According to the principles of special relativity, an object with mass cannot travel at the speed of light. As an object with mass approaches the speed of light, its energy increases infinitely, making it impossible to reach or exceed the speed of light.
So, in the scenario you described, where two rockets are a light-year apart and traveling exactly toward each other at speeds approaching the speed of light, we need to consider how their relative velocities are calculated.
Let's assume that the rockets are traveling toward each other at a constant velocity of 0.9 times the speed of light (0.9c). Relative velocities in special relativity are not calculated by simple addition or subtraction but rather through a formula known as the relativistic velocity addition formula:
Relative velocity (v) = (v1 + v2) / (1 + (v1 * v2 / c^2))
where v1 and v2 are the velocities of the two rockets, and c is the speed of light.
Given that v1 = v2 = 0.9c, we can plug these values into the formula:
v = (0.9c + 0.9c) / (1 + (0.9c * 0.9c / c^2)) v = (1.8c) / (1 + 0.81) v = (1.8c) / 1.81 v ≈ 0.9945c
So, the relative velocity between the two rockets would be approximately 0.9945 times the speed of light.
Now, let's consider how fast they "see" each other. Since light travels at the speed of light in a vacuum, the light emitted from one rocket takes one year to reach the other rocket, which is a light-year away. However, due to the relative velocity between the rockets, the light is Doppler shifted, and its frequency changes. The Doppler effect causes the light to appear more blue-shifted (higher frequency) than if the rockets were stationary relative to each other.
The precise calculation of the frequency shift depends on the exact geometry and relative motion, but the relative velocity of 0.9945c would result in a significant blue-shift of the light each rocket "sees" from the other. This effect is a consequence of the relativistic Doppler effect, which is different from the classical Doppler effect experienced at everyday speeds.