No, this scenario does not disprove Einstein's theory of relativity. According to special relativity, the speed of light in a vacuum is an absolute constant, denoted as 'c,' and it is the same for all observers regardless of their relative motion. The theory of relativity does not permit anything to travel faster than the speed of light.
In the scenario you described, where two flashlights are shot in opposite directions and the spots they illuminate are 2 light-years apart, it may seem like the photons from each flashlight need to travel at twice the speed of light relative to each other to achieve that distance in one year. However, this is not the case.
In special relativity, distances and times can become distorted for observers in relative motion. As an observer approaches the speed of light, space and time undergo a phenomenon known as time dilation and length contraction. These effects ensure that the speed of light remains constant for all observers, regardless of their relative motion.
When considering the scenario of the two flashlights, an observer at rest relative to the starting point of the flashlights would measure the spots of light to be 2 light-years apart after one year. However, an observer moving at a high velocity relative to the starting point, but still slower than the speed of light, would measure the distance between the spots to be shorter due to length contraction. From their perspective, the spots would not be 2 light-years apart after one year. This apparent discrepancy in distance is a result of the relativity of simultaneity and the distortion of space and time at relativistic speeds.
It's important to note that no information or causality is violated in this scenario. While the spots of light may appear to be separated by 2 light-years for one observer, another observer moving relative to them would measure a different distance due to the relativistic effects. Nevertheless, neither observer would witness any object or information traveling faster than the speed of light. The theory of relativity remains consistent in this situation.