@PPenguin, any source that to a black hole scale will be spinning. At this time, I don't know how to deal with spinning with my proposal. I appreciate that you are of the opinion that because cross product-like elements do not appear in the differential line element, my proposal will never be able to deal with spin. At this point, for me I think it remains an open question because the cross product terms are where cross product terms should be.

A far simpler and controllable on this very Earth experiment is to create a system with a super precise measurement of the Poynting vector along 3 axes. Measure that at the surface of the Earth and in a mine shaft. GR predicts no change due to the properties of an exterior derivative. My proposal predicts one of the three will change. Clean, precise, controllable by experimenters. They will have to calculate if they could measure the difference because otherwise GR wins by default.

In my one minor effort at outreach, I searched the pre-print server for authors who wrote papers that had "Poynting" in the title. I wrote two of them to ask if anything like this had been done, but they were not aware of it. Not an exhaustive search by any means.

"At this point, for me I think it remains an open question because the cross product terms are where cross product terms should be." But those terms are always there, so if you want to use them for rotating terms, it screws up your flat spacetime claims.

At the end of the day, you only have 4 degrees of freedom. That is not enough to allow you to describe the diagonal terms correctly and the off diagonal terms correctly. This is not a wait and see moment. This particular approach clearly cannot work.

It sounds like you even see that it doesn't currently work, but confusingly will not accept what is preventing it from working. Once you accept this particular approach has failed, you can move onto your next idea.

Regarding the poynting vector thing, that is not a prediction of your theory. This is another of those things you just state. You need to actually develop a theory which is well defined enough that any physicist can calculate the predictions. For example, in your theory you proclaim the E and B fields are no longer part of a tensor ... that would have huge effects and can probably already be ruled out if you state the full theory with your version of electrodynamics.

@sweetser However, work on the electrodynamics part is pointless if the particular approach to obtaining line elements has already failed on more general grounds. Learn from this result and counting the degrees of freedom should give you hints for your next idea.

If you are willing to modify the values of dq before multiplying like you have been doing, that gives you 4 degrees of freedom. If you added another quaternion G and have like: dq G dq, where in flat space-time G is (1,0,0,0) and maybe different in curved spacetime, now you have 8 degrees of freedom. Now maybe you could handle some off diagonals.

I don't know what's next. My point is just that it is clear your previous ideas for extracting a line element cannot succeed, so it is time to learn from what we found when exploring that. Denying, ignoring, or glossing over the failures is not conducive to learning. Exploring new ideas will have tons of failures, learning from them is key.