In special relativity, what is "now" for people who are very far apart? In The Fabric of the Cosmos page 136, it says that if I and another man are sitting still 10 billion light years apart (and assuming away any movement of the planets, or space expansion, etc) then we both have the same view of "now".  If he starts moving away from me at 10mph, then the events on earth that belong on his new now list are events that happened 150 years ago, according to me. If he starts moving towards me at 10mph, then the events on earth that belong on his new now list would coincide with 150 years in the future according to me.   The "now slice" for the moving observer rotates into the past or future of the stationary observer.
I need help to understand this.  Why does it happen at all, and why does it take such a small increase in his speed, when his distance away is very far?  And why does the time move from past to future, depending on if he is moving away or towards me?
 A: Three good questions.  The first is ambiguous so answering both senses:
1a)  Because that's the way it is, only God knows why.
1b)  We know because we know the speed of light is a physical constant unaffected by the movement of an observer.  Time and space measurements must be distorted in three ways.  Time is dilated, lengths are shrunk, and time is changed by an amount proportional to distance.
2)  This last feature is referred to as relativity of simultaneity, and is 'how' each observer sees a clock of the other running slow.  The faster you go, or the farther away you look in the direction of travel, the bigger the time difference in the other observer's frame between his time here and his time over there.  If the movement is very slow, you see the difference at very large distances.
3)  The time difference needed is direction specific.  If you could see the watches on the passengers of a passing train, subject to all the obvious assumptions you would see an earlier time on the watches near the front of the train than on those at the rear.  That is why, when they compare notes after passing your watch, they see it running slow even though you see their watches running slow.  Your example of past and present is the right way round - so well done on that.  It can be confusing.
The effect is referred to as a rotated time slice, assuming the frames are inertial and moving at constant relative speed to one another.  That is because the time difference increases by a constant amount for each unit of distance.  You can think of it as a time gradient.  The gradient is proportional to the speed.
A: I had a similar question, but taking the basic result as given. If we are in the same time slice when we are both sitting still, and then he starts walking away, and his new time slice is 150 years in the past, does this mean he observes what happens on my planet in reverse? One moment, he see me sitting in a chair, the very next (when he walks away) he see the beginning of the American revolution or whatever…? Or more precisely, after he seees me at 30 years old sitting in the chair, the next moment, I am in the kitchen before I sit down in the chair??
