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I was watching an old cartoon movie where a scientist makes a gadget, which when bound on the wrist, freezes the movement of the whole world. So, that one may do 100s of things in a single second. (The beagle boys later use the gadget to rob a whole bank in 1 second).

The scientist explains the working that the gadget helps in moving "very very fast". So fast that the person who wears it, sees the world, he finds everything frozen. As whatever he is doing is going on 1000s of times faster than the people in the world are doing.

The above explanation is quite logical and easily understandable actually. So, with this logic, if movement occurs faster, time freezes. More the movement, more the time freezes.

With this thing in mind, how come then theory of relativity claims that "moving faster may transport you ahead in time".

What I am assuming here is "transporting ahead in time" is probably something exactly opposite of "freezing the time". Am I right?

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Yes. Special Relativity satisfies your question somehow. Probably, there are many discussions in SE based on the topic. But, there are several cons I'd like to correct in your question...

If movement occurs faster, time freezes. More the movement, more the time freezes.

According to Einstein - Length, mass, time and space are interdependent variables. These motions depend on the speed of light $c$ in vacuum which is the only constant here and is what the second postulate says. OK, Let's take your "time" part...

Time doesn't freeze out anytime. It may freeze if you move at $c$ which is impossible (practically & theoretically) to achieve in vacuum. And, you say, "More the movement, more it freezes". It's not a state of matter. And, No laws have ever proved that "Time depends upon temperature". But, Time really slows down when you approach a good velocity comparable to $c$. Time dilation could be predicted indeed (taking Lorentz factor into account) using $$T=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$$ where $t_0$ is the time measured by an observer at its rest frame. $T$ is the apparent time measured which is always less than $t_0$ and it reduces with increasing velocity. (freezes more - in your words)

How come then theory of relativity claims that "moving faster may transport you ahead in time"?

Short answer - No, There's no such thing in Relativity. But, its the postulates that made the physicists think about the plausibility of time travel. These guys thought it in a different way. If there's a possibility for objects to travel faster than light (assuming hypothetical particles called Tachyons to study the behaviour), maybe it would forward time into future (like that)... But, I think it is unsuccessful..!

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Simple answer: the theory of relativity does not claim that "moving faster may transport you ahead in time".

It (special relativity) does, however, state that as you move faster time moves slower. This is known as time dilation. You have have become confused by the twin paradox which is a corollary of this.

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