In the story Escape! the Brain sends Gregory Powell and Mike Donovan to space. When they get back, everything seems normal, like time has not passed since they went away.
According to the Twin Paradox, Powell and Donovan should have returned later in the Earth point of view (e.g. people would be older and waiting for them for years).
Did Isaac Asimov ignore this concept, or maybe the Brain did not send the technicians far away enough to feel the concequences of the concept?
As for the distance travelled,
The needle stood stiff and proud at the hairline whose figure read 300,000 parsecs.
Unless the dial lied, they had travelled quite a distance.
So the only option is that Asimov chose to ignore it. He left the precise method of travel vague enough.
As Dr. Alfred Lanning states:
"From a scientific standpoint the situation, while not entirely clear, is subject to intelligent analysis. The question of interstellar travel under present conditions of physical theory is... uh... vague. The matter is wide open -- and the information given by Consolidated to its thinking machine, assuming these we have to be the same, was similarly wide open. Our mathematical department has given it a thorough analysis, and it seems Consolidated has included everything. Its material for submission contains all known developments of Franciacci's space-warp theory, and, apparently, all pertinent astrophysical and electronic data. It’s quite a mouthful."
Also, Powell and Donovan die during warp, coming back alive once arrived. Who knows what strange mechanisms may be at work that keep time from dilating and causality from being violated.
Also see:
How does “warp drive” not violate Special Relativity causality constraints?
This question on the Physics Stack shows that physicists don't yet agree on what happens, either.
The interstellar travel in Asimov's works is not translational (at least in this story). It involves a concept of hyperspace (scroll down for a paragraph on Asimov's take on this), through which protagonists travel without obeying traditional laws of motion. This excerpt is from Risk, which shares its concept of travel with Escape!:
The ordinary laws of motion do not apply to hyperspace
. . .
Hyper-forces are not limited by the speed of light. It is very possible that they have no limit of velocity at all.
Since the main postulate upon relativity rests is the speed limit in the universe being equal to speed of light, we can safely assume that the Twin Paradox does not apply here.
