Are there any hard science based designs that show the possibility/impossibility of space launch for the cheela?
Given that Robert L Forward was a professional physicist who did a lot of work on things like space fountains (the method used by the Cheela to leave Egg the first time around) I'm inclined to think that the methods in the book are reasonably realistic... if reliant on some materials science that is completely beyond us. Since we're unlikely to survive anywhere close to an actual neutron star I don't imagine that we're too worried here about what humans are capable of.
The book states that:
We do not know the propulsion technique used by the cheela to lift their spacecraft off the surface of the neutron star (the escape velocity of Dragon's Egg is 39% the speed of light). ... they suspect that the cheela used some sort of antigravity catapult to get off the star
Actually, that's just the humans in the story not knowing what's going on down on the surface. The story itself goes into quite a bit of detail from the perspective of the Cheela.
There are two main technologies the Cheela use to escape from (and return to) Egg:
- Space Fountain (Dragon's Egg)
- Gravity Catapult (Starquake)
In "Dragon's Egg" the Cheela develop a space fountain composed of stabilized magnetic monopole blocks fired nearly to orbit by a magnetic accelerator. They magnetically couple to the fountain to raise a platform at a sedate pace until the reach a height limit based on their need for a high gravity field to keep their body together. From there they can make it the rest of the way into orbit relatively cheaply.
The science behind this is reasonably sound, except perhaps for the magnetic monopoles. While these are theoretically plausible we have yet to find any examples of them in the real world. The energy requirements are fairly insane, but that's just an engineering problem. A species that developed on the surface of neutron star will probably have different limits than we do, so I'm happy to cut them some slack on the little details.
Space fountains are an alternative to a space elevator, and are better suited to a neutron star. A space elevator requires a high-tensile tether material even in relatively low-gravity environments like Earth. We'd need something about on par with scrith to do the job on a neutron star.
In the second novel "Starquake" the space fountain is destroyed, leaving a number of Cheela stranded in space with no way to return to the surface of Egg. After several generations the space-going Cheela manage to recreate an older technology for manipulating gravity via counter-rotating charged singularity (black hole) dust. They land a small gravity catapult by throwing it at the surface and firing it downward at the right time to slow it enough to land safely, then reverse it to catch small ships. They build a larger version to launch the ships again, establishing two-way traffic between orbit and the surface.
Ignoring the material science requirements, the idea of twisting space into pretzels uses some of the more interesting ideas from general relativity. The effect of large rotating masses were predicted by general relativity, and was demonstrated when the effect turned out to perfectly explain the discrepancy in Mercury's orbit. The Cheela's gravity catapult could operate on a related set of principles. If you could make specific "shapes" in the curvature of spacetime - our best current guess as to the underlying mechanism of gravity - then it's conceivable that we could use gravity to move pretty much anything anywhere. Hell, with enough power you could open the event horizon of a black hole.
Other ideas (as I think of them)...
The magnetic levitation idea isn't impossible either, again dependent on some material science that we don't have access to here.
A big deal is made throughout both novels about the strength of the magnetic field on Egg, and its pervasive nature. It's strong enough that the locals are shaped by it, and build all of their stuff along the field lines. The field lines are 'frozen' into the star's structure, giving a nice strong and reliable field to push off.
Given that the Cheela have the ability to build things out of magnetic monopoles it could be possible to create a field rider that engages with the field lines and travels along them. If the magnetic field lines aren't too constrained you might make it high enough to reach orbit, or hopefully get close enough to make the rest of the way under power.
The downside to this idea is that a neutron star rotates very quickly. If the magnetic poles are aligned with the spin poles then Coriolis is going to rip you off the field shortly after you leave the vicinity of the pole. For this to work you're going to need a very specific type of neutron star where the magnetic poles are on the equator. On the plus side, you'd probably get some assistance from Eotvos forces. Still, it'd be a hell of a ride.
Warp Drive/Exotic Matter
Alcubierre drive, as Willk suggests, is an interesting option. We'll hand-wave the exotic matter requirements. It would need to be extremely powerful to warp space sufficiently to counter the existing warpage of the gravity well itself. The energy and the sheer amount of exotic matter required to effect that sort of field would be... tremendous.
If you think about it though, there's a more direct way to do it. The Alcubierre drive requires exotic matter with negative mass. Negative mass means the force of gravity is reversed. Instead of being crushed by millions of gravities the exotic matter will be forced upwards with force proportional to its mass. This could be used directly to provide lift without all that fooling around with spacial curvature.