The problems with the "Hollow Earth" theory

One of the things the controversial author Lobsang Rampa is known for is a theory that the Earth is actually hollow. According to this theory, rather than having a crust floating on a mantle over a molten outer core and solid inner core of iron (see, for instance, here and here), there is an inner "sun", then a gap of air, and then a shell about 800 to 1,000 miles thick around this (see here, for instance).

Now, there are a lot of problems with this theory - particularly as it was presented at the time I read it, with claims that there haven't been flights over the poles, etc: there have been a lot of such flights, and satellite passes, and no evidence of the claimed entrance at the poles. 

More fundamentally, this is why I have a problem with the claims: 

• the speed of sound varies depending on the density of the matter it is travelling through, and there are various effects (refraction and reflection) at changes of density. As a result of this, over dozens of years and many earthquakes (we have dozens of minor earthquakes every day, and moderate ones relatively frequently as well), geologists have been able to build up a good picture of the earth - the mostly solid core, liquid mantle, etc;
• if the Earth truly was hollow as claimed, the real observations would need to be accounted for. Maybe that would suggest something like a denser layer in the middle of the outer shell, for instance (although I would hate to be the one to try to work out such a distribution of density and match it to hundreds of thousands of observations); 
• now, the next problem this runs into is gravity - and in particular, the orbits and spins of planets.
  The preceding alternative could possibly account for total mass by compressing the outer shell, but then that would change the spin of the earth - think of an ice skater, who spins more quickly by bring their arms in closer.
  Before you say "perhaps that has already been included in the derivation of calculations", it would also have to apply to gaseous planets. Also, we know what the gravitational constant is from lots of small scale experiments - including measuring the sideways gravitational pull of a mountain. 

I cannot see any way this theory stands up.

There may, of course, be smaller caverns (albeit caverns much larger than we have come across) in the Earth, and people may be confusing or rationalising having passed through some sort of dimensional portal, but those are different matters. (And I trust someone has ruled out that the claims of confusion on the part of some polar explorers was not just ergot poisoning or simple disorientation from stress, etc.)

I'm putting this theory in the category of disproved, and have done so pretty much from the time I first came across it.