Main Page: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
mNo edit summary
No edit summary
 
(396 intermediate revisions by more than 100 users not shown)
Line 1: Line 1:
'''Semiclassical gravity''' is the approximation to the theory of [[quantum gravity]] in which one treats [[Field (physics)|matter fields]] as being quantum and the [[Gravitation|gravitational field]] as being classical.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


In semiclassical gravity, matter is represented by quantum matter fields that propagate according to the theory of [[quantum field theory in curved spacetime|quantum fields in curved spacetime]]. The spacetime in which the fields propagate is classical but dynamical. The curvature of the spacetime is given by the ''semiclassical Einstein equations'', which relate the curvature of the spacetime, given by the [[Einstein tensor]] <math>G_{\mu\nu}</math>, to the expectation value of the [[Stress–energy tensor|energy–momentum tensor]] operator, <math>T_{\mu\nu}</math>, of the matter fields:
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


:<math> G_{\mu\nu} = \frac{ 8 \pi G }{ c^4 } \left\langle \hat T_{\mu\nu} \right\rangle_\psi </math>
Registered users will be able to choose between the following three rendering modes:  


where ''G'' is [[Gravitational constant|Newton's constant]] and <math>\psi</math> indicates the quantum state of the matter fields.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==Stress–energy tensor==
<!--'''PNG''' (currently default in production)
There is some ambiguity in regulating the stress–energy tensor, and this depends upon the curvature. This ambiguity can be absorbed into the [[cosmological constant]], [[Newton's constant]], and the [[f(R) gravity|quadratic couplings]]<ref>See Wald (1994) Chapter 4, section 6 "The Stress-Energy Tensor".</ref>
:<math forcemathmode="png">E=mc^2</math>
:<math>\int d^dx \,\sqrt{-g} R^2</math> and <math>\int d^dx\, \sqrt{-g} R^{\mu\nu}R_{\mu\nu}</math>.
There's also the other quadratic term
:<math>\int d^dx\, \sqrt{-g} R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}</math>,
but (in 4-dimensions) this term is a linear combination of the other two terms and a surface term. See [[Gauss–Bonnet gravity]] for more details.


Since the theory of quantum gravity is not yet known, it is difficult to say what is the regime of validity of semiclassical gravity. However, one can formally show that semiclassical gravity could be deduced from quantum gravity by considering ''N'' copies of the quantum matter fields, and taking the limit of ''N'' going to infinity while keeping the product ''GN'' constant. At diagrammatic level, semiclassical gravity corresponds to summing all [[Feynman diagram]]s which do not have loops of gravitons (but have an arbitrary number of matter loops). Semiclassical gravity can also be deduced from an axiomatic approach.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


==Experimental status==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
There are cases where semiclassical gravity breaks down. For instance,<ref>See Page and Geilker; Eppley and Hannah; Albers, Kiefer, and Reginatto.</ref> if ''M'' is a huge mass, then the superposition
:<math>\frac{1}{\sqrt{2}} \left( \left| M \text{ at } A \right\rangle + \left| M \text{ at } B \right\rangle \right)</math>
where ''A'' and ''B'' are widely separated, then the expectation value of the stress–energy tensor is ''M/2'' at ''A'' and ''M/2'' at ''B'', but we would never observe the metric sourced by such a distribution. Instead, we [[decohere]] into a state with the metric sourced at ''A'' and another sourced at ''B'' with a 50% chance each.


==Applications==
==Demos==
The most important applications of semiclassical gravity are to understand the [[Hawking radiation]] of [[black hole]]s and the generation of random gaussian-distributed perturbations in the theory of [[cosmic inflation]], which is thought to occur at the very beginnings of the [[Big Bang|big bang]].


==Notes==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
{{Reflist}}


==References==
* Birrell, N. D. and Davies, P. C. W., ''Quantum fields in curved space'', (Cambridge University Press, Cambridge, UK, 1982).
* Don N. Page, and C. D. Geilker, "Indirect Evidence for Quantum Gravity."  ''Phys. Rev. Lett.'' '''47''' (1981) 979–982. DOI:[http://dx.doi.org/10.1103/PhysRevLett.47.979 10.1103/PhysRevLett.47.979]
* K. Eppley and E. Hannah, "The necessity of quantizing the gravitational field." ''Found. Phys.'' '''7''' (1977) 51–68. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1007/BF00715241 10.1007/BF00715241]
* Mark Albers, Claus Kiefer, Marcel Reginatto, "Measurement Analysis and Quantum Gravity." ''Phys.Rev.D'' '''78''' 6 (2008) 064051, [http://dx.doi.org/10.1103/PhysRevD.78.064051 DOI:10.1103/PhysRevD.78.064051]. Eprint [http://arxiv.org/abs/0802.1978 arXiv:0802.1978] [gr-qc].
* Robert M. Wald, ''Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics''. University of Chicago Press, 1994.
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=-43587 Semiclassical gravity on arxiv.org]


{{theories of gravitation}}
* accessibility:
{{quantum gravity}}
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


[[Category:Theories of gravitation]]
==Test pages ==
[[Category:Quantum field theory]]
 
[[Category:Quantum gravity]]
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

Retrieved from "https://en.formulasearchengine.com/index.php?title=Main_Page&oldid=329754"