<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/df85d8fc-83a8-4255-bef6-0200b8cbce19/A19E5475-39A1-487C-91B9-5E212D1A503F/" 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"><P>The equations governing ordinary gravity in free space can be obtained from an action, by applying the <A title="Variational principle" href="http://en.wikipedia.org/wiki/Variational_principle">variational principle</A> to a certain <A title="Action (physics)" href="http://en.wikipedia.org/wiki/Action_%28physics%29">action</A>. Let <I>M</I> be a (<A title="Pseudo-Riemannian manifold" href="http://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold">pseudo-</A>)<A title="Riemannian manifold" href="http://en.wikipedia.org/wiki/Riemannian_manifold">Riemannian manifold</A>, which may be taken as the <A title="Spacetime" href="http://en.wikipedia.org/wiki/Spacetime">spacetime</A> of <A title="General relativity" href="http://en.wikipedia.org/wiki/General_relativity">general relativity</A>. If <I>g</I> is the <A title="Metric (mathematics)" href="http://en.wikipedia.org/wiki/Metric_%28mathematics%29">metric</A> on this manifold, one defines the <A title="Action (physics)" href="http://en.wikipedia.org/wiki/Action_%28physics%29">action</A> <SPAN class="texhtml"><I>S</I>(<I>g</I>)</SPAN> as</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"><DD><IMG alt="S(g)=\int_M R(g) \mbox{vol}(g)\," src="http://upload.wikimedia.org/math/b/f/a/bfa662867010323c4a2d3dd1f456f34e.png" class="tex" /></DD></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 <I>R(g)</I> is the <A title="Scalar curvature" href="http://en.wikipedia.org/wiki/Scalar_curvature">scalar curvature</A> and vol(<I>g</I>) is the <A title="Volume element" href="http://en.wikipedia.org/wiki/Volume_element">volume element</A>. By applying the <A title="Variational principle" href="http://en.wikipedia.org/wiki/Variational_principle">variational principle</A> to the action</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"><DD><IMG alt="\frac{\delta S(g)}{\delta g} = 0" src="http://upload.wikimedia.org/math/c/8/0/c801e8dad308967f7e5199d55871a70c.png" class="tex" /></DD></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>one obtains precisely the <A title="Einstein equation" href="http://en.wikipedia.org/wiki/Einstein_equation">Einstein equations</A> for free space:</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"><DD><IMG alt="R_{ij} - \frac{1}{2}g_{ij}R = 0" src="http://upload.wikimedia.org/math/b/2/e/b2e75e02ade04120446936c9a73d3c62.png" class="tex" /></DD></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>Here, <SPAN class="texhtml"><I>R</I><SUB><I>i</I><I>j</I></SUB></SPAN> is the <A title="Ricci tensor" href="http://en.wikipedia.org/wiki/Ricci_tensor">Ricci tensor</A>.</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"><H3><SPAN class="editsection">[<A title="Edit section: The Maxwell equations" href="http://en.wikipedia.org/w/index.php?title=Kaluza%E2%80%93Klein_theory&action=edit&section=5">edit</A>]</SPAN> <SPAN class="mw-headline">The Maxwell equations</SPAN></H3></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>By contrast, the <A title="Maxwell equation" href="http://en.wikipedia.org/wiki/Maxwell_equation">Maxwell equations</A> describing <A title="Electromagnetism" href="http://en.wikipedia.org/wiki/Electromagnetism">electromagnetism</A> can be understood to be the <A title="De Rham cohomology" href="http://en.wikipedia.org/wiki/De_Rham_cohomology">Hodge equations</A> of a <A title="Principal bundle" href="http://en.wikipedia.org/wiki/Principal_bundle">principal <I>U</I>(1)-bundle</A> or <A title="Circle bundle" href="http://en.wikipedia.org/wiki/Circle_bundle">circle bundle</A> <IMG alt="\pi:P\to M" src="http://upload.wikimedia.org/math/c/0/c/c0cb05909f93dbdd0468e264865bbda1.png" class="tex" /> with fiber <A title="U(1)" href="http://en.wikipedia.org/wiki/U%281%29"><I>U</I>(1)</A>. This is, the <A title="Electromagnetic field" href="http://en.wikipedia.org/wiki/Electromagnetic_field">electromagnetic field</A> <I>F</I> is a <A title="Harmonic form" href="http://en.wikipedia.org/wiki/Harmonic_form">harmonic 2-form</A> in the space <SPAN class="texhtml">Ω<SUP>2</SUP>(<I>M</I>)</SPAN> of differentiable <A title="2-form" href="http://en.wikipedia.org/wiki/2-form">2-forms</A> on the manifold <I>M</I>. In the absence of charges and currents, the free-field Maxwell equations 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"><DD><IMG alt="\mathrm{d}F=0\,\quad" src="http://upload.wikimedia.org/math/f/6/f/f6f0c3d6d7bf63d6f6b0f19a920e16ae.png" class="tex" /> and <IMG alt="\quad \mathrm{d}*F=0\," src="http://upload.wikimedia.org/math/a/6/2/a622ada196dff039e8aceac5b5bd37ba.png" class="tex" /></DD></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 * is the <A title="Hodge star" href="http://en.wikipedia.org/wiki/Hodge_star">Hodge star</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/A19E5475-39A1-487C-91B9-5E212D1A503F/blog/" title="blog or email this clip"><img src="http://content6.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>

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