Nothing in physics has yet unequivocally ruled out travel back in time, but if it was possible, it would really mess up our understanding of pretty much everything. Travel forwards in time is different. You’re doing it now. At one second per second, you’re travelling forward into the future.
So is everyone else, obviously, so that’s not particularly interesting. But what if you could travel into the future at a different speed to everyone else?
You can. From Einstein’s 1905 Special Theory of Relativity it can be shown that time passes more slowly for a moving object. It’s actually been tried: atomic clocks flown in aircraft are “slow” when they’re brought back and compared with a stationary one. But only by a tiny amount. It takes relative speeds close to the speed of light to make a big difference, and nobody knows how to do that yet. One day though.
Einstein’s later General Theory of Relativity proved that gravity affects time as well. Again, in any situation that we can set up today, the difference is minimal, but it’s still real. The atomic clocks on the satellites which broadcast GPS navigation signals run slightly fast because they’re further from the Earth’s centre than we are. GPS calculations have to allow for it.
But what I’d really like is to age a lot slower. To quote from Wikipedia: “It has been calculated that, under General Relativity, a person could travel forward in time at a rate four times faster by residing at the bottom of a 5 meter tall funnel with the mass of Jupiter. For such a person, every one second of their “personal” time would correspond to four seconds for external observers. Of course, squeezing the mass of a large planet into a non-spherical object five meters in length is not expected to be within our technological capabilities in the near future.”