Editing Wormhole (section)

== Metrics ==
Theories of ''wormhole metrics'' describe the spacetime geometry of a wormhole and serve as theoretical models for time travel. An example of a (traversable) wormhole [[metric tensor|metric]] is the following:<ref>{{cite book |last1=Raine |first1=Derek |last2=Thomas |first2=Edwin |date=2009 |title= Black Holes: An Introduction |url=https://archive.org/details/blackholesintrod00rain |url-access=limited |publisher=Imperial College Press |page=[https://archive.org/details/blackholesintrod00rain/page/n156 143] |isbn=978-1-84816-383-6 |edition=2nd|doi=10.1142/p637 }}</ref>
{{block indent|<math>ds^2= - c^2 \, dt^2 + d\ell^2 + (k^2 + \ell^2)(d \theta^2 + \sin^2 \theta \, d\varphi^2),</math>}}

first presented by Ellis (see [[Ellis wormhole]]) as a special case of the [[Ellis drainhole]].

One type of non-traversable wormhole [[metric tensor|metric]] is the [[Schwarzschild metric|Schwarzschild solution]] (see the first diagram):

{{block indent|<math>ds^2= - c^2 \left(1 - \frac{2GM}{rc^2}\right) \, dt^2 + \frac{dr^2}{1 - \frac{2GM}{rc^2}} + r^2(d \theta^2 + \sin^2 \theta \, d\varphi^2).</math>}}

The original Einstein–Rosen bridge was described in an article published in July 1935.<ref>{{cite journal |last1=Einstein |first1=A. |last2=Rosen |first2=N. |title=The Particle Problem in the General Theory of Relativity |journal=Physical Review |date=1 July 1935 |volume=48 |issue=1 |pages=73–77 |doi=10.1103/PhysRev.48.73 |bibcode = 1935PhRv...48...73E |doi-access=free}}</ref><ref>{{Cite web |url=https://www.youtube.com/watch?v=OBPpRqxY8Uw |archive-url=https://ghostarchive.org/varchive/youtube/20211211/OBPpRqxY8Uw |archive-date=2021-12-11 |url-status=live |title=Leonard Susskind &#124; "ER = EPR" or "What's Behind the Horizons of Black Holes?" |date=4 November 2014 |via=www.youtube.com}}{{cbignore}}</ref>

For the Schwarzschild spherically symmetric static solution 
{{block indent|<math>ds^2 = - \frac{1}{1 - \frac{2m}{r}} \, dr^2 - r^2(d\theta^2 + \sin^2 \theta \, d\varphi^2) + \left(1 - \frac{2m}{r} \right) \, dt^2,</math>}}
where <math>ds</math> is the proper time and <math>c = 1</math>.

If one replaces <math>r</math> with <math>u</math> according to <math>u^2 = r - 2m</math>

{{block indent|<math>ds^2 = -4(u^2 + 2m)\,du^2 - (u^2 + 2m)^2(d\theta^2 + \sin^2 \theta \, d\varphi^2) + \frac{u^2}{u^2 + 2m} \, dt^2 </math>}}

{{Blockquote|text=The four-dimensional space is described mathematically by two congruent parts or "sheets", corresponding to <math>u>0</math> and <math>u< 0</math>, which are joined by a hyperplane <math>r = 2m</math> or <math>u= 0</math> in which <math>g</math> vanishes. We call such a connection between the two sheets a "bridge".|author=A. Einstein, N. Rosen, "The Particle Problem in the General Theory of Relativity"|source=}}

For the combined field, gravity and electricity, Einstein and Rosen derived the following Schwarzschild static spherically symmetric solution
{{block indent|<math>\varphi_1 = \varphi_2 = \varphi_3 = 0, \varphi_4 = \frac{\varepsilon}{4},</math>}}
{{block indent|<math>ds^2 = - \frac{1}{\left( 1 - \frac{2m}{r} - \frac{\varepsilon^2}{2 r^2}\right)} \, dr^2 - r^2 (d\theta^2 + \sin^2 \theta \, d\varphi^2) + \left(1 - \frac{2m}{r} - \frac{\varepsilon^2}{2 r^2}\right) \, dt^2,</math>}}
where <math>\varepsilon</math> is the electric charge.

The field equations without denominators in the case when <math>m = 0</math> can be written
{{block indent|<math>\varphi_{\mu \nu} = \varphi_{\mu,\nu} - \varphi_{\nu,\mu}</math>}}
{{block indent|<math>g^2 \varphi_{\mu\nu;\sigma}g^{\nu\sigma} = 0</math>}}
{{block indent|<math>g^2 (R_{ik} + \varphi_{i\alpha}\varphi_k^\alpha - \frac{1}{4} g_{ik} \varphi_{\alpha\beta}\varphi^{\alpha\beta}) = 0</math>}}

In order to eliminate singularities, if one replaces <math>r</math> by <math>u</math> according to the equation:
{{block indent|<math>u^2 = r^2 - \frac{\varepsilon^2}{2}</math>}}

and with <math>m = 0</math> one obtains<ref>{{Cite web|url=https://www.sciencedaily.com/releases/2015/09/150903081506.htm|title=Magnetic 'wormhole' connecting two regions of space created for the first time|website=ScienceDaily}}</ref><ref>{{Cite web|url=http://www.uab.cat/web/newsroom/news-detail/magnetic-wormhole-created-for-first-time-1345668003610.html?noticiaid=1345689132054|title=Magnetic wormhole created for first time|website=UAB Barcelona}}</ref>

{{block indent|<math>\varphi_1 = \varphi_2 = \varphi_3 = 0</math> and <math>\varphi_4 = \frac{\varepsilon}{\left( u^2 + \frac{\varepsilon^2}{2} \right)^{1/2}}</math>}}

{{block indent|<math>ds^2 = - du^2 - \left(u^2 + \frac{\varepsilon^2}{2}\right)(d \theta^2 + \sin^2 \theta \, d\varphi^2) + \left(\frac{2 u^2}{2 u^2 + \varepsilon^2}\right) \, dt^2</math>}}

{{Blockquote|text=The solution is free from singularities for all finite points in the space of the two sheets|author=A. Einstein, N. Rosen, "The Particle Problem in the General Theory of Relativity"}}