Does Special Relativity Imply Multiple Realities? 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?
Thanks
 A: 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. 
A: Relax, take a deep breath :-)  to me it seems that you do not realise that Alice and Bob have two different times. That is, what Alice calls time, Bob calls a mixture of space and time. So, If you draw a single chart showing all the events in space and time on one page, then draw Alice's time you will find it has a different direction on the chart from Bob's time. To grok relativity, it is best to get a handle on this point first. Alice and Bob disagree on what is simultaneous. So, there is no conflict here, and hence no need to two realities. Although you might be thinking that two directions of time is the same thing, but the whole chart is a single compatible background reality.
A: The analogy that helped me grok this (somewhat) is that of a Doppler effect. Say you're in a car, driving at a constant speed. You hear the cars engine humming at some medium frequency. A bystander however will hear the car humming a high frequency when approaching, and then a low frequency when departing.
The reality is the same for everyone, but the measurements about it (the frequency of the sound) disagree. However our understanding of reality allows us to calculate these differences, and the bystander will be able to tell what the driver hears and vice versa.
Same about relativity. When we measure something about it (the "time"), our results differ, although the underlying reality is the same. And we can use our understanding of the laws of physics to calculate what the other person measures.
A: This "twin paradox" is not so much a paradox. I'll try an analogy... 
It's more like both Alice and Bob are measuring their age with watches, and it happens to be watches that are hand made by the clumsy apprentice. They have some accuracy issues and Alice's watch is running a bit slower than the one of Bob.
So - at some point they claim that they're the same age, and sync their watches. Then Bob flopps on the sofa, has a beer and watches a movie, while Alice is going for a run. When she's back, Alice's watch says that she's a bit younger than Bob, because her watch ran a bit slower. 
Now - the difference with special relativity is like the watches have been made by the Switzerland's best watchmaker -a real master of his craft- , and they are accurate. The way these watches go wrong then isn't the watchmaker's lack of skill, but rather at the speed at which Alice ran.
This way Alice stays young and wrinkle-free (which is what we all want).  
No need for alternate realities here. IIRC the theory about different realities is more linked to superposed quantum states (https://en.wikipedia.org/wiki/Many-worlds_interpretation)
A: I would say it rather implies that there is no "reality". That is: there is no meaningful, universal "t" coordinate differentiating which you could get the time which has passed for a subject passing their way to the future. And for your

while Alice sees herself age by $t$ years, she sees Bob age by $t\sqrt{1-\frac{v^2}{c^2}}$ years

yes Alice can see her aged by $t$ years, but there is no certain "while" which she could see as Bob's count. Both $t\sqrt{1-\frac{v^2}{c^2}}$ to $\frac{t}{\sqrt{1-\frac{v^2}{c^2}}}$ would be close enough (spacelike) points to consider as simultaneous to her $t$ moment.
A: Let's further specify that Alice and Bob start at the same location, and they later meet up and compare their viewpoints. Bob just keeps going, while Alice turns around and rushes to catch up.
Alice's perspective:


*

*Bob's clock is running slow. She maintains steady speed for an hour, and calculates that Bob has aged less than an hour.

*Alice turns around. To do so, she must accelerate rapidly towards Bob. She calculates that, during her acceleration, Bob's clock ran fast. Very fast. How fast Alice's viewpoint thinks Bob's clock is running during her acceleration depends not just on how much Alice accelerates, but also on how far Bob is from her.

*Alice maintains steady speed approaching Bob. Bob's clock is again running slow.

*They meet and compare notes. Bob's clock running fast during Alice's acceleration actually made a greater difference than it running slow the rest of the time, so Bob has aged more than Alice.


Bob's perspective:


*

*Alice's clock is running slow. He maintains steady speed for over an hour, and calculates that Alice has aged one hour.

*Alice turns around. This takes a short amount of time, and makes little difference.

*Alice's clock, after momentarily running as fast as Bob's, is running slow again. He continues to maintain steady speed as Alice catches up.

*They meet and compare notes. Alice's acceleration had negligible effect on total time, so Alice's clock running slow the rest of the time means Alice has aged less than Bob.


End result, after meeting back up they both agree that Alice aged less and Bob aged more. They will even agree on the magnitude of the difference.
You might then ask why is Alice the one that aged less? The reason for that is that she is the one that accelerated. When Alice accelerated, she changed her reference frame. She was in one inertial reference frame before accelerating, and a different one after, and in a non-inertial frame while accelerating.
Calculating using the before-and-after frames, they have different definitions of what time is "now" at any location other than Alice's. Calculating using the acceleration frame, clocks run faster at locations she is accelerating towards, and slower or even backwards at locations she is accelerating away from, proportional to the distance between her and the clock.
She never actually sees a clock run backwards, however, because the effect scales at a rate that always allows the increase in travel time of the light from the clock to her to outweigh it - to overcome this limit and see a clock running backwards, Alice would have to move faster than light.
Another followup question is what happens if neither Alice nor Bob accelerate, and they instead compare notes by sending messages? Answer: exact same thing, just with a message taking Alice's or Bob's place for part of the trip. They can't compare notes until the message from one is in the same location as the other, and achieving that will involve the message in question going through the acceleration frame transition that Alice did in the original version. One way or another, the difference will get resolved by something accelerating before it can affect any physical interactions.
