Time travel is possibly the most interesting problem in all of physics. On the other hand, there's not much physicists can do about it; it's just not yet possible to do much work on the subject. Aside from a very few very specific experiments "around the edges," it isn't possible to investigate time the way physicists are able to examine most other aspects of this universe.
There are many puzzling issues in the study of time. Many physicists even claim that time is an illusion, that there is no such "thing" as time. Such scientists, sounding more like philosophers, believe that time is an illusion of the way our human consciousness is structured. Among the philosophical community there are two distinctly different views on time that have become popular. The first, dubbed Newtonian Time because Isaac Newton subscribed to the theory, is that time is a fundamental dimension of the universe in which events occur in sequence. The second idea, held by the likes of Immanuel Kant, suggests that time is neither an event nor a thing, and therefore cannot be traversed or measured.
Another fundamental problem that perplexes those studying time concerns its unidirectional flow. Most aspects of our universe are symmetrical, or in balance, but time is one of the glaring exceptions which is decidedly one way and no other. It seems impossible to go backwards in time, yet no one can venture exactly why. Of course, it would be illogical to travel into the past and kill oneself or one's forebears - but that in itself does not explain why time cannot flow backwards, as in the case of a broken vase "reordering" itself or a glass of milk "unspilling" itself. Such occurrences would be illogical, but lack of logic is no explanation in and of itself.
Though the tangible nature of time will continue to be the subject of debate, perhaps forever, it is undeniable that its measurement - even if in crude or arbitrary terms - is essential to life on Earth. Measuring time by the repetitions of a cyclical event (say, a hypothetic pendulum) during a fixed period to establish "seconds" or any other unit of time has been utterly vital to the functions of human life.
The closest physicists are able to come to explaining this puzzle involves the related concept of entropy. Time's unidirectional flow onwards to the future and the fact of entropy in our world seem to be intimately linked. For some reason, "things break down," the natural progress of the universe is to break down, wind down, unwind, fall apart. It's almost certain that no time travel into the past is possible. But no one is able to decisively prove it, in large part due to the whole mystery over just what time is.
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