Let’s say that immediately after the Big Bang, 2 identical clocks were placed next to each other.
As the expansion of space continued, they quickly moved apart relative to each other, and eventually ended up in areas that later became the Andromeda and Milky Way galaxies (each one on a planet within a star system in each galaxy).
Fast forward 18 billion years and, as a result of the galaxies merging, the 2 clocks find themselves together again. Would the clocks show the same time? If not, how much would their times expect to differ?
I don’t think so. Time dilation slows down clocks. They probably haven’t experienced identical acceleration throughout the ages. Clocks on Earth and on satellites run slightly differently.
No, they wouldn’t show the same time. The rate of time passage is affected by gravity as shown by General Relativity.
For that matter, if a clock had been put on the Moon and one on the Earth, they also would not have the same time when you brought them together. Or put a clock at the top of a tower and one below it on the ground, and their times will begin to diverge. You would need an atomic clock to measure it, though.
How large a difference would there be? That totally depends on where the clocks are. On planets, in space, etc.
The two clocks would have to move at exactly the same speed. If one was on a planet that rotated faster/slower or orbited faster/slower or the solar system orbited faster/slower not to mention how the whole galaxy moves then the clocks would not agree and diverge the longer they ran.
The chances of getting all these bits to line up exactly equal to each other is close to zero I would think.
Doesn’t it follow that they would also have different calculations of the age of the universe? Does that make sense that 2 objects in the exact same place have different calculations of this?
Two clocks which each remained in the rest frame of their respective locally comoving coordinate frames, and which were both far away from any significant mass, would record the same age for the Universe. Galaxies are excellent approximations for such frames, but they’re not perfect, so in the experiment you propose, there would doubtless be some slight differences in the two clocks. Most likely, the clock in the Andromeda galaxy would be slightly faster, due to the greater mass of that galaxy. But there’d also be random noise in both of them, so it’s hard to say for sure.
If the two objects took different paths to get there? Sure, that makes perfect sense. By analogy, suppose that you and a friend each have a huge ball of string. You both start from the same place, and tie off one end of the string there. You then both go to some other place, along what each of you thinks is the best route to get there, unwinding your string as you go. But you disagree about which route is better, so you take different routes. Most likely, one of you will have unwound more string than the other. The difference probably won’t be much, since both of you will probably be very close to the true shortest route, but there will still be some difference.