselves.
One would think that the speed of the bullet relative to the outside observer is just the speed of the train plus the speed of the bullet on the train…but it actually *isn’t*. It’s slightly less. The difference is imperceptibly (and honestly near-immeasurably) small at lower speeds, but as you get to higher and higher speeds the deviation from just adding the speeds together will get greater and greater.
In the most extreme example (the bullet being a photon traveling at the speed of light), the bullet is observed as traveling at ***precisely*** c to both observers, despite their different velocities relative to one another.
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