Consider an object traveling across the universe and consider two observers billions of light years away from each other at rest with respect to their local universes. The object will pass by the first observer at time $t=0$ and the object will pass by the second observer at time $t=T$. Would the expansion of space cause any difference in the relative velocity between the object and the first observer at t=0 and the relative velocity between the object and the second observer at t=T?
General Relativity – Is Relative Velocity Affected by the Expansion of Space?
cosmologygeneral-relativityobserversspace-expansion
Best Answer
Yes, the object would move slower with respect to the second observer. The more general way of framing this is to define the object's peculiar velocity as its velocity with respect to its local universe, wherever it is. Peculiar velocities decay proportionally to $1/a$, where $a$ is the expansion factor.
This is essentially the cosmological redshift for massive particles. It can be understood as a self-sorting effect. All of the observers "at rest with respect to their local universe" are moving away from you, with more distant observers receding more quickly. If you start to move in any direction, you will gradually overtake slower-moving observers in that direction, asymptotically approaching the observer moving at the same velocity as you. So your velocity with respect to the instantaneously local observer decreases over time, asymptotically approaching zero.