Editing Anthropic principle (section)

=== Dimensions of spacetime ===
[[File:Spacetime dimensionality.svg|thumb|300px|Properties of {{nowrap|(''n'' + ''m'')}}-dimensional spacetimes<ref>{{cite journal |last=Tegmark |first=Max |date=1997-04-01 |title=On the dimensionality of spacetime |journal=Classical and Quantum Gravity |volume=14 |issue=4 |pages=L69–L75 |doi=10.1088/0264-9381/14/4/002 |arxiv=gr-qc/9702052 |bibcode=1997CQGra..14L..69T |s2cid=250904081 |issn=0264-9381}}</ref>]]

There are two kinds of dimensions: [[spatial dimension|spatial]] (bidirectional) and [[temporal dimension|temporal]] (unidirectional).<ref name="Skow2007">{{Cite journal |author-link=Bradford Skow|last1=Skow |first1=Bradford |title=What makes time different from space? |journal=Noûs |year=2007 |volume=41 |issue=2 |pages=227–252 |url=http://web.mit.edu/bskow/www/research/temporality.pdf |access-date=13 April 2018 |doi=10.1111/j.1468-0068.2007.00645.x |citeseerx=10.1.1.404.7853 |archive-url=https://web.archive.org/web/20160824025031/http://web.mit.edu/bskow/www/research/temporality.pdf |archive-date=2016-08-24 }}</ref> Let the number of spatial dimensions be ''N'' and the number of temporal dimensions be ''T''. That {{nowrap|1=''N'' = 3}} and {{nowrap|1=''T'' = 1}}, setting aside the compactified dimensions invoked by [[string theory]] and undetectable to date, can be explained by appealing to the physical consequences of letting ''N'' differ from 3 and ''T'' differ from 1. The argument is often of an anthropic character and possibly the first of its kind, albeit before the complete concept came into vogue.

The implicit notion that the dimensionality of the universe is special is first attributed to [[Gottfried Wilhelm Leibniz]], who in the [[Discourse on Metaphysics]] suggested that the world is "[[wikiquote:Gottfried Leibniz|the one which is at the same time the simplest in hypothesis and the richest in phenomena]]".<ref name="Leibniz1686">{{Cite book |last1=Leibniz |first1=Gottfried |title=Die philosophischen schriften von Gottfried Wilhelm Leibniz |volume=4 |year=1880|publisher=Weidmann |pages=427–463 |chapter-url=https://en.wikisource.org/wiki/Discourse_on_Metaphysics |access-date=13 April 2018 |chapter=Discourse on metaphysics }}</ref> [[Immanuel Kant]] argued that 3-dimensional space was a consequence of the inverse square [[law of universal gravitation]]. While Kant's argument is historically important, [[John D. Barrow]] said that it "gets the punch-line back to front: it is the three-dimensionality of space that explains why we see inverse-square force laws in Nature, not vice-versa" (Barrow 2002:204).<ref group=note>This is because the law of gravitation (or any other [[inverse-square law]]) follows from the concept of [[flux]] and the proportional relationship of flux density and field strength. If {{nowrap|1=''N'' = 3}}, then 3-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of [[radius]] ''r'' has a surface area of 4''πr''{{sup|2}}. More generally, in a space of ''N'' dimensions, the strength of the gravitational attraction between two bodies separated by a distance of ''r'' would be inversely proportional to ''r''{{sup|''N''−1}}.</ref>

In 1920, [[Paul Ehrenfest]] showed that if there is only a single time dimension and more than three spatial dimensions, the [[orbit]] of a [[planet]] about its Sun cannot remain stable. The same is true of a star's orbit around the center of its [[galaxy]].<ref>{{Cite journal |last=Ehrenfest |first=Paul |author-link=Paul Ehrenfest |title=Welche Rolle spielt die Dreidimensionalität des Raumes in den Grundgesetzen der Physik? |trans-title=How do the fundamental laws of physics make manifest that space has 3 dimensions? |journal=Annalen der Physik |volume=61 |pages=440–446 |year=1920 |doi=10.1002/andp.19203660503 |issue=5 |bibcode=1920AnP...366..440E |url=https://zenodo.org/record/1424351 }}. Also see Ehrenfest, P. (1917) "In what way does it become manifest in the fundamental laws of physics that space has three dimensions?" ''Proceedings of the Amsterdam academy'' 20:200.</ref> Ehrenfest also showed that if there are an even number of spatial dimensions, then the different parts of a [[wave]] impulse will travel at different speeds. If there are <math>5 + 2k</math> spatial dimensions, where ''k'' is a positive whole number, then wave impulses become distorted. In 1922, [[Hermann Weyl]] claimed that [[James Clerk Maxwell|Maxwell]]'s theory of [[electromagnetism]] can be expressed in terms of an action only for a four-dimensional manifold.<ref>Weyl, H. (1922). ''Space, time, and matter''. Dover reprint: 284.</ref> Finally, Tangherlini showed in 1963 that when there are more than three spatial dimensions, electron [[atomic orbital|orbitals]] around nuclei cannot be stable; electrons would either fall into the [[atomic nucleus|nucleus]] or disperse.<ref>{{Cite journal |last=Tangherlini |first=F. R. |title=Schwarzschild field in ''n'' dimensions and the dimensionality of space problem|journal=Nuovo Cimento |volume= 27|issue= 3|pages= 636–651|year=1963 |doi=10.1007/BF02784569 |bibcode=1963NCim...27..636T |s2cid=119683293 }}</ref>

[[Max Tegmark]] expands on the preceding argument in the following anthropic manner.<ref name="tegmark-dim">{{Cite journal |last=Tegmark |first=Max |author-link=Max Tegmark |title=On the dimensionality of spacetime |journal=Classical and Quantum Gravity |volume=14 |issue=4 |pages=L69–L75 |date=April 1997 |url=https://space.mit.edu/home/tegmark/dimensions.pdf |doi=10.1088/0264-9381/14/4/002 |access-date=2006-12-16 |arxiv=gr-qc/9702052 |bibcode=1997CQGra..14L..69T |s2cid=15694111 }}</ref> If ''T'' differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant [[partial differential equation]]s. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if {{nowrap|''T'' > 1}}, Tegmark maintains that [[proton]]s and [[electron]]s would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.)<ref name="tegmark-dim" /> Lastly, if {{nowrap|''N'' < 3}}, gravitation of any kind becomes problematic, and the universe would probably be too simple to contain observers. For example, when {{nowrap|''N'' < 3}}, [[nerve]]s cannot cross without intersecting.<ref name="tegmark-dim" /> Hence anthropic and other arguments rule out all cases except {{nowrap|1=''N'' = 3}} and {{nowrap|1=''T'' = 1}}, which describes the world around us.

On the other hand, in view of creating [[black hole]]s from an ideal [[monatomic gas]] under its self-gravity, Wei-Xiang Feng showed that {{nowrap|(3 + 1)}}-dimensional spacetime is the marginal dimensionality. Moreover, it is the unique [[dimensionality]] that can afford a "stable" gas sphere with a "positive" [[cosmological constant]]. However, a self-gravitating gas cannot be stably bound if the mass sphere is larger than ~10<sup>21</sup> solar masses, due to the small positivity of the cosmological constant observed.<ref>{{cite journal |last=Feng |first=W.X. |date=2022-08-03 |title=Gravothermal phase transition, black holes and space dimensionality |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.106.L041501 |journal=Physical Review D |volume=106 |issue=4 |pages=L041501 |arxiv=2207.14317 |doi=10.1103/PhysRevD.106.L041501 |bibcode=2022PhRvD.106d1501F |s2cid=251196731 }}</ref>

In 2019, James Scargill argued that complex life may be possible with two spatial dimensions. According to Scargill, a purely scalar theory of gravity may enable a local gravitational force, and 2D networks may be sufficient for complex neural networks.<ref>{{cite journal |last=Scargill |first=J. H. C. |date=2020-02-26 |title=Existence of life in 2 + 1 dimensions |url=https://link.aps.org/doi/10.1103/PhysRevResearch.2.013217 |journal=Physical Review Research |volume=2 |issue=1 |pages=013217 |arxiv=1906.05336 |doi=10.1103/PhysRevResearch.2.013217 |bibcode=2020PhRvR...2a3217S |s2cid=211734117 }}</ref><ref>{{Cite web |title=Life could exist in a 2D universe (according to physics, anyway) |url=https://www.technologyreview.com/2019/06/20/102942/life-could-exists-in-a-2d-universe-according-to-physics-anyway/ |access-date=2021-06-16 |website=technologyreview.com |language=en}}</ref>