en>MZMcBride: -:Category:Article Feedback 5

2012-12-24T10:41:27Z

← Older revision		Revision as of 12:41, 24 December 2012
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	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Andrew~~ and ~~my spouse doesn~~'~~t like it at all~~. ~~He is an purchase clerk~~ and it'~~s something he truly enjoy. To perform lacross~~ is ~~something he would never give up~~. ~~Ohio~~ is ~~where his home~~ is ~~and his family members loves~~ it.<br><br>~~My web site~~: [~~http~~://~~www~~.~~sirudang~~.~~com~~/~~siroo_Notice/2110 tarot readings~~]		This article contains proof of [[List of formulas in Riemannian geometry\|formulas in Riemannian geometry]] that involve the [[Christoffel symbols]].

			==Contracted Bianchi identities==

			===Proof===
			Start with the [[Curvature form#Bianchi identities\|Bianchi identity]]<ref name="syngeandschild">{{cite book \|author=Synge J.L., Schild A.\|title=Tensor Calculus\|year= 1949\|pages=87–89-90}}</ref>
			:<math> R_{abmn;l} + R_{ablm;n} + R_{abnl;m} = 0\,\!</math>.<!-- The symbols \,\! exist to force PNG rendering; do not remove them. -->

			[[Tensor contraction\|Contract]] both sides of the above equation with a pair of [[metric tensor]]s:
			:<math> g^{bn} g^{am} (R_{abmn;l} + R_{ablm;n} + R_{abnl;m}) = 0,\,\!</math>

			:<math> g^{bn} (R^m {}_{bmn;l} - R^m {}_{bml;n} + R^m {}_{bnl;m}) = 0,\,\!</math>

			:<math> g^{bn} (R_{bn;l} - R_{bl;n} - R_b {}^m {}_{nl;m}) = 0,\,\!</math>

			:<math> R^n {}_{n;l} - R^n {}_{l;n} - R^{nm} {}_{nl;m} = 0.\,\!</math>
			The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a [[mixed tensor\|mixed]] Ricci tensor,
			:<math> R_{;l} - R^n {}_{l;n} - R^m {}_{l;m} = 0.\,\!</math>
			The last two terms are the same (changing dummy index ''n'' to ''m'') and can be combined into a single term which shall be moved to the right,
			:<math> R_{;l} = 2 R^m {}_{l;m},\,\!</math>
			which is the same as
			:<math> \nabla_m R^m {}_l = {1 \over 2} \nabla_l R\,\!</math>.
			Swapping the index labels ''l'' and ''m'' yields
			:<math> \nabla_l R^l {}_m = {1 \over 2} \nabla_m R\,\!</math>,      ''[[Q.E.D.]]''     (''[[Riemannian geometry cheat sheet#Ricci and scalar curvatures\|return to article]]'')

			==The covariant divergence of the Einstein tensor vanishes==

			===Proof===
			The last equation in Proof 1 above can be expressed as
			:<math> \nabla_l R^l {}_m - {1 \over 2} \delta^l {}_m \nabla_l R = 0\,\!</math>
			where δ is the [[Kronecker delta]]. Since the mixed Kronecker delta is equivalent to the mixed metric tensor,
			:<math> \delta^l {}_m = g^l {}_m,\,\!</math>
			and since the [[covariant derivative]] of the metric tensor is zero (so it can be moved in or out of the scope of any such derivative), then
			:<math> \nabla_l R^l {}_m - {1 \over 2} \nabla_l g^l {}_m R = 0.\,\!</math>
			Factor out the covariant derivative
			:<math> \nabla_l \left(R^l {}_m - {1 \over 2} g^l {}_m R\right) = 0,\,\!</math>
			then raise the index ''m'' throughout
			:<math> \nabla_l \left(R^{lm} - {1 \over 2} g^{lm} R\right) = 0.\,\!</math>
			The expression in parentheses is the [[Einstein tensor]], so <ref name="syngeandschild" />
			:<math> \nabla_l G^{lm} = 0,\,\!</math>     ''[[Q.E.D.]]''    (''[[Riemannian geometry cheat sheet#Einstein tensor\|return to article]]'')

			this means that the covariant divergence of the Einstein tensor vanishes.

			==See also==
			*[[Four-gradient]]
			*[[Ricci calculus]]

			==References==

			{{reflist}}

			==Books==
			* {{citation \| last1=Bishop\|first1=R.L.\|last2=Goldberg\|first2=S.I. \| title = Tensor Analysis on Manifolds\| publisher=The Macmillan Company \| year=1968\|edition=First Dover 1980\|isbn=0-486-64039-6}}
			*{{cite book
			\| last = Danielson
			\| first = Donald A.
			\| title = Vectors and Tensors in Engineering and Physics
			\| edition = 2/e
			\| year= 2003
			\| publisher = Westview (Perseus)
			\| isbn = 978-0-8133-4080-7
			}}
			*{{cite book
			\| last = Lovelock
			\| first = David
			\| coauthors = Hanno Rund
			\| title = Tensors, Differential Forms, and Variational Principles
			\| year= 1989
			\| publisher = Dover
			\| isbn = 978-0-486-65840-7
			\| origyear = 1975
			}}
			* {{cite book \|author=Synge J.L., Schild A. \|title=Tensor Calculus \|publisher=first Dover Publications 1978 edition \|year= 1949
			\|isbn=978-0-486-63612-2}}
			* {{citation \| author=J.R. Tyldesley\| title = An introduction to Tensor Analysis: For Engineers and Applied Scientists\| publisher=Longman\| year=1975\|isbn=0-582-44355-5}}
			* {{citation \| author=D.C. Kay\| title = Tensor Calculus\| publisher=Schaum’s Outlines, McGraw Hill (USA)\|year=1988\|isbn=0-07-033484-6}}
			* {{citation \| author=T. Frankel\| title = The Geometry of Physics\| publisher=Cambridge University Press\|edition=3rd\|year=2012\|isbn=978-1107-602601}}

			{{DEFAULTSORT:Christoffel Symbols/Proofs}}
			[[Category:Riemannian geometry\|*]]
			[[Category:Article proofs]]
			[[Category:Differential geometry]]
			[[Category:Tensors]]


			{{differential-geometry-stub}}

en>Michael Hardy at 03:09, 9 May 2012

2012-05-09T03:09:12Z

New page

I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. He is an purchase clerk and it's something he truly enjoy. To perform lacross is something he would never give up. Ohio is where his home is and his family members loves it.<br><br>My web site: [http://www.sirudang.com/siroo_Notice/2110 tarot readings]

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en>MZMcBride: -:Category:Article Feedback 5

en>Michael Hardy at 03:09, 9 May 2012