The
very first word of this thread (which appears in the title) is "
IF".
How much clearer can that be?
By starting off with that, the rest of the question is clearly a thought experiment - so all the practicalities of the physical possibility of such a void and/or the survivability of the environment are completely, totally and utterly irrelevant.
For those still champing at the bit on that - you have no imagination. The purpose of a thought experiment is to look past these limitations and focus on some elements of the system being considered in isolation - for the sole purpose of understanding how these components work within themselves. Even if there is no immediate practical application being pursued, such exercises are still valid as basic research.
So, please, let's give that silliness a rest.
As for the matter of the Earth's geometry - yes, it is an oblate spheroid, not a perfect sphere, but that, in itself, does not make any discussion meaningless. One source I found stated that the equatorial radius is 6,378 km and the polar radius is 6,356 km - a difference of 22 km or about 0.35%. While you couldn't really call this totally insignificant, it is certainly not a huge variation - and I would suggest the variation from the spherical is not going to be particularly noticeable. In fact, for a body at the centre of the Earth, there will be no difference at all. So long as the Earth maintains point symmetry around that body, all gravitational forces from every constituent part of the Earth will cancel out - and this would still hold true if the Earth was a cube (Please don't start about how the Earth could never be a cube - this is a
thought experiment for the purpose of illustrating some maths!)
What happens if we move off-centre within an oblate spheroid? Now when it comes to the mathematically pure answer to that - I don't know, but for the practical case of the Earth, the variation is not going to be significant, because the eccentricity is so small. If anyone wants to do the math, please feel free.
Now there has also been a reference to the fact that the Earth does not have a consistent mass distribution. While this is technically true, so long as each concentric shell around the centre has a consistent density, then the gravity at the centre will still be zero. It does not matter if one shell layer is ten times the density of another and twenty times as thick, it all works out.
Where this does result in some perturbations of the calculations is where this density is not consistent through a shell - and the Earth's crust is a valid example. But even here, there is a qualification of one of the basic parameters of the original question that can "save the day" as it were. That qualification is "Define the 'centre' ".
There are two definitions that immediately come to mind. The first is the geometric centre - find the middle point of all the points within the volume of the Earth. The second is the centre of gravity - that magical point towards which the Earth attracts all bodies outside its volume. In the case of an infinite set of shells having their own consistent density, these two points are the same, but if there is any variation, then these two points may separate.
If they were different, for the purposes of the original question, the centre of gravity would be the point to choose - and the answer to the original question would, again, be zero - for the very reasons that this point was determined to be the centre of gravity.
What happens if you were to move off-centre within such a scenario? Again, I don't have the mathematically pure answer, but I will stick my neck out and say that, for all intents and purposes there is not going to be much of a variation from the ideal spheroid model.
.... IF our 'Earth' was hollow, (or was ANY other object in Space),
and WE were INSIDE such a 'hollow' sphere, then our gravitational attraction
(internally) to ANY point of the 'Earths' shell, no matter where we are within
that 'void', will be equal, so we would stay in the position we are in.
This is so close to being perfectly correct (especially for the ideal spherical model), I hesitate to qualify it - but I feel I must.
As I mentioned in a previous post, this observation is correct - so long as the mass of the contents of the void is zero. Since this is a reasonable assumption for a thought experiment, it does not deserve direct criticism - but my thinking went a little further.....
If there is anything in that void that has any mass whatsoever, then there will be a centrally attracting gravitational force on any body within that void. For a body at a distance of
r from the centre, the gravitational force come from that of the mass of a sphere of radius
r from that same centre. This will result in the body accelerating towards the centre - but, as it does, the radius of the sphere exerting the gravity will decrease - and so will the acceleration. As the body reaches the centre, the increase in velocity will drop to zero, but the velocity it has gained will carry it through to the other side, where acceleration in the opposite direction from the now growing central sphere will cause it to slow, stop and then reverse it's travel. The result would be an oscillation. Air resistance would be the only dampening factor, otherwise this motion would continue indefinitely.
To put this into proper perspective, however, one needs to consider the mass of the air and just how much acceleration that would impart. From my quick calculations, for a mass of air equivalent to that at standard temperature and pressure within the 15m void mentioned earlier - for a 70kg person starting at rest 10m from the centre, the initial acceleration would be 3.6 nm/
s2. At that rate, it would take more than 77 hours to have moved 1mm. I don't even want to think of the period of the oscillation.
Just remember ... this is - and always has been - a thought experiment.