Otherspace
The higher-order dimension that most people call “otherspace” is the discovery that birthed the Foundation as it exists today. Access to it is what allowed humanity to colonise the stars, but that access is still limited and poorly understood. Centuries of research have led to the development of better and faster drives, but the deeper mysteries of otherspace seem resistant to scientific inquiry, and FTL travel is still subject to hard limits.
The technology used to transition between realspace and otherspace is strongly affected by gravitational fields, to the extent that shifting an entire ship into otherspace is impossible within a star’s sphere of influence. Likewise, ships that “collide” with an unexpected gravity well in otherspace are torn back to the lower level of reality - often with serious damage as a result.
The stronger the gravitational fields you are subject to, the more energy you need to shift even a small object into otherspace. By expending huge amounts of power, it’s possible to send electromagnetic energy into otherspace within a star’s gravity well - but only energy, not matter. This has allowed for the creation of ships that can redirect their energy signatures into an extradimensional heat sink, but it seems that is as far as the technology will go for now.
Astrogation
In order to travel between stars, a captain needs a navkey: a precalculated set of extradimensional vectors and formulae for a destination. Without one, the ship’s onboard computer can’t perform the calculations required for a safe voyage through otherspace.
Navkeys require immense computing power to generate, and they are made up of an enormous quantity of information. That means that sending one via radio would take weeks; they generally need to be distributed in person, on a dataslab. That gives transit authorities some control over who goes where, and makes it possible for “secret” colonies and bases to hide themselves from the Foundation.
Of course, navkeys don’t just appear out of thin air. The only way to make a navkey is by mapping out the extradimensional route with an exploratory vessel. This is a slow and dangerous process, requiring many “blind jumps” that carry you closer and closer to your destination in a series of short hops. It is during these trailblazing missions that many of the greatest of otherspace’s hazards manifest.
Strange Horizons
Otherspace is two things: it’s the medium that enables faster-than-light travel, and it’s a mysterious and poorly understood dimension filled with strange and sometimes dangerous phenomena that defy explanation. Otherspace has captured the popular imagination as a haunted, beautiful and deadly place; it’s no surprise, then, that the Foundation is full of ghost stories and urban legends about it.
When you’re travelling through otherspace with a Llewellyn Drive, the area around your ship is an envelope of reality in an unreal world. Seen through viewports and cameras, it appears as an eerie iridescent bubble whose mottled surface is a constantly-shifting swirl of fizzing colours. Anything that passes beyond the surface of that bubble is simply gone - people, debris, probes. No one knows what happens to things which leave the envelope, and nothing has ever been recovered once it passes through.
The jury’s still out on the reverse, though. Interstellar veterans have all kinds of stories about things they’ve seen in otherspace. Strange shapes, ominous visions, and even the faces of loved ones have been glimpsed hanging in the veil of colour. There are even scattered reports of so-called otherspace horrors emerging from that bubble and hitching a ride on unsuspecting ships.
Otherspace visions are generally considered by the mainstream scientific community to be a delusional syndrome caused by the reality warping properties of the envelope. The official advice is “don’t look”.