What is time dilation?
I am a high school teacher. In books on time dilation I see many examples of completed problems of what would happen in such journeys out into space. An example is the twins paradox - is there a formula where I could calculate how much time has passed on the Earth while the traveler is flying in space ? Other examples I have seen in programs: "It is possible to journey to the center of our galaxy, a flight time of 50 yrs (?) and when you return, 4 million yrs have gone by." OR "12 year journey and the Brother is now 80 years old". I'm told that such a formula exists - what is it and what are the variables that I can calculate ?
Length contraction and time dilation are both effects of Special Relativity, which take place when an object is travelling close to the speed of light. It's really not considered time travel, except to the extent that we all travel through time inexorably into the future. Nevertheless, these effects are certainly real. You can indeed travel very near the speed of light for a short time and come back to Earth, where some millions of years have passed. The explanation for this is an entire physics course on its own, and can be found in introductory texts on special relativity. Essentially, it is an immediate consequence of the speed of light being a constant for all observers, no matter what their own speed.
Moving clocks run slow and moving sticks are shortened by a factor
gamma = 1/Sqrt(1 - v2/c2).
So, let's say we're thinking about the "twin paradox," and that our intrepid traveller is moving at speed v equal to 0.9 times the speed of light, c. In this case,
gamma = 1/Sqrt( 1 - 0.92 ) = 2.29.
So the twin on Earth sees the spaceship's clock running 2.29 times slower than her own, i.e. every 2.29 years on Earth corresponds to one year by the spaceship's clock. The Earthbound twin also observes that the spaceship travelling at a speed 0.9c is 2.29 times shorter than it was before the launch. The faster the spaceship travels, the more pronounced the effect. Cool, eh?
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