Kaluza-Klein theory part 3 - Kaluza-Klein's theory Part 2

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	Kaluza-Klein theory part 3 - Kaluza-Klein's theory Part 2

link_hyrule5

6-30-2007 1:53 PM

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tags:

einstein, relativity, maxwell, em, e&m, electromagetism, kaluza-klein, quantum theory of gravity

link_hyrule5 says:

Part 3 of 3

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<div style="margin: 12px 0px; font-family: arial; color: #333333; background: #ffffff; border: solid 4px #e5e5e5; width: 100%; clear: left;"><div class="CM_CTB_Content_Wrap" style="margin: 0px; padding: 0px;background-color: #ffffff;"><div style="border-bottom: solid 1px #dcdcdc; white-space: nowrap; margin-bottom: 8px; background-color: #eeeeee ;background-image: url(http://clipmarks.com/images/source-bg.gif); background-repeat: repeat-x; height: 24px; line-height: 24px; vertical-align: middle; padding-bottom: 4px; color: #666666; font-size: 10px;" ><a href="http://clipmarks.com/clip-to-blog/" title="see clips that are hot right now"><img src="http://content.clipmarks.com/blog_embed/7dc5a70d-3d9a-4f61-b774-9c100a95e5fb/CB3116CB-9CB0-44E6-82AD-6BD198E51464/" alt="" width="19" height="19" border="0" style="vertical-align: middle; margin: 0px 4px; display: inline; border: none; float:none;" /></a>clipped from <a title="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971" href="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971" style="font-size: 11px;">en.wikipedia.org</a></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971"><SPAN class="texhtml">π <SUP>*</SUP> <I>F</I> = d<I>A</I></SPAN></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971"><P>That there always exists such a connection, even for fiber bundles of arbitrarily complex topology, is a result from <A title="Homology (mathematics)" href="http://en.wikipedia.org/wiki/Homology_%28mathematics%29">homology</A> and specifically, <A title="K-theory" href="http://en.wikipedia.org/wiki/K-theory">K-theory</A>. Applying <A title="Fubini's theorem" href="http://en.wikipedia.org/wiki/Fubini%27s_theorem">Fubini's theorem</A> and integrating on the fiber, one gets</P></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971"><div align="center"><img src="http://content8.clipmarks.com/blog_cache/en.wikipedia.org/img/5F25778C-A7E6-4D52-8BA4-A8480E62D389" alt="S(\widehat{g})=\Lambda \int_M  \left( R(g) - \frac{1}{\Lambda^2} \vert F \vert^2  \right)  \;\mbox{vol}(g)" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971"><P>Varying the action with respect to the component <I>A</I>, one regains the Maxwell equations. Applying the variational principle to the base metric <I>g</I>, one gets the Einstein equations</P></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971"><div align="center"><img src="http://content9.clipmarks.com/blog_cache/en.wikipedia.org/img/FE820627-1C4D-4FAC-8365-F254042D57C7" alt="R_{ij} - \frac{1}{2}g_{ij}R = \frac{1}{\Lambda^2} T_{ij}" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971"><P>with the <A title="Stress-energy tensor" href="http://en.wikipedia.org/wiki/Stress-energy_tensor">stress-energy tensor</A> being given by</P></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971"><div align="center"><img src="http://content6.clipmarks.com/blog_cache/en.wikipedia.org/img/27203FE2-1CB0-4DCB-BFDE-F167A8857979" alt="T^{ij} = F^{ik}F^{jl}g_{kl}  - \frac{1}{4}g^{ij} \vert F \vert^2" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971"><P>sometimes called the <B><A title="Maxwell stress tensor" href="http://en.wikipedia.org/wiki/Maxwell_stress_tensor">Maxwell stress tensor</A></B>.</P></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&oldid=137155971"><P>The original theory identifies <SPAN class="texhtml">Λ</SPAN> with the fiber metric <SPAN class="texhtml"><I>g</I><SUB>55</SUB></SPAN>, and allows <SPAN class="texhtml">Λ</SPAN> to vary from fiber to fiber. In this case, the coupling between gravity and the electromagnetic field is not constant, but has its own dynamical field, the <A title="Radion" href="http://en.wikipedia.org/wiki/Radion">radion</A>.</P></blockquote></div><div style="margin: 0px 6px 6px 4px;"><table style="font-size: 11px;border-spacing: 0px;padding: 0px;" cellpadding="0" cellspacing="0" width="100%"><tr><td style="background:transparent;border-width:0px;padding:0px;">&nbsp;</td><td align="right" style="background:transparent;border-width:0px;padding:0px;width:107px" width="107"><a href="http://clipmarks.com/share/CB3116CB-9CB0-44E6-82AD-6BD198E51464/blog/" title="blog or email this clip"><img src="http://content7.clipmarks.com/images/c2b-foot.png" border="0" alt="blog it" width="107" height="17" style="border-width:0px;padding:0px;margin:0px;" /></a></td></tr></table></div></div>