I already mentioned Greg Egan's "Incandescence" once before, but I'll mention it again. Egan has a fictional race living on a small tidelocked object orbiting a black hole. The race doesn't have vision as we know it, and aren't even familiar with light, much less special relativity. Nonetheless, they manage to stumble onto a theory that's equivalent to General Relativity based on observations of "weight", the sort of weight one might measure with a spring-based scale that measures the force required to hold the body in place in the "frame" of the tidelocked body. The weight will be zero at the center of the tidelocked body, and will change as one moves away from the center. It turns out that the weight is proportional to the distance away from the center, the proportionality constant depends on the direction one moves in. (Choices of direction are radial, orbital, and perpendicular to both].
Just calculating these quantities is quite a difficult exercise already knowing General Relativity. I wrote an Insight article on the topic (of the calculation already knowing GR) at
https://www.physicsforums.com/insights/a-problem-from-incandescence/
I'll summarize the end results.
The weight can be viewed as the sum of the tidal forces and the centrifugal forces. The tidal forces for a stationary object in geometric units in an orthonormal basis frame would be:
$$\left[ -\frac{2M}{r^3} \quad \frac{M}{r^3} \quad \frac{M}{r^3} \right]$$
The sign convention is that tidal stretching forces have a minus sigh, tidal compressive forces have a plus sign.
For an orbiting body, though, the weights due to the tidal forces are different. A lengthly calculation gives:
$$ \left[ -\frac{2M}{r^3} \left( \frac{1 – \frac{3M}{2r} } {1 – \frac{3M}{r} }\right) \quad \frac{M}{r^3} \left( \frac{1}{1- \frac{3M}{r}} \right) \quad \frac{M}{r^3} \right] $$
And the centrifugal force "weights" add to the above, and are for a tidelocked body
$$ \left[ -\frac{M}{r^3} \quad 0 \quad -\frac{M}{r^3} \right] $$
Summing these, we get the observed weights:
$$\left[ -\frac{3M}{r^3} \left( \frac{1 – \frac{2M}{r} } {1 – \frac{3M}{r} } \right) \quad \frac{M}{r^3} \left( \frac{1}{1- \frac{3M}{r}} \right) \quad 0\right]$$
This does not explain the interesting topic of the principles of the derivation of GR, but it demonstrates that there are non-trivial and measurable GR effects as simple as measuring the "weight" of a body with a spring scale.
The Newtonian results can be derived from the above by taking the limit where r goes to infinity, we see the ratio of the two nonzero components is -3:1:0, the GR calculation gives slightly different results sufficiently close to the black hole.
The fun doesn't stop there though - the black hole in Egan's book rotating, so further refinements need to be (and are) made. It's rather interesting that the alien physicists stumble on the notion of a limiting velocity from these simple measurements, without having directly observed it.
Egan makes the interesting claim that GR is equivalent to the alien physicist's principle called "Zach's principle", which is that the sum of the weights in a non-rotating frame of reference is zero. I'm not aware of the derivation of this purported equivalence. (Note that I'm easy, and I personally trust the author on this point, but since it's a work of fiction that's a dangerous thing to do, and probably not actually a good idea.].
