There are a few methods by which we could travel into the future that are based on Einsteins relativity. In fact we, and by we I mean most of us, have already traveled into the future a tiny bit using the first method I'll talk about. It exploits the fact that time moves more slowly for an object moving at a high velocity. If you want to understand how this works, here is a video. Essentially, its a result of the speed of light being constant, and if you understand how vectors work, you know that if light is moving at the speed of light in the Y direction, and it is moving at all in the X direction, then the light will be actually moving faster than light. To compensate for this, time is slowed down. Ok, so we can slow time down by moving fast, but how does that help us move into the future? We just more really REALLY fast. For example, (more info here) if you took a rocket ship flight at near the speed of light for about ten years, you would come back to the earth and it would be some 1000 years later (this is not the actual amount but you get the idea). So now you have essentially moved into the future 990 years. But this method sort of sucks. First of all, any self respecting time machine is instant. I don't want to wait ten years for my time machine to work, I want to press a button and be in the future. Also, this sounds extremely expensive an impractical. So lets discard this method.
The next method by which you can travel into the future is by occupying a point in space near a massive object. Einstein taught us that time slows in the presence of a gravitational field. How strong does this need to be to amount to a noticeable difference? Black hole strong. So we would have to fly around a black hole, careful not to fall past the event horizon, and hang out for a while. This method sucks for all the same reasons the first one does.
There is however a final method, one which would be relatively inexpensive, and that has nothing to do with relativity (thank god), although to demonstrate why I consider it time travel, I will exploit one of Einsteins most insightful principles. First let me define what I consider time travel into the future. I'll tell it as if I am the time traveler. I walk into a room, or some machine. I am probably hooked up to some biofeedback machines, to monitor my vitals. I possibly am put asleep, so that I don't notice the loudness/unpleasant side effects of traveling in time. I disappear from the rest of humanity and then I wake up, exit the room/machine and I am in the future. I honestly don't care what method they used to transport me into the future, so long as what I described above happens, and this is why I believe we already have/are approaching the age of time machines.
Lets back up a few steps first. Today I had an upper endoscopy done and I was put asleep. I was talking to the doctor, and the next thing I knew I was finished. It felt like I had been transported 10 minutes into the future. Of course this is not the case, my body had aged those ten minutes, so I did not really travel in time. However, what if I could somehow preserve the state of my body? What if I could be put to sleep for 1000 years and wake up in the same body I have now? would I have traveled in time? Not in the really cool science fictiony way that we usually associate with time travel, but for all intents and purposes, yes. I could follow the exact same narrative I outlined above with this method. In fact, it even loosely follows Einstein's Equivalence Principle. The Equivalence principle basically states that if something, for example, feels like gravity and behaves like gravity, such as acceleration, then we can assume that it is gravity, and has all the properties of it. While this principle doesn't strictly apply to my argument, I dont think its a stretch. If it looks like time travel, feels like time travel, has all the effects of time travel, then for all intents and purposes, it is time travel.
How, though, do we go about preserving someone's body, (and their mind) for 1000 years? Walt Disney might know something about that actually. The process of vitrification promises that we can basically freeze ourselves and be thawed out in a later point in time. If this actually works, then we could use this process for my version of time travel. Who would have thought that Walt Disney would be a pioneer of the 4th dimention?! Also there is another process, which is far more promising, less expensive, and within out technological reach called "gorking." Its rather involved, so follow that link to a TED Talk about the subject. Basically, they can put organisms into a state of suspended animation, where they slow all metabolic activity to a standstill, essentially preserving the bodies. If they can perfect this technique, and be able to keep something in suspended animation indefinitely, then we will be able to use this to travel forward in time.
So do time machines exist? yes and no. There is no machine, but techniques to move you forward in time theoretically do exist, and their reality is approaching. Would I be interested in trying it? Of course! I'd kill to see where we are even 100 years from now. Fortunately, I don't think I'll have to travel in time to see the distant future, but that is another topic for another post.
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