The physical possibility of human travel to the future is well accepted, but travel to the past is more controversial, and time travel that changes either the future or the past is generally considered to be impossible.
Our understanding of time travel comes mostly from the implications of Einstein’s general theory of relativity. This theory has never failed any of its many experimental tests, so we trust its implications for human time travel.
Einstein’s theory permits two kinds of future time travel—either by moving at high speed or by taking advantage of the presence of an intense gravitational field. Actually any motion produces time travel (relative to the clocks of those who do not travel), but if you move at extremely high speed, the time travel is more noticeable; you can travel into the future to the year 2,300 on Earth (as measured by clocks fixed to the Earth) while your personal clock measures that merely, let’s say, ten years have elapsed. You can participate in that future, not just view it; you can meet your twin sister’s descendants. But you cannot get back to the twenty-first century on Earth by reversing your velocity. If you get back, it will be via some other way.
It's not that you suddenly jump into the Earth's future of the year 2,300. Instead you have continually been traveling forward in both your personal time and the Earth’s external time, and you could have been continuously observed from Earth’s telescopes during your voyage.
How about travel to the past, the more interesting kind of time travel? This is not allowed by either Newton's physics or Einstein's special relativity, but is allowed by general relativity. In 1949, Kurt Gödel surprised Albert Einstein by discovering that in some unusual worlds that obey the equations of general relativity—but not in our world—you can continually travel forward in your personal time but eventually arrive into your own past.
Unfortunately, even if you can travel to the past in the actual world you cannot do anything that has not already been done, or else there would be a contradiction. In fact, if you do go back, you would already have been back there. For this reason, if you go back in time and try to kill your childhood self, you will fail no matter how hard you try.
While attempting this assassination, you will be in two different bodies at the same time.
Here are some philosophical arguments against past-directed time travel. I suggest that none of these are convincing. The last one is subtle.
1. If past time travel were possible, then you could be in two different bodies at the same time, which is ridiculous.
2. If you were presently to go back in time, then your present events would cause past events, which violates our concept of causality.
3. Time travel is impossible because, if it were possible, we should have seen many time travelers by now, but nobody has encountered any time travelers.
4. If there were time travel, then when time travelers go back and attempt to change history they must always botch their attempts to change anything, and it will appear to anyone watching them at the time as if Nature is conspiring against them. Since observers have never witnessed this apparent conspiracy of Nature, there is no time travel.
5. Travel to the past is impossible because it allows the gaining of information for free. Here is a possible scenario. Buy a copy of Darwin's book The Origin of Species, which was published in 1859. In the 21st century, enter a time machine with it, go back to 1855 and give the book to Darwin himself. He could have used your copy in order to write his manuscript which he sent off to the publisher. If so, who first came up with the knowledge about evolution? Neither you nor Darwin. Because this scenario contradicts what we know about where knowledge comes from, past-directed time travel isn't really possible.
6. The philosopher John Earman describes a rocket ship that carries a time machine capable of firing a probe (perhaps a smaller rocket) into its recent past. The ship is programmed to fire the probe at a certain time unless a safety switch is on at that time. Suppose the safety switch is programmed to be turned on if and only if the “return” or “impending arrival” of the probe is detected by a sensing device on the ship. Does the probe get launched? It seems to be launched if and only if it is not launched. However, the argument of Earman’s Paradox depends on the assumptions that the rocket ship does work as intended—that people are able to build the computer program, the probe, the safety switch, and an effective sensing device. Earman himself says all these premises are acceptable and so the only weak point in the reasoning to the paradoxical conclusion is the assumption that travel to the past is physically possible.I recommend an alternative solution to Earman’s Paradox. Nature conspires to prevent the design of the rocket ship just as it conspires to prevent anyone from building a gun that shoots if and only if it does not shoot. I cannot say what part of the gun is the obstacle, and I cannot say what part of Earman’s rocket ship is the obstacle.