Here's my take on how to mathematically make sense of this:
First case is rather simple: we model the situation by transforming the space locally around the portals into a curved manifold where the positions of the two portals is a single point. We can make the transformation so that any singularities created by the transformation is in the far field of the system and therefore any vector fields are fully analytic everywhere in the region of interest.
Now we have a loop of some sort in this manifold. And a quick calculation shows that the gravitational field should also be loopy (that or I got my numbers wrong). Given the way the loops are set out initially, the gravitational field is continuous through the portal point, and so should accelerate. It and even if the portals started in different orientation wrt the gravitation field (ie we get some a discontinuity), we can just do a Gaussian Integral around the point
The model becomes kinda strained in the second example. Our manifold now continuously deforms. But nothing in the local space continuously deforms except maybe the rod.
Now instead if we consider our original space as some sort of complex space that is everywhere zero except along the rod, with singularities at the portals, and do a contour integral of sorts around the rod acting as a branch cut connecting the two singularities. Then as long as the nature of the singularity of the portal does not change (as in our small loop integral around the portals yields the same result regardless of the position of the portals), we should find that the overall length of the rod decreases. This lets us ignore what actually happens at the portals. So since we have zero loss of actual material, the rod is compressed.
And as for the 3rd situation, I think I agree with most people that the rod is cut when the portal changes.
Actually...ignore the complicated maths. I was using this as relativity revision. A friend looked over and gave a much better argument:
We instead consider the rod as a continuous set of rigidly linked points/lumps of matter (or disks if you prefer) initially at rest. I think everyone agrees that when we point a single lump object with mass in this portal configuration, it will accelerate continuously though the loop.
Now back our continuous line of lump/disk masses. We have every lump accelerating at the exact same rate (g) downwards. So they should not exert any extra force on teach other (given no other external forces) despite being rigidly linked. So we can consider each lump separately. The rod therefore accelerates down as a whole.
In the second case, we now have the same number of lumps moving a shorter span of space. Therefore the rod is compressed. How it is compressed depends on how fast the portals moved closer together.
3rd case. We just have to consider the situation immediately after the orange portal changes- just the rod at the same length above the blue portal. It's now been cut at where it enters the portals
Why do people keep coming up with paradoxes if the original premise does not make sense? You can make any weird **** happen if the original premise is wrong.
Actually, it's not a thought experiment. You can't do a thought experiment on something that doesn't make sense.
Like, 'imagine if 1=0' is not a thought experiment.
A) If they are vertically aligned as in the image, it would accelerate downwards. gravity is affecting it still.
B) You would be unable to move the orange portal's location, as you would essentially be trying to squeeze the rod together.
C) The rod has a finite length. Chances are, upon moving the portal it would either slice the rod off where the orange portal used to be, or seriously damage the solder and break the link.