Editing Hierarchy problem (section)

==== Extra dimensions ====
No experimental or observational evidence of [[extra dimensions]] has been officially reported. Analyses of results from the [[Large Hadron Collider]] severely constrain theories with [[large extra dimensions]].<ref name="ATLAS_blackholes">{{cite journal |last1=Aad |first1=G. |last2=Abajyan |first2=T. |last3=Abbott |first3=B. |last4=Abdallah |first4=J. |last5=Abdel Khalek |first5=S. |last6=Abdinov |first6=O. |last7=Aben |first7=R. |last8=Abi |first8=B. |last9=Abolins |first9=M. |last10=Abouzeid |first10=O. S. |last11=Abramowicz |first11=H. |display-authors=6 |year=2014 |title=Search for Quantum Black-Hole Production in High-Invariant-Mass Lepton+Jet Final States Using Proton-Proton Collisions at {{sqrt|s}} = 8 TeV and the ATLAS Detector |journal=Physical Review Letters |volume=112 |issue=9 |pages=091804 |arxiv=1311.2006 |bibcode=2014PhRvL.112i1804A |doi=10.1103/PhysRevLett.112.091804 |pmid=24655244 |last12=Abreu |first12=H. |last13=Abulaiti |first13=Y. |last14=Acharya |first14=B. S. |last15=Adamczyk |first15=L. |last16=Adams |first16=D. L. |last17=Addy |first17=T. N. |last18=Adelman |first18=J. |last19=Adomeit |first19=S. |last20=Adye |first20=T. |last21=Aefsky |first21=S. |last22=Agatonovic-Jovin |first22=T. |last23=Aguilar-Saavedra |first23=J. A. |last24=Agustoni |first24=M. |last25=Ahlen |first25=S. P. |last26=Ahmad |first26=A. |last27=Ahmadov |first27=F. |last28=Aielli |first28=G. |last29=Åkesson |first29=T. P. A. |last30=Akimoto |first30=G.|s2cid=204934578 }}</ref> However, extra dimensions could explain why the gravity force is so weak, and why the expansion of the universe is faster than expected.<ref>{{cite web |date=20 January 2012 |title=Extra dimensions, gravitons, and tiny black holes |url=http://home.web.cern.ch/about/physics/extra-dimensions-gravitons-and-tiny-black-holes |access-date=13 December 2015 |publisher=CERN}}</ref>

If we live in a 3+1 dimensional world, then we calculate the gravitational force via [[Gauss's law for gravity]]:

<math display="block">\mathbf{g}(\mathbf{r}) = -Gm\frac{\mathbf{e_r}}{r^2} \qquad (1)</math>

which is simply [[Newton's law of gravitation]]. Note that Newton's constant {{mvar|G}} can be rewritten in terms of the [[Planck mass]].

<math display="block">G = \frac{\hbar c}{M_{\mathrm{Pl}}^{2}}</math>

If we extend this idea to {{mvar|&delta;}} extra dimensions, then we get:

<math display=block>\mathbf{g}(\mathbf{r}) = -m\frac{\mathbf{e_r}}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta}r^{2+\delta}} \qquad (2)</math>

where <math display="inline">M_{\mathrm{Pl}_{3+1+\delta}}</math> is the {{math|3+1+<math display="inline">\delta</math>}}-dimensional Planck mass. However, we are assuming that these extra dimensions are the same size as the normal 3+1 dimensions. Let us say that the extra dimensions are of size {{math|''n'' ≪}} than normal dimensions. If we let {{math|''r'' ≪ ''n''}}, then we get (2). However, if we let {{math|''r'' ≫ ''n''}}, then we get our usual Newton's law. However, when {{math|''r'' ≫ ''n''}}, the flux in the extra dimensions becomes a constant, because there is no extra room for gravitational flux to flow through. Thus the flux will be proportional to {{mvar|n{{sup|&delta;}}}} because this is the flux in the extra dimensions. The formula is:

<math display="block">\begin{align}
  \mathbf{g}(\mathbf{r}) &= -m \frac{\mathbf{e_r}}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta} r^2 n^\delta} \\[2pt]
  -m \frac{\mathbf{e_r}}{M_\mathrm{Pl}^2 r^2} &= -m \frac{\mathbf{e_r}}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta}r^2 n^\delta}
\end{align}</math>

which gives:

<math display="block">\begin{align}
  \frac{1}{M_\mathrm{Pl}^2 r^2} &= \frac{1}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta} r^2 n^\delta} \\[2pt]
  \implies \quad M_\mathrm{Pl}^2 &= M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta} n^\delta
\end{align}</math>

Thus the fundamental Planck mass (the extra-dimensional one) could actually be small, meaning that gravity is actually strong, but this must be compensated by the number of the extra dimensions and their size. Physically, this means that gravity is weak because there is a loss of flux to the extra dimensions.

This section is adapted from ''Quantum Field Theory in a Nutshell'' by A. Zee.<ref>{{cite book |last=Zee |first=A. |title=Quantum field theory in a nutshell |publisher=Princeton University Press |year=2003 |isbn=978-0-691-01019-9 |bibcode=2003qftn.book.....Z}}</ref>