The scenario you've described, where one entangled particle travels on a spaceship at relativistic speeds while the other remains on Earth, introduces several complex factors involving special relativity and quantum entanglement. The interplay between these two domains is an area of ongoing research and theoretical exploration.
According to our current understanding, relativistic effects such as time dilation can indeed introduce complications for entangled particles that experience significant differences in proper time due to their relative motion. Time dilation occurs when an object moves at speeds approaching the speed of light, causing time to pass more slowly for the moving object compared to a stationary observer.
In this situation, the entangled particles would experience different rates of time due to the relative motion between them. As a result, their perceived "timelines" would differ. However, it is important to note that time dilation alone does not break the entanglement between the particles.
Quantum entanglement is a non-local correlation that persists even across large distances and can transcend differences in time and space. As long as the entangled particles remain isolated from external influences, their entangled state should be maintained despite their temporal separation.
However, the situation becomes complex when we consider the measurement or observation of the entangled particles. The act of measuring or observing one of the entangled particles collapses its quantum state and, according to the principles of quantum mechanics, instantaneously affects the state of the other entangled particle, regardless of their temporal separation.
In the scenario you described, if a measurement is made on either particle, it would break the entanglement between them, and the observed results would depend on the specific measurements made in their respective timelines. The entangled state cannot be used to communicate information faster than the speed of light, and the results of measurements would not be correlated between the two timelines.
It is worth noting that the combination of relativistic effects and quantum entanglement is a complex and active area of research, and there are ongoing theoretical investigations into understanding the precise nature of entanglement in relativistic scenarios.