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_rpd comments on Estimating the probability of human extinction - Less Wrong
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Naively, the required condition is v + dH > c, where v is the velocity of the spaceship, d is the distance from the threat and H is Hubble's constant.
However, when discussing distances on the order of billions of light years and velocities near the speed of light, the complications are many, not to mention an area of current research. For a more sophisticated treatment see user Pulsar's answer to this question ...
http://physics.stackexchange.com/questions/60519/can-space-expand-with-unlimited-speed/
... in particular the graph Pulsar made for the answer ...
http://i.stack.imgur.com/Uzjtg.png
... and/or the Davis and Lineweaver paper [PDF] referenced in the answer.
Wow. It looks like light from James' spaceship can indeed reach us, even if light from us cannot reach the spaceship.
Yes, until the distance exceeds the Hubble distance of the time, then the light from the spaceship will red shift out of existence as it crosses the event horizon. Wiki says that in around 2 trillion years, this will be true for light from all galaxies outside the local supercluster.