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I'm a math PhD student, and far from an expert on physics. If anything is ambiguous or uses the wrong terminology, please correct it or let me know.

About a decade ago, a friend gave me a paradoxical thought experiment that I still haven't been able to resolve, although I've asked around here and there:

Suppose that earth and outer space were "flipped" in such a way that the vacuum of space became an earth-like solid and earth became an earth-sized hole in this solid: a "Swiss cheese universe". How much gravity would a person experience if they were standing inside of the earth-sized hole with their feet on the boundary?

I have two mutually exclusive answers which are equally convincing to me. I'm interested to hear an explanation of which answer (if any!) is correct, and where the reasoning breaks down in the wrong answer.

Answer #1: The person experiences earth-like gravity.

This answer argues that for almost every particle in the universe (shown below in blue and red) there is a particle in the "opposite" position which creates an equal and opposite pull. Except for that the particles in an earth-sized sphere right underneath the person (shown in yellow) don't have "opposite" particles, so the net force the person experiences is due to the particles in the earth-sized sphere underneath them.

answer 1

Answer #2: The person experiences no gravity.

Due to the shell theorem, if the person is inside of a hollow sphere of any mass and thickness, they experience no force due to gravity. In particular, in the limit, as the thickness goes to infinity, the force due to gravity remains zero.

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  • $\begingroup$ #2 would be correct, if density was equal in all directions of the shell, as shell theorem does not have a maximum shell thickness $\endgroup$
    – Adrian Howard
    Commented Aug 31, 2019 at 6:42
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    $\begingroup$ The problem is that the force on your observer is calculated by summing the forces from all the infinitesimal masses in the infinite universe and this is not a convergent series so it doesn't have a well defined value. What you're finding is that you get different values depending on your starting point. It's like the old saw of summing 1-1+1-1+... and getting either 1 or 0 as the result depending on how you do the sum. $\endgroup$
    – John Rennie
    Commented Aug 31, 2019 at 6:46
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    $\begingroup$ Possible duplicate: physics.stackexchange.com/q/490829 $\endgroup$
    – safesphere
    Commented Aug 31, 2019 at 7:04
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    $\begingroup$ @safesphere Wouldn't you agree that the issue is that it is not sufficiently defined and not really that it is unphysical or involves infinities? We deal with idealized unphysical situations all the time that involve infinities--and we can meaningfully deal with them as long as the properties of the infinities are pinned down in a well-defined manner, which is not the case here. $\endgroup$
    – user87745
    Commented Aug 31, 2019 at 7:40
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    $\begingroup$ To echo what has already been said by others, the issue is that the question uses the notion of infinity in a blanket manner. In physics, if we only know that something is infinite, it is rarely good enough. The way to deal with infinities in physics is to see them as limits of certain well-defined sequences. Unless we know as to which sequence is behind creating the given infinity in a limiting case, the situation is insufficiently described. And thus, one can imagine completing this gap in information via considering two different sequences which lead to two different answers. $\endgroup$
    – user87745
    Commented Aug 31, 2019 at 7:45

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