1.) Travel at speeds reaching the speed of light, then time relative to the outside world will slow down. (Einstein’s theory of special relativity)
Traveling 90% the speed of light in a spaceship, time will pass about 2.6 times slower than it was back on Earth.
It has been measured! Physicists took twin atomic clocks, one flown in a jet aircraft and the other stationary on Earth. The flying clock ticked slower, because of its speed.
More proof: "The highest speeds achieved through any human technology are probably the protons whizzing around the Large Hadron Collider at 99.9999991% of the speed of light. Using special relativity, we can calculate one second for the proton is equivalent to 27,777,778 seconds, or about 11 months, for us."
Large Hadron Collider- the world's largest and highest-energy particle accelerator contained in a circular tunnel, with a circumference of 17 mi, at a depth ranging from 164 to 574 ft. Underground.
2.) Once again, according to Einstein's theory of general relativity, the stronger the gravity you feel, the slower the time moves around you. The strength of gravity increases as you get closer to the center of the Earth. "Time runs slower for your feet than your head."
It has been measured! "In 2010, physicists at the US National Institute of Standards and Technology (NIST) placed two atomic clocks on shelves, one 33 centimeters above the other, and measured the difference in their rate of ticking. The lower one ticked slower because it feels a slightly stronger gravity."
To travel to the future, we need a region of extremely strong gravity like a black hole.
Black Hole- "a place in space where gravity pulls so much that even light cannot get out"
3.) Slow down or stop your bodily processes to slow your perception of time and then restart them later. This is pretty much what happened to Captain America. To help the war cause, he flew his plane into the freezing ocean with a deadly weapon. Decades later, he was discovered to be alive, but frozen. When he came to, he woke up to a different century. Yeah, I get this is just a movie, but it could happen. For example, hibernation. Some animals can slow their bodily processes for a certain amount of time and wake up much later. Maybe humans can do this too, but to more of an extent like years or decades.
Right now, completely stopping the metabolism is too advanced for our technology but scientists "are working towards achieving inducing a short-term hibernation state lasting at least a few hours" which could help greatly in the medical field. For example, this could be enough time for a medical emergency, like a cardiac arrest, to make it to the hospital in time by slowing the bodily processes, ultimately slowing the rate of death.
Proof: "In 2005, American scientists demonstrated a way to slow the metabolism of mice (which do not hibernate) by exposing them to minute doses of hydrogen sulphide, which binds to the same cell receptors as oxygen. The core body temperature of the mice dropped to 13 °C and metabolism decreased 10-fold. After six hours the mice could be reanimated without ill effects."
4.) Wormholes "General relativity also allows for the possibility for shortcuts through spacetime, known as wormholes, which might be able to bridge distances of a billion light years or more, or different points in time. Many physicists, including Stephen Hawking, believe wormholes are constantly popping in and out of existence at the quantum scale, far smaller than atoms. The trick would be to capture one and inflate it to human scales - a feat that would require a huge amount of energy, but which might just be possible, in theory."
5.) Light "Another idea, put forward by the American physicist Ron Mallet, is to use a rotating cylinder of light to twist spacetime. Anything dropped inside the swirling cylinder could theoretically be dragged around in space and in time, in a similar way to how a bubble runs around on top your coffee after you swirl it with a spoon. According to Mallet, the right geometry could lead to time travel into either the past and the future."