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

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

link_hyrule5

6-30-2007 1:46 PM

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tags:

einstein, relativity, maxwell, em, e&m, electromagnetism, quantum theory of gravity, 5th dimension

link_hyrule5 says:

Part 2 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/7aefce51-822c-4075-a9d2-569fddd1af92/213836E9-E08D-443A-B09C-F77C7F863613/" 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="mw-headline">The Kaluza–Klein geometry</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>To build the Kaluza–Klein theory, one picks an invariant metric on the circle <SPAN class="texhtml"><I>S</I><SUP>1</SUP></SPAN> that is the fiber of the <I>U</I>(1)-bundle of electromagnetism. In this discussion, an <I>invariant metric</I> is simply one that is invariant under rotations of the circle. Suppose this metric gives the circle a total length of <SPAN class="texhtml">Λ</SPAN>. One then considers metrics <IMG alt="\widehat{g}" src="http://upload.wikimedia.org/math/6/2/9/6297bb38f5af8db97918365d207b4c99.png" class="tex" /> on the bundle <I>P</I> that are consistent with both the fiber metric, and the metric on the underlying manifold <I>M</I>. The consistency conditions are:</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"><li style="margin-left:16px;padding-left: 0px;">The projection of <IMG alt="\widehat{g}" src="http://upload.wikimedia.org/math/6/2/9/6297bb38f5af8db97918365d207b4c99.png" class="tex" /> to the vertical subspace <IMG alt="\mbox{Vert}_pP \subset T_pP" src="http://upload.wikimedia.org/math/9/e/4/9e499e6d7158b164850b2e9a62cf18f9.png" class="tex" /> needs to agree with metric on the fiber over a point in the manifold <I>M</I>.</LI></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"><li style="margin-left:16px;padding-left: 0px;">The projection of <IMG alt="\widehat{g}" src="http://upload.wikimedia.org/math/6/2/9/6297bb38f5af8db97918365d207b4c99.png" class="tex" /> to the horizontal subspace <IMG alt="\mbox{Hor}_pP \subset T_pP" src="http://upload.wikimedia.org/math/5/8/f/58f46e9736cfa77168e55858af03c155.png" class="tex" /> of the <A title="Tangent space" href="http://en.wikipedia.org/wiki/Tangent_space">tangent space</A> at point <IMG alt="p\in P" src="http://upload.wikimedia.org/math/9/1/3/913555e444ef02353e16ac6d9fecb58d.png" class="tex" /> must be isomorphic to the metric <I>g</I> on <I>M</I> at <SPAN class="texhtml">π(<I>p</I>)</SPAN>.</LI></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 Kaluza–Klein action for such a metric is 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://content7.clipmarks.com/blog_cache/en.wikipedia.org/img/9D3932AB-2A82-4ADD-B49B-93B7716BF173" alt="S(\widehat{g})=\int_P R(\widehat{g}) \;\mbox{vol}(\widehat{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>The scalar curvature, written in components, then expands to</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/3B8B8CB6-965B-4862-A2AD-1CE10003BA38" alt="R(\widehat{g}) = \pi^*\left(  R(g) - \frac{\Lambda^2}{2} \vert F \vert^2 \right)" /></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>where <SPAN class="texhtml">π <SUP>*</SUP></SPAN> is the <A title="Pullback (differential geometry)" href="http://en.wikipedia.org/wiki/Pullback_%28differential_geometry%29">pullback</A> of the fiber bundle projection <IMG alt="\pi:P\to M" src="http://upload.wikimedia.org/math/c/0/c/c0cb05909f93dbdd0468e264865bbda1.png" class="tex" />. The connection <I>A</I> on the fiber bundle is related to the electromagnetic field strength as</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/213836E9-E08D-443A-B09C-F77C7F863613/blog/" title="blog or email this clip"><img src="http://content9.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>