Greg Egan's latest novel Incandescence is a great thought experiment exploring how a primitive alien race living inside the tunnels of an asteroid orbiting around a neutron star can find evidence for Newtonian physics and even general relativity using simple experiments inside their closed world. The motion around the star causes a nontrivial field of forces inside the asteroid. There is no weight at the center. Away from the center, the amount and direction of the weight depends on the location relative to the center. Zak, the alien Newton, starts to explain this state of affairs with a simple assumption: "Weight is the difference between preferred and actual motion."
For example, when an object is placed north or south of the center, it is pulled towards the center. Here is what an object left to free fall from the north of the center looks like to an observer inside the asteroid:
However from a different perspective, one can see that the object is actually moving around its own slightly tilted orbit around the star. And if the object is supported by a floor and is prevented from moving toward the center, it will experience a gravitational pull toward the center -- thus an example of the maxim: "Weight is the difference between preferred and actual motion."
When two object are placed toward and away from the sun with respect to the center of the asteroid, they feel a pull away from the center:
Again, from a different perspective, this can be seen as the objects trying to follow their own natural orbits, and weight being the difference between preferred and actual motion:
As a final example, suppose an object at the center is given a radial push. Unlike the objects in the previous example, this object will not accelerate away from the center, but keep cycling in an elliptical course as seen from inside the asteroid:
The standard explanation is in terms of Coriolis force due to the object's motion balancing the tidal acceleration. From the orbital perspective, our radial push places the object in an elliptical orbit crossing the circular orbit of the planet:
Starting with these and many other interesting observations, Zak and his friends uncover the secrets of their universe before ever stepping outside their closed world and ever seeing their sun, their orbit, or fixed stars. The book is a treat for aficionados of gedankenexperiments and qualitative physics.
P.S. I'd like to thank Greg Egan for allowing the use of these animations from his page, which hosts a treasure of information about science and science fiction.