phinds said:
Which would DECREASE the gravity in the room and make time move slightly faster than on the surface of the Earth. Do you not understand how gravity works? Google the Shell Theorem.
Unfortunately I do not think the Shell Theorem can be validly applied to this problem. See below.
Bandersnatch said:
The gravity would decrease, but gravitational time dilation depends on how deep in the potential well you are, not on the magnitude of gravitational acceleration.
I don't think the gravity would decrease inside the room.
First, as I said above, I don't think the Shell Theorem applies to this problem. The Shell Theorem says that a spherically symmetric distribution of matter and energy outside some spherical shell has zero effect on the spacetime geometry inside the shell. If our "room" is a spherical shell 10 km deep inside the Earth, the distribution of matter and energy outside it is
not spherically symmetric; it is skewed towards the Earth's center. So the Shell Theorem does not apply.
(Another way to see that it can't apply is to note that, if it did apply, it would predict that, in the absence of any gravitating masses inside the shell, spacetime would be flat there. Flat means
zero gravity, not just decreased gravity; in other words, if you excavated a spherical chamber 10 km underground, the Shell Theorem, if it applied, would say you could float in it in free-fall. If this were actually true, then, for example, miners in deep mine chambers should at least notice a significant drop in gravity, and they don't.)
I think a sort of generalized "Shell Theorem" could be applied to this case, but the generalized theorem would say, basically, that a spherical shell in a gravitational field has zero effect on either the gravitational potential (this part would be the same as the standard Shell Theorem--see further comments below) or the "acceleration due to gravity" inside the shell. In simple Newtonian terms, the "gravitational pull" of all the parts of the room on any object inside the room, when summed, would cancel out, leaving only the "gravitational pull" due to the Earth that was there in the first place.
As far as gravitational potential and "rate of time flow" is concerned, the standard Shell Theorem already says that the shell has zero effect on the gravitational potential inside it; and, as noted above, I would expect any generalization of it that applied to this problem to say the same thing. So clocks inside the room would go at the same rate as clocks just outside the room.
Note that all of the above applies regardless of what the room's walls are made of. A room with neutronium walls inside a neutron star would behave the same as a room with osmium walls (or walls of any other substance found on Earth) 10 km deep inside the Earth.
