A giant hole exists in the earth surrounded by a super material that can withstand the extreme temperatures and pressures from the crust to crust through the center core of the earth. It is perfectly straight. The material that keeps the hole open maintains an even cave temperature throughout.
How would atmospheric pressure change from the surface to the core? If the hole joined the North and South pole, wouldn't that be different than if it was opposite ends of the Equator, given centripetal forces of the earths rotation?
I'm not holding answers to the hard-science tag, but I would appreciate seeing the math or explanations.
This question is only about atmospheric pressure. We can imagine this hole is perfectly maintained (no rain gets in) and keeps temperature steady (unless having temperature that matches the earths layers makes the question easier to answer).
My interest in this question is to imagine a person traversing such a hole. When would it be lethal? Looks like it would be around 3-4 bars or about 1000km down the hole? The best "real world" data we have on this is from the Galileo spacecraft's probe: