WE HAVE A PARADOX!! or do we?

Let's set up the problem, see why it seems like a paradox, then attempt to see whether it really is a paradox. To be honest with you I'm not assed enough to explain frequently used concepts and equations, so I'll explain it once and for all here. 

Special Relativity:

1. The speed of light is the same for every single observer, no matter how fast/slow the observer is, whether the observer is moving towards light or away from it, the speed is always c
2. Inn all inertial frames of reference, the laws of physics are always the same and can be stated in their simplest form.  

Event:

Something happening at a point in time, somewhere in space. 

Lorentz transformations:

\(t=v \gamma x' + \gamma t'\)

\(x= \gamma x' + v \gamma t'\)

Converting coordinates (t',x') in the frame moving along the x-axis to coordinates (t,x) in the laboratory frame.

If we wish to go the other way, we just make v -> -v and we're good. Note that \(\gamma = \dfrac{1}{\sqrt{1-v^2}}\) what we will call the Lorenz factor. 

Time dilation and length contraction:

\(\Delta t = \gamma \Delta t_0\) Where t_0 is the proper time

\(L = \dfrac{L_0}{\gamma}\) Where L_0 is the proper length

Proper length is just the length of o the object in its own rest frame, proper time is conceptually the same. 

Spacetime intervals:

\(\Delta s^2 = \Delta t ^2 -\Delta x^2\)

This is only true for motion along one axis, it's x here but it would've been identical ha it been the y axis. 

we also know that 

\(\Delta s = \Delta s'\)

Imagine 2 planets, Homey and destination, both positioned along the y-axis. The distance between them measured in PRF (Planet reference frame) is 200 light years, remember that in PRF the planets are stationary. 

How long would it take Lisa, an astronaut who is travelling at 99% the speed of light, or 0.99v in short, to travel that distance? This is an easy task, we first calculate the time in PRF (dividing distance by speed) we get then approximately that 202 years have passed in the planet's frame of reference. 

How long time has it taken on Lisa's wristwatch (non-faulty and synchronised)? We insert the values we have in our equation for time dilation, where the proper time is the only time unknown, we get 28.5 years. Knowing that the journey to is 28.5 and 202 for the two FR, we know that the journey back would just be the double, due to symmetry. 

Let's switch frames of reference and look at it from Lisa's perspective AFTER WE REACHED DESTINY AND TURN BACK, her frame of reference is stationary and all else is in motion, meaning that here she doesn't leave homey, homey leaves here. She doesn't go to destiny, destiny comes to her. That's what is it mesans when we talk about her frame of reference. 

Let's now find out how long it will take in the planet's frame of reference, we again set our values in the time dilation formula where the only unknown is the proper time. We then get 4 years. So it it 202 years or 4? Confusing I know. Remember, this happened when we switched RF, we also know that all laws should be the same in all inertial systems (Lorentz transformation are derived for constant motion)! Wait a minute .. maybe our system is not inertial????? Well for any body to change direction you have to undergo acceleration, and that opposes our definition of inertial systems. In short, the frame of reference she has on her way to destiny is not the same as when she's on her way home! 

To take a closer look at this "paradox" we must introduce another astronaut in another spaceship, this will be important later so you can just ignore it for now, but it will be included in the illustrations. We'll get back to this later. 

Imagine that instead of Lisa travelling with her spaceship, she goes aboard a continuous train, the destiny express, this train can be thought of as carriages of spaceships if you will, it will become apparent later why this is of importance.

Now in that train, there the outgoing carriage which is the one Lisa is on, there happens to be a carriage behind, let's call it observer carriage, at a distance L_0 behind (200 light years), as soon as Lisa reaches the planet Destiny, Event B' happens: the observer carriage checks the clocks (stopwatches if you will) of the planet Homey, to see the time elapsed, also at the same time she jumps off of that train and hops on a train going the opposite direction now bound to homey, the homey express. Read this one more time. Quick recap event A is Lisa hopping on the train, B is her reaching Destiny, B' is the observer reading the clocks and finding out how much time has gone by, by looking at the clocks. I understand this might not make sense but let these drawings and table inspire you. 

Using Lorentz Transformations, we fill in the table. One must remember that if something is moving account for length contraction and time dilation. We also used that Event B and B' happen at the same time SRF. Filling in the values we got

Check the red values, that's where it gets trippy because even though the event B is supposed to be at the same time as B', we see then that from the reference frame of the planet it took 4 years for Lisa to arrive at destiny, and a whole 198 years just to jump from the train on to the other train. What in tarnation? AHA! Remember what we talked about earlier? That special relativity is made for constant velocity RFs? That's what happened in our case, because she accelerated (quite fast at that). But now we know that it is 4 years, we know we had it wrong up there, swapping the frames of reference and getting the same results is a postulate that makes up the theory of special relativity. Lisa changed her frame of reference, that was the issue. 

Using symmetry, we can say that the trip was indeed 404 years, but it only took 4 years going and 4 years coming back, however, the rest of the time was spent on changing frames of reference (396), or accelerating if you will. It's very important for you to understand this concept, if the planet had experience the same acceleration we would have gotten consistent results!

OKAY now let's introduce our good friend Peter, Event D: on the spacecraft all the way at the top (on planet Beyond), that is on its way down the homey express, he gets on that train at time 0, and position 0, in his frame of reference, SS-FR we'll call it. I am avoiding the usually x' and x'' that is usually used because I think it's easier to understand like this, it gets too messy with the apostrophes, but it's also more compact. 

There happens to be an observer a distance L_0 under him, and as soon as he reaches destiny, the observer checks the clock at homey, this is event B'', if confusing check the figure below:

Let's do what we did a tad bit earlier here again, namely find out what the observer should see, except this time we can't use Lorentz Transformations, for they don't apply when the frames of reference are not correctly synchronised, they also don't start at the same time at the same position. 

First we must find the time it takes for Peter to get to destiny using the invariance of the spacetime interval. We end up getting 28.5 years which is expected due to symmetry (same velocity in their respective reference frames). Now we'll try to find to find what time it the observer reads, and using the invariance of spacetime again we obtain an equation that gives us 400 years when we insert the values. This is, again, to be expected. We earlier used symmetry to argue for why that would be the case, if we subtract 4 years (the years it took Lisa from homey to destiny) we get 396 which is the exact same, we might have gotten lucky now that I think about it, but it makes sense.

A lot of the so called paradoxes are usually not taking into account acceleration, things just magically change direction and speed. 

That's the solution folks, thanks for coming to my ted talk.