I understand it might happen in a weak gravitational field. If I have twins on Earth, but one of them suddenly gets transported to intergalactic medium (without any great mass nearby), then the transported twin would grow old faster and would get back to Earth as an older person (Tell me if I'm wrong, please).
But are there any other ways (scientifically speaking) that a twin on Earth would stay younger than a twin that have "gone elsewhere"?
Here's a solution involving traversable wormholes, which Ash already mentioned as a possibility, but I'll give a little more detail to show how it can be done without violating causality (no one can do anything to exert a causal influence on their own past self).
One key thing to understand about traversable wormholes is that time hooks up differently through the wormhole than the time measured by external observers (as discussed for example on p. 501-502 of the book Black Holes and Time Warps by Kip Thorne, the physicist who discovered that traversable wormholes were theoretically allowed in Einstein's theory of general relativity, and explored how they would behave according to the theory). Suppose we have two wormhole mouths that are initially at rest relative to each other, and clocks are placed next to each mouth, and synchronized by sending signals through the wormhole. So, assuming the trip through the wormhole is quasi-instantaneous, if you pass by one clock reading 9:00 AM, Jan 1, 2100 as you enter one mouth, then as you exit the other mouth you'll see the clock there also reading 9:00 AM, Jan 1, 2100. Now suppose one of the mouths is accelerated to a significant fraction of the speed of light relative to the other mouth. Assuming an asymptotically flat spacetime (no significant spacetime curvature except for wormholes themsleves), an external observer who is at rest relative to one mouth is going to see the clock next to the moving mouth tick significantly slower than the clock next to the mouth they see as being at rest, due to velocity-based time dilation. So from their perspective, the clocks are going to get increasingly out-of-sync. However, anyone traveling through the wormhole will see the clocks remain in sync, because of the way time threads through the wormhole according to general relativity. So, if someone dives into one mouth when the clock next to it reads 3:15 PM, Jan. 17, 2100, they'll exit the other mouth when the clock next to that one reads 3:15 PM, Jan. 17, 2100, even if this implies the external observer will measure them to enter and exit at very different times, or even measure them exit one mouth earlier than they entered the other.
Suppose there is a planet, call it Planet X, which is 1000 light years away from Earth, and at rest in the Earth's frame. And suppose we have a wormhole connecting the two, with one mouth initially at rest near each planet, and clocks next to each mouth are initially seen as synchronized in the reference frame where both are at rest. When the clock on the one near Earth reads 9:00 AM, Jan 1, 2100, twin B jumps through the wormhole while twin A remains on Earth, so twin B exits the other mouth near Planet X on 9:00 AM, Jan 1, 2100, both according to the clock next to that mouth and according to the coordinate time in the Earth/Planet X rest frame.
Now suppose that while twin B remains living on Planet X for many years, the wormhole mouth that was formerly at rest near Planet X is accelerated to a large fraction of light speed in Planet X's rest frame, travels away from Planet X for some time, then is accelerated to turn around and travel back towards Planet X at a large fraction of light speed, and finally accelerated a third time to come to rest again near Planet X. In this case, if twin B compares the time elapsed on his own clock to the time elapsed on the clock that traveled along with the wormhole mouth, he will find his clock shows more time has elapsed than the clock that took the trip--this is the standard twin paradox scenario. For example if the wormhole mouth had a speed of about 0.994987437 times the speed of light in Planet X's frame for nearly all of its trip, then twin B should measure the mouth's clock to have elapsed only about 0.1 the amount elapsed on his own clock. Say twin B exited the mouth near Planet X at 9:00 AM, Jan 1, 2100, and very shortly after that they saw the wormhole mouth begin its journey, accelerating to 0.994987437c at 11:15 AM, Jan. 1, 2100. Twin B remained on Planet X for the next 60 years, and then at 11:15 AM, Jan. 1, 2160 according to their own clock, they saw the wormhole mouth return from its journey. But because of the velocity-based time dilation effect, the clock that had traveled along with the wormhole mouth only showed an elapsed time of 6 years rather than 60, so its time would be 11:15 AM, Jan. 1, 2106.
But because of time threading differently through the wormhole than outside it, this means that if twin B now hops through the wormhole mouth to return to Earth, they will exit the mouth near Earth when its clock also reads 11:15 AM, Jan. 1, 2106. But since that wormhole mouth has just remained at rest near Earth, not taking any high-speed journeys, it has remained in sync with the personal clock of twin A on Earth. So twin A on Earth has really only experienced 6 years between the moment twin B first hopped through the wormhole to depart for planet X, and the moment twin B emerged back from the mouth. But twin B experienced 60 years between those wormhole journeys, so he's really 60 years older, i.e. he has aged 54 more years than twin A.
In the coordinate system where Earth and Planet X are both at rest, it's true that the event of twin B emerging from the wormhole mouth back on Earth happens at an earlier time-coordinate than the event of twin B traveling into the wormhole mouth at Planet X. However, this is not a true causality violation because a light signal sent from the first event won't be able to reach the second event, since the second event happens 54 years after the first one in this frame, but I said earlier that Earth and Planet X are 1000 light-years apart so it'll take 1000 years for a signal to travel from one to the other through normal space. So this type of story could be consistent with the chronology protection conjecture which supposes that although it might be possible for such time differences to accumulate between separated wormhole mouths, if you try to move them close enough to allow for causality violations (say, slowly moving the two mouths in this example towards each other until they get to be 54 light-years apart while retaining the time difference of 54 years in their rest frame), quantum effects will build up and destroy the wormhole. I talked more about what the assumption of wormhole travel + the chronology protection conjecture might imply for a science fiction universe in this answer.
Yes, but probably not on the scale you want.
Time dilation exists on earth. According to a study from the National Institute of Standards and Technology, "if one twin spent 79 years living at an altitude 1 foot higher than her sister, the first twin would end up approximately 90 billionths of a second older." We can extrapolate from that that a person ages 1.13 billionths of a second per year per foot of altitude relative to a control subject. So, if one twin was living at the peak of Everest (the highest peak on earth) and the other was living in the Krubera cave (the deepest cave on earth), the twin on Everest would gain approximately .004 seconds of age every year over the twin in Krubera.
Although we can talk about situations involving speeding up or slowing time and hit an answer, there's a whole other scientific realm available to this answer: biology. And while we don't know how to stop aging exactly (or exactly what causes it), a biology-based answer requires no less handwavium than physics (where we can say what to do but fundamentally lack things like exotic matter to actually execute on the idea).
Ultimately, the process of aging is an evolved characteristic that was deemed to be beneficial to us (though not to all creatures). One idea on why this is the case is that there may be evolutionary changes that disproportionately benefit us earlier in our lives but ultimately result in aging and death.
While the exact biological causes of aging have not been worked out yet, one idea is that telomeres are involved. Here is a basic explanation of what telomeres are and what they do:
Inside the nucleus of a cell, our genes are arranged along twisted, double-stranded molecules of DNA called chromosomes. At the ends of the chromosomes are stretches of DNA called telomeres, which protect our genetic data, make it possible for cells to divide, and hold some secrets to how we age and get cancer.
Telomeres have been compared with the plastic tips on shoelaces, because they keep chromosome ends from fraying and sticking to each other, which would destroy or scramble an organism's genetic information.
Yet, each time a cell divides, the telomeres get shorter. When they get too short, the cell can no longer divide; it becomes inactive or "senescent" or it dies. This shortening process is associated with aging, cancer, and a higher risk of death. So telomeres also have been compared with a bomb fuse.
Thus there exists a theory that if one could lengthen telomeres - likely by an engineered retrovirus - aging itself could be slowed down, stopped, or (in some ways) even reversed.
While this theory would likely solve only part of the overall puzzle, there is reason to believe that therapies related to telomeres could reduce some of the symptoms associated with aging and (on average) lengthen ones remaining life - effectively making one "younger". Here's an interesting Wikipedia entry on some research along those lines.
Of course, there is the opportunity that messing with Telomeres will also benefit a number of forms of cancer, so the targeting mechanism will need to be good. At our current level of understanding that's where the handwavium comes in - though unlike a physics-based answer that requires forms of matter we can't conceive of generating at usable scale, this is a potentially solvable problem.
The twin on earth enjoyed the use of a new anti-aging therapy that was invented while their counterpart was zooming around space.
Time travel, either back in time from the journey location to the return arrival time or farther along their own personal timeline. If the "traveler" twin is displaced in time as well as space, or goes somewhere where the passage of time differs from that on Earth (in this case it would have to go faster), then they may actually experience more time than the "stay home" twin.
In many ways, time is just another dimension, one that we humans can't traverse it with modern technology. In fact, if our current understanding of physics is correct then the universe can only traverse it in one direction, but if the traveler is being assisted by ASBs it may be a different story. He could travel back in time such that his arrival back on Earth in the present would mean he experiences time his twin does not. The mode of transportation could also be a source of temporal displacement in the case of a Traversable wormhole since the time-like dimension of the wormhole exit is not necessarily fixed. Either of these scenarios will result in the traveler being older than the stayer if the traveler spends more time "elsewhere" than the twin spends on Earth during the traveler's trip, even if the two are experiencing time at the same rate.
We also know that small changes to our experience of time occur when we leave Earth's gravity well, but they're miniscule. We don't know for certain that there are not places where that difference is far more pronounced, although current evidence suggests not. A trip to such a location would also have the desired effect; the traveler would age at a faster rate than the stay-at-home twin.
Here's how it works. If you want the twin on Earth to be much older than the twin that left, just have the twin be close to something extremely heavy, i.g. a black hole. Or, have them travel somewhere extremely fast where time would slow down for them. This is about the Theory of Relativity, so the faster you go, the slower time is for you. Use this how you want. For keeping the earth twin younger, I have an idea. Based on the speed = slower principle, it may be possible that our earth and everything around it could be moving forward at a high speed, we just can't tell because everything around us is moving similarly fast. If a twin were to somehow "stop" moving as fast, they could be older than the other twin, however, you would have to instantaneously transport them to a "future" destination of the earth, so as to not reverse the effect by having her travel quickly to catch back up.
Going another way:
The twin on Earth takes advantage of cryostorage (greatly improved over current capabilities) and suspends life for a few years before being unfrozen.
