|Hi! When looking at the theories of special and general relativity, it becomes evident that the theoretical possibility of "traveling" into the past using various suitable geometries that may be created is a real one. Also, relativity seems to suggest that traveling faster than light will result to moving backwards in time. Does traveling faster than light result in time travel to the past (according to relativity)? And what mathematical, etc. evidence does relativity provide to support that idea? Furthermore, how would a person be able to calculate the extent to which one would travel into the past (sort of like being able to calculate time dilation between a object moving at speeds just under the speed of light and a non-moving object)? Is there a specific formula or equation that I could use to calculate that? I would assume that in the case of traveling into the past, the "time dilation" would be used in a negative sense, because it is usually meant to calculate how "far" into the future a person would go relative to the earth (in most cases) due to the person's very high speed (close to the speed of light)? I would like to compose a graph of time dilation to the "future" and "past" relative to an object that is moving at speeds close to and beyond the speed of light and a "non-moving" object (time as perceived on earth), if the above ideas suggested by relativity are valid. Of course, that graph would be completely theoretical in its purpose, because an object with mass could never reach or past the speed of light. Nevertheless, I think it would be an interesting way to depict theoretical time travel in a more mathematically and visually understandable way. Thanks for your help!
Traveling faster than the speed of light would cause an object to travel to the past. General relativity provides a very great framework for how objects interact with each other via gravity. Although we can't travel in the past in our own universe, we can change the parameters of the equations of general relativity to get a hint of what could be. As far as traveling faster than the speed of light, by construction relativity sets the speed limit at the speed of light. So in regards to our physically observable universe, this is the root of why we don't expect to be able to travel faster than the speed of light. However, we can ask ourselves what would happen if we were to attempt to plug in a velocity greater than the speed of light in to relativistic equations. One such equation for the relativistic energy of an object is:
E = M c2 / sqrt( 1 - (v/c)2 )
Note that if we have a value of 'v' greater than 'c', the denominator of that equation will get bigger as the velocity increases. This means that the energy of the object will get SMALLER the faster it goes. (Additionally, since we're taking the square root of a negative number, we must have an imaginary mass (in the sense of M = i * m, where i = sqrt(-1) ) to cancel out the factor from the square root. This is even less well understood, and an unnecessary tangent for this explanation)
Our faster than light particle ends up going faster and faster, traveling further and further back in time, in order to lower its energy. Although we can see this by investigating the math of relativity, physicists always have to do a 'reality check' with their results, and here is unfortunately where things breaks down for us. There would be nothing to stop the tachyon (faster than light particle) from traveling all the way to the beginning of the big bang, which tells us essentially that the particle could not exist in the first place.
Time dilation is another facet of relativity that isn't completely relevant to time travel in this instance, and it would further complicate things. However, if you're interested in time travel prospects in general relativity, the most compelling topic currently would be worm holes. Although these are speculated to exist, and they could potentially make time travel possible, for reasons of causality (what does it mean for a future self to interact with its past self? is that even possible?) it's still unlikely that actual time traveling devices could be made.
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