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Bill Wilker asked the Naked Scientists: Gravity pulls me towards the Earth because it is the closest massive object. Now, if the very core of the Earth was hollowed out into a sphere 20 feet in diameter and I was placed in the middle, what would the effect of gravity be on me? (We can assume I'm indestructible!) Would I float in the centre? Gravity should be pulling me nearly equally in each direction, right?Thanks,Bill WilkerAtlanta, Georgia, USWhat do you think?
Would I float in the centre? Gravity should be pulling me nearly equally in each direction, right?
I can't agree that the gravitational field is zero everywhere in a hollow sphere. As soon as you move off center, the distance to that side of the sphere will be closer, and the distance to the other side farther. The 1/R˛ effects apply and you get drawn to the closer side. I confirmed my theory by computing a simple piecewise approximation using Excel.
Then we're talking about a two-body problem involving a perfectly evacuated shell and an object of infintely small mass. It could not involve the hollowness filled with mass (not even air). It could not involve an object "floating around inside" whose outer diameter is only, say, 1 micron smaller than the inner diameter of the shell.
We are assuming there is a vacuum within the sphere presumably if it was filled with fluid or gas objects would gravitate to the centre
Given the assumptions, non-rotating, non-accelerating, perfect spherical (hollow) shell with uniform density and, for the purposes of GR, locally flat space, then it is true in General Relativity too.
Kiran, we do know, both from theory and practice, that the field resulting from a spherical shell and inside the shell, for forces obeying the inverse square law, is evrywhere zero. The practical result is obtainable from electrostatics. I doubt there has been any measurement to confirm this with gravity because the apparatus would have to be very large because gravity is such a small force. Nonetheless it has been shown that gravity obeys the inverse square law to high accuracy. It is sufficient to say that it would be very likely to be well behaved and predictable (accurate even with Newtonian mechanics) except in very extreme circumstances. I doubt that even a perfectly hollowed out sphere the size/mass of the earth would show anything significantly different from Newtonian mechanics. Certainly a comet would not and, in any case, where would you get a comet that was perfectly spherical and of perfectly uniform density?I see no reason to depart from the accepted theories, whether Newtonian or Relativistic, without some reasoned justification for doubting them. I have no doubt that Relativity will get superseded at some point, but I don't think it is likely to be on this basis.
We are talking about a gravitational field inside a perfect, non-rotating, spherical shell of any thickness but uniform density. The field due to the shell, within the shell, is everywhere zero. There are no fields, no tidal effects. The size of the body being acted upon is not relevent.
RAJ1, when you study tripple integrals in college, one of your homework assignments will be to integrate Newton's law of gravity for a uniform hollow sphere. When you do so, you will discover for yourself that the field inside the sphere is zero, and outside it is the same as if all the mass were concentrated at the center. Until you discover it for yourself, you'll just have to take the word of your seniors. As for general relativity, it yields exactly the same results as Newtonian physics except in extreme cases, like black holes and the whole universe. A hollow Earth is not such an extreme case. The reason the Earth can't be hollow is because two halves of the hollow sphere would be attracted to one another. The pressure where two halves meet would be greater than the strength of the material. So pieces would break off and float inside, where they would be attracted to each other's gravity and form a solid ball. That ball would attract other chuncks from the near side of the hollow sphere, and very quickly the whole thing would collapse into one solid ball. A small sphere can be made of just about any solid, but when you get to the radius of Earth, it would need to be quite strong. You can do the math yourself; it's pretty simple. Consider two hemispherical shells of a given thickness and radius. The center of mass of a thin hemisphere is located midway between the center and the surface. Calculate the mass of each hemisphere (m = 2πρr˛δr), use Newton's equation (F = G(m˛/r˛)) to get the force of gravity and divide by the area (a = 2πrδr) where the two hemispheres meet. That will give you the compressive pressure. Compare that to a table of the compressive strenghts of various materials.
As I move toward the side (4 feet from me) the pull will only become stronger and the pull from the side 100,000 light years away will grow a little weaker.