Editing Inverse-square law (section)

== Non-Euclidean implications ==

The inverse-square law, fundamental in [[Euclidean geometry|Euclidean]] spaces, also applies to [[Non-Euclidean geometry|non-Euclidean geometries]], including [[Hyperbolic geometry|hyperbolic space]]. The curvature present in these spaces alters physical laws, influencing a variety of fields such as [[cosmology]], [[general relativity]], and [[string theory]].<ref>{{Cite journal |last=Barrow |first=John D |date=2020 |title=Non-Euclidean Newtonian cosmology |journal=[[Classical and Quantum Gravity]] |volume=37 |issue=12 |pages=125007 |arxiv=2002.10155 |bibcode=2020CQGra..37l5007B |doi=10.1088/1361-6382/ab8437 }}</ref>

[[John D. Barrow]], in his 2020 paper "Non-Euclidean Newtonian Cosmology," expands on the behavior of force (F) and potential (Φ) within hyperbolic 3-space (H3). He explains that F and Φ obey the relationships F ∝ 1 / R² sinh²(r/R) and Φ ∝ coth(r/R), where R represents the curvature radius and r represents the distance from the focal point.

The concept of spatial dimensionality, first proposed by Immanuel Kant, remains a topic of debate concerning the inverse-square law.<ref>{{Cite journal |last1=Gatzia |first1=Dimitria Electra |last2=Ramsier |first2=Rex D. |date=2021 |title=Dimensionality, symmetry and the Inverse Square Law |journal=Notes and Records: The Royal Society Journal of the History of Science |volume=75 |issue=3 |pages=333–348 |doi=10.1098/rsnr.2019.0044 }}</ref> Dimitria Electra Gatzia and Rex D. Ramsier, in their 2021 paper, contend that the inverse-square law is more closely related to force distribution symmetry than to the dimensionality of space.

In the context of non-Euclidean geometries and general relativity, deviations from the inverse-square law do not arise from the law itself but rather from the assumption that the force between two bodies is instantaneous, which contradicts [[special relativity]]. General relativity reinterprets gravity as the curvature of spacetime, leading particles to move along geodesics in this curved spacetime.<ref>{{Cite web |last=Guth |first=Alan |date=2018 |title=Introduction to Non-Euclidean General Relativity |url=https://web.mit.edu/8.286/www/lecn18/ln05-euf18.pdf |access-date=2023-07-30 |publisher=[[MIT OpenCourseWare]]}}</ref>