# 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?

• Did you study the Lorentz transformation? (Or a bit of special relativity) Commented Jan 3, 2020 at 21:12
• I have a basic understanding of how movement can change relative time, but I have not read this idea in particular, where it is distance alone that changes relative time. If he was sitting next to me, then there would be only a tiny change in time. But because he is 10 billion light years away, there is a huge change in time. Commented Jan 3, 2020 at 21:20
• As Ofek said, this is a straightforward application of the Lorentz transformation. Penrose calls this scenario the Andromeda paradox. And of course, in the real universe it's a little more complicated trying to talk about such huge spacelike slices, since spacetime is curved. Commented Jan 3, 2020 at 21:30
• @foolishmuse I really encourage you watch the following minute physics videos (I'm sending a link to a list). Almost no math, but really gets the details. youtube.com/playlist?list=PLoaVOjvkzQtyjhV55wZcdicAz5KexgKvm Commented Jan 3, 2020 at 21:31
• I think to really answer your question you should see videos 1-4 , but It is recommended to watch the rest as well Commented Jan 3, 2020 at 21:34

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.

• Yes, in your point 1b, this all works because the speed of light is constant and everything else changes. The videos mentioned above make this graphically clear. Commented Jan 3, 2020 at 22:55
• In most discussions I've seen about Lorentz transformation, they concentrate on observers moving at relativistic speeds and the distance issue is barely mentioned. This question has clarified that both speed and distance are factors. I'm going to post a follow up question on entangled particles. Commented Jan 5, 2020 at 21:20