Thank you for reading. Before you answer my question, I feel I need to let you know that I'm still a beginner in special relativity…so the more thorough the answer, the better. Thank you! Alright, here goes:
Alice and Bob are moving at speed $V$ relative to one another.
Lets first look at Alice's point of view.
From Alice's point of view, Bob's clock runs slower than hers. That is, for any amount of time which Bob travels, while Alice sees herself age by $t$ years, she sees Bob age by $t\sqrt{1-\frac{v^2}{c^2}}$ years.
Let's say that Bob travels a distance $D$ as measured by Alice.
To justify the fact that Alice thinks that Bob aged less than she did as he traveled that distance $D$, she would say that from Bob's point of view, he must've only felt like he traveled a distance of $D\sqrt{1-\frac{v^2}{c^2}}$.
The reason she concludes this is because the way that the time which passes is defined is the distance which light moves away from someone divided by the velocity of light. Light must've moved less away from Bob as it did from her, and thus less time must've passed for him.
However, from Bob's point of view, it would've been Alice moving at a velocity of $V$ relative to him, he would've said that she aged by an amount of $t\sqrt{1-\frac{v^2}{c^2}}$ for any time $t$ that passed for him, and therefore he would've concluded that from her point of view the distance $D$ must've felt like $D\sqrt{1-\frac{v^2}{c^2}}$.
Does this mean that the Bob in Alice's reality and the Bob in his own reality are…two different Bobs?
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Best Answer
I'm not going to really touch what one could mean by "multiple realities", but I think this can still be useful.
Let's say we are facing each other. I see a building to my left. You see a building to your right. Does this mean there are two different buildings? Let's say I then point to the right, and you say I pointed to the left. Did we just create two new realities?
If I am driving down the road at a constant velocity, I will see a building move past me and my car at rest relative to me. However, if you are in the building you see a moving car and a stationary building. Does this mean we exist in two different realities?
Going a bit further... Let's say we are both floating in space with nothing else that is visible around us. Let's say I see you moving past me at a constant speed. Then you will see me moving by you at a constant speed. If we both got together to meet up later, there would be absolutely no way for either of us to say one was moving at the other was at rest. Does this mean we existed in two different realities just because we were in different inertial frames?
No, it's all the same "reality". It's just what different observers see is relative to the frame they are in. This is why it is called relativity. Of course when we start talking about how lengths and times are relative it becomes more counterintuitive since we don't experience it at our slow relative speeds, but the idea is essentially the same.