Relativistic time dilation provides a kind of one-way time travel into the future, which is featured in this story.
1931: "Out Around Rigel", a novelette by Robert H. Wilson, first published in Astounding Stories, December 1931, available at the Internet Archive, also available from Project Gutenberg. I cited this same story in my answer to the old question Earliest use of Relativity in SF?.
The narrator, Dunal, was an inhabitant of the moon, way back when it had air and water and life. He goes on a trip to Rigel with his friend Garth in a spaceship constructed by the latter, who believes that it can go faster than light:
"You see, since your mathematical friends derived their identical formulas for gravity and electro-magnetism, my job was pretty easy. As you know, a falling body follows the line of least resistance in a field of distortion of space caused by mass. I bend space into another such field by electromagnetic means, and the Comet flies down the track. Working the mercury disintegrators at full power, I can get an acceleration of two hundred miles per second, which will build up the speed at the midpoint of my trip to almost four thousand times that of light. Then I'll have to start slowing down, but at the average speed the journey will take only six months or so."
"But can anyone stand that acceleration?" Kelvar asked.
"I've had it on and felt nothing. With a rocket exhaust shoving the ship, it couldn't be done, but my gravitational field attracts the occupant of the Comet just as much as the vessel itself."
Returning alone from the trip to Rigel, Dunal finds the moon strangely changed:
With a jerk, I got to my feet and climbed up the sloping floor to the atmosphere tester. My fingers slipped off the stopcock, then turned it. And the air-pressure needle scarcely moved. It was true. Somehow, as the scientists had always told us would be the case eventually, the air on the moon, with so little gravity to hold it back, had evaporated into space.
But in six months? It was unthinkable. Surely someone had survived the catastrophe. Some people must have been able to keep themselves alive in caves where the last of the atmosphere would linger. Kelvar must still be alive. I would find her and bring her to the Comet. We would go to some other world.
[. . . .]
Then I knew.
In our argument as to the possible speed of the Comet, Garth and I had both been right. In our reference frame, the vessel had put on an incredible velocity, and covered the nine-hundred-odd light-years around Rigel in six months. But from the viewpoint of the moon, it had been unable to attain a velocity greater than that of light. As the accelerating energy pressed the vessel's speed closer and closer toward that limiting velocity, the mass of the ship and of its contents had increased toward infinity. And trying to move laboriously with such vast mass, our clocks nd bodies had been slowed down until to our leaden minds a year of moon time became equivalent to several hours.
The Comet had attained an average velocity of perhaps 175,000 miles per second, and the voyage that seemed to me six months had taken a thousand years. A thousand years! The words went ringing through my brain. Kelvar had been dead for a thousand years. I was alone in a world uninhabited for centuries.
I threw myself down and battered my head in the sand.