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I'm reading a book introducing gravity and find something I don't understand. Please check the attached image. The sentence in red bracket claims that if one adding up gravitational fields of infinitely many objects, the total gravitational field depends on the order in which we add them.

I'm just wondering how do summation order affect the total gravitational field? Is this some mathematical magic in infinite series?enter image description here

Qmechanic
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shinotakatoshi
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  • I assume this is a reference to the fact the Madelung Constant is conditionally convergent. – jacob1729 Jul 19 '19 at 20:24
  • The issue pointed out in the text has been discussed on this site here and here. – knzhou Jul 19 '19 at 20:29
  • @jacob1729 Thank you. In Madelung constant case there exists both positive and negative charges so I can understand why order matters. But for gravity the masses should all be positive – shinotakatoshi Jul 19 '19 at 20:34
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    It's because the gravitational forces due to these masses can be in different directions, and hence cancel. – knzhou Jul 19 '19 at 20:38
  • @knzhou the specific issue arising with constant uniform density might be what they meant, but its somewhat hard to tell since the text sounds like its referring to a countably infinite number of discrete bodies. – jacob1729 Jul 19 '19 at 20:39
  • @jacob1729 Sure, but I think the same conceptual issues arise whether it's continuous or countably infinite, as long as on large scales the density is approximately uniform. – knzhou Jul 19 '19 at 20:44
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    Please do not post images of texts you want to quote, but type it out instead so it can be indexed by search engines. For formulae, use MathJax instead. – ACuriousMind Jul 19 '19 at 20:49

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