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Christoffel symbols

From Wikipedia, the free encyclopedia
Array of numbers describing a metric connection

Inmathematics andphysics, theChristoffel symbols are an array of numbers describing ametric connection.[1] The metric connection is a specialization of theaffine connection tosurfaces or othermanifolds endowed with ametric, allowing distances to be measured on that surface. Indifferential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow:parallel transport,covariant derivatives,geodesics, etc. also do not require the concept of a metric.[2][3] However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how thetangent space is attached to thecotangent space by themetric tensor.[4] Abstractly, one would say that the manifold has an associated (orthonormal)frame bundle, with each "frame" being a possible choice of acoordinate frame. An invariant metric implies that thestructure group of the frame bundle is theorthogonal groupO(p,q). As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold.[5][6] The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.

In general, there are an infinite number of metric connections for a givenmetric tensor; however, there is a unique connection that is free oftorsion, theLevi-Civita connection. It is common in physics andgeneral relativity to work almost exclusively with the Levi-Civita connection, by working incoordinate frames (calledholonomic coordinates) where the torsion vanishes. For example, inEuclidean spaces, the Christoffel symbols describe how thelocal coordinate bases change from point to point.

At each point of the underlyingn-dimensional manifold, for any local coordinate system around that point, the Christoffel symbols are denotedΓijk fori,j,k = 1, 2, ...,n. Each entry of thisn ×n ×narray is areal number. Underlinearcoordinate transformations on the manifold, the Christoffel symbols transform like the components of atensor, but under general coordinate transformations (diffeomorphisms) they do not. Most of the algebraic properties of the Christoffel symbols follow from their relationship to the affine connection; only a few follow from the fact that thestructure group is the orthogonal groupO(m,n) (or theLorentz groupO(3, 1) for general relativity).

Christoffel symbols are used for performing practical calculations. For example, theRiemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their firstpartial derivatives. Ingeneral relativity, the connection plays the role of thegravitational force field with the correspondinggravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of theΓijk arezero.

The Christoffel symbols are named forElwin Bruno Christoffel (1829–1900).[7]

Note

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The definitions given below are valid for bothRiemannian manifolds andpseudo-Riemannian manifolds, such as those ofgeneral relativity, with careful distinction being made between upper and lower indices (contra-variant and co-variant indices). The formulas hold for eithersign convention, unless otherwise noted.

Einstein summation convention is used in this article, with vectors indicated by bold font. Theconnection coefficients of theLevi-Civita connection (or pseudo-Riemannian connection) expressed in a coordinate basis are calledChristoffel symbols.

Preliminary definitions

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Main article:Tangent space

Given amanifoldM{\displaystyle M}, anatlas consists of a collection of chartsφ:U→Rn{\displaystyle \varphi :U\to \mathbb {R} ^{n}} for eachopen coverU⊂M{\displaystyle U\subset M}. Such charts allow the standardvector basis(e→1,⋯,e→n){\displaystyle ({\vec {e}}_{1},\cdots ,{\vec {e}}_{n})} onRn{\displaystyle \mathbb {R} ^{n}} to bepulled back to a vector basis on the tangent spaceTM{\displaystyle TM} ofM{\displaystyle M}. This is done as follows. Given some arbitrary real functionf:M→R{\displaystyle f:M\to \mathbb {R} }, the chart allows agradient to be defined:

∂if≡∂(f∘φ−1)∂xifor i=1,2,…,n{\displaystyle \partial _{i}f\equiv {\frac {\partial \left(f\circ \varphi ^{-1}\right)}{\partial x^{i}}}\quad {\mbox{for }}i=1,\,2,\,\dots ,\,n}

This gradient is commonly called apullback because it "pulls back" the gradient onRn{\displaystyle \mathbb {R} ^{n}} to a gradient onM{\displaystyle M}. The pullback is independent of the chartφ{\displaystyle \varphi }. In this way, the standard vector basis(e→1,⋯,e→n){\displaystyle ({\vec {e}}_{1},\cdots ,{\vec {e}}_{n})} onRn{\displaystyle \mathbb {R} ^{n}} pulls back to a standard ("coordinate") vector basis(∂1,⋯,∂n){\displaystyle (\partial _{1},\cdots ,\partial _{n})} onTM{\displaystyle TM}. This is called the "coordinate basis", because it explicitly depends on the coordinates onRn{\displaystyle \mathbb {R} ^{n}}. It is sometimes called the "local basis".

This definition allows a commonabuse of notation. The∂i{\displaystyle \partial _{i}} were defined to be in one-to-one correspondence with the basis vectorse→i{\displaystyle {\vec {e}}_{i}} onRn{\displaystyle \mathbb {R} ^{n}}. The notation∂i{\displaystyle \partial _{i}} serves as a reminder that the basis vectors on the tangent spaceTM{\displaystyle TM} came from a gradient construction. Despite this, it is common to "forget" this construction, and just write (or rather, define) vectorsei{\displaystyle e_{i}} onTM{\displaystyle TM} such thatei≡∂i{\displaystyle e_{i}\equiv \partial _{i}}. The full range of commonly used notation includes the use of arrows and boldface to denote vectors:

∂i≡∂∂xi≡ei≡e→i≡ei≡∂i{\displaystyle \partial _{i}\equiv {\frac {\partial }{\partial x^{i}}}\equiv e_{i}\equiv {\vec {e}}_{i}\equiv \mathbf {e} _{i}\equiv {\boldsymbol {\partial }}_{i}}

where≡{\displaystyle \equiv } is used as a reminder that these are defined to be equivalent notation for the same concept. The choice of notation is according to style and taste, and varies from text to text.

The coordinate basis provides a vector basis forvector fields onM{\displaystyle M}. Commonly used notation for vector fields onM{\displaystyle M} include

X=X→=Xi∂i=Xi∂∂xi{\displaystyle X={\vec {X}}=X^{i}\partial _{i}=X^{i}{\frac {\partial }{\partial x^{i}}}}

The upper-caseX{\displaystyle X}, without the vector-arrow, is particularly popular forindex-free notation, because it both minimizes clutter and reminds that results are independent of the chosen basis, and, in this case, independent of the atlas.

The same abuse of notation is used topush forwardone-forms fromRn{\displaystyle \mathbb {R} ^{n}} toM{\displaystyle M}. This is done by writing(φ1,…,φn)=(x1,…,xn){\displaystyle (\varphi ^{1},\ldots ,\varphi ^{n})=(x^{1},\ldots ,x^{n})} orx=φ{\displaystyle x=\varphi } orxi=φi{\displaystyle x^{i}=\varphi ^{i}}. The one-form is thendxi=dφi{\displaystyle dx^{i}=d\varphi ^{i}}. This is soldered to the basis vectors asdxi(∂j)=δji{\displaystyle dx^{i}(\partial _{j})=\delta _{j}^{i}}. Note the careful use of upper and lower indexes, to distinguish contravariant and covariant vectors.

The pullback induces (defines) ametric tensor onM{\displaystyle M}. Several styles of notation are commonly used:gij=ei⋅ej=⟨e→i,e→j⟩=eiaejbηab{\displaystyle g_{ij}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}=\langle {\vec {e}}_{i},{\vec {e}}_{j}\rangle =e_{i}^{a}e_{j}^{b}\,\eta _{ab}}where both the centerdot and the angle-bracket⟨,⟩{\displaystyle \langle ,\rangle } denote thescalar product. The last form uses thetensorηab{\displaystyle \eta _{ab}}, which is understood to be the "flat-space" metric tensor. ForRiemannian manifolds, it is theKronecker deltaηab=δab{\displaystyle \eta _{ab}=\delta _{ab}}. Forpseudo-Riemannian manifolds, it is the diagonal matrix having signature(p,q){\displaystyle (p,q)}. The notationeia{\displaystyle e_{i}^{a}} serves as a reminder that pullback really is a linear transform, given as the gradient, above. The index lettersa,b,c,⋯{\displaystyle a,b,c,\cdots } live inRn{\displaystyle \mathbb {R} ^{n}} while the index lettersi,j,k,⋯{\displaystyle i,j,k,\cdots } live in the tangent manifold.

Thematrix inversegij{\displaystyle g^{ij}} of the metric tensorgij{\displaystyle g_{ij}} is given bygijgjk=δki{\displaystyle g^{ij}g_{jk}=\delta _{k}^{i}}This is used to define the dual basis:ei=ejgji,i=1,2,…,n{\displaystyle \mathbf {e} ^{i}=\mathbf {e} _{j}g^{ji},\quad i=1,\,2,\,\dots ,\,n}

Some texts writegi{\displaystyle \mathbf {g} _{i}} forei{\displaystyle \mathbf {e} _{i}}, so that the metric tensor takes the particularly beguiling formgij=gi⋅gj{\displaystyle g_{ij}=\mathbf {g} _{i}\cdot \mathbf {g} _{j}}. This is commonly done so that the symbolei{\displaystyle e_{i}} can be used unambiguously for thevierbein.

Definition in Euclidean space

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InEuclidean space, the general definition given below for the Christoffel symbols of the second kind can be proven to be equivalent to:Γkij=∂ej∂xi⋅ek=∂ej∂xi⋅gkmem{\displaystyle {\Gamma ^{k}}_{ij}={\frac {\partial \mathbf {e} _{j}}{\partial x^{i}}}\cdot \mathbf {e} ^{k}={\frac {\partial \mathbf {e} _{j}}{\partial x^{i}}}\cdot g^{km}\mathbf {e} _{m}}

Christoffel symbols of the first kind can then be found viaindex lowering:Γkij=Γmijgmk=∂ej∂xi⋅emgmk=∂ej∂xi⋅ek{\displaystyle \Gamma _{kij}={\Gamma ^{m}}_{ij}g_{mk}={\frac {\partial \mathbf {e} _{j}}{\partial x^{i}}}\cdot \mathbf {e} ^{m}g_{mk}={\frac {\partial \mathbf {e} _{j}}{\partial x^{i}}}\cdot \mathbf {e} _{k}}

Rearranging, we see that (assuming the partial derivative belongs to the tangent space, which cannot occur on a non-Euclideancurved space):∂ej∂xi=Γkijek=Γkijek{\displaystyle {\frac {\partial \mathbf {e} _{j}}{\partial x^{i}}}={\Gamma ^{k}}_{ij}\mathbf {e} _{k}=\Gamma _{kij}\mathbf {e} ^{k}}

In words, the arrays represented by the Christoffel symbols track how the basis changes from point to point. If the derivative does not lie on the tangent space, the right expression is the projection of the derivative over the tangent space (seecovariant derivative below). Symbols of the second kind decompose the change with respect to the basis, while symbols of the first kind decompose it with respect to the dual basis. In this form, it is easy to see the symmetry of the lower or last two indices:Γkij=Γkji{\displaystyle {\Gamma ^{k}}_{ij}={\Gamma ^{k}}_{ji}} andΓkij=Γkji,{\displaystyle \Gamma _{kij}=\Gamma _{kji},}from the definition ofei{\displaystyle \mathbf {e} _{i}} and the fact that partial derivatives commute (as long as the manifold and coordinate systemare well behaved).

The same numerical values for Christoffel symbols of the second kind also relate to derivatives of the dual basis, as seen in the expression:∂ej∂xi=−Γijkek,{\displaystyle {\frac {\partial \mathbf {e} ^{j}}{\partial x^{i}}}=-{\Gamma ^{i}}_{jk}\mathbf {e} ^{k},}which we can rearrange as:Γijk=−∂ei∂xj⋅ek.{\displaystyle {\Gamma ^{i}}_{jk}=-{\frac {\partial \mathbf {e} ^{i}}{\partial x^{j}}}\cdot \mathbf {e} _{k}.}

General definition

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The Christoffel symbols come in two forms: the first kind, and the second kind. The definition of the second kind is more basic, and thus is presented first.

Christoffel symbols of the second kind (symmetric definition)

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The Christoffel symbols of the second kind are the connection coefficients—in a coordinate basis—of theLevi-Civita connection.In other words, the Christoffel symbols of the second kind[8][9]Γkij (sometimesΓk
ij or{k
ij})[7][8] are defined as the unique coefficients such that∇iej=Γkijek,{\displaystyle \nabla _{i}\mathrm {e} _{j}={\Gamma ^{k}}_{ij}\mathrm {e} _{k},}where∇i is theLevi-Civita connection onM taken in the coordinate directionei (i.e.,∇i ≡ ∇ei) and whereei = ∂i is a local coordinate (holonomic)basis. Since this connection has zerotorsion, and holonomic vector fields commute (i.e.[ei,ej]=[∂i,∂j]=0{\displaystyle [e_{i},e_{j}]=[\partial _{i},\partial _{j}]=0}) we have∇iej=∇jei.{\displaystyle \nabla _{i}\mathrm {e} _{j}=\nabla _{j}\mathrm {e} _{i}.}Hence in this basis the connection coefficients are symmetric:[8]Γkij=Γkji.{\displaystyle {\Gamma ^{k}}_{ij}={\Gamma ^{k}}_{ji}.}For this reason, a torsion-free connection is often calledsymmetric.

The Christoffel symbols can be derived from the vanishing of thecovariant derivative of themetric tensorgik:0=∇lgik=∂gik∂xl−gmkΓmil−gimΓmkl=∂gik∂xl−2gm(kΓmi)l.{\displaystyle 0=\nabla _{l}g_{ik}={\frac {\partial g_{ik}}{\partial x^{l}}}-g_{mk}{\Gamma ^{m}}_{il}-g_{im}{\Gamma ^{m}}_{kl}={\frac {\partial g_{ik}}{\partial x^{l}}}-2g_{m(k}{\Gamma ^{m}}_{i)l}.}

As a shorthand notation, thenabla symbol and the partial derivative symbols are frequently dropped, and instead asemicolon and acomma are used to set off the index that is being used for the derivative. Thus, the above is sometimes written as0=gik;l=gik,l−gmkΓmil−gimΓmkl.{\displaystyle 0=\,g_{ik;l}=g_{ik,l}-g_{mk}{\Gamma ^{m}}_{il}-g_{im}{\Gamma ^{m}}_{kl}.}

Using that the symbols are symmetric in the lower two indices, one can solve explicitly for the Christoffel symbols as a function of the metric tensor by permuting the indices and resumming:[10]Γikl=12gim(∂gmk∂xl+∂gml∂xk−∂gkl∂xm)=12gim(gmk,l+gml,k−gkl,m),{\displaystyle {\Gamma ^{i}}_{kl}={\frac {1}{2}}g^{im}\left({\frac {\partial g_{mk}}{\partial x^{l}}}+{\frac {\partial g_{ml}}{\partial x^{k}}}-{\frac {\partial g_{kl}}{\partial x^{m}}}\right)={\frac {1}{2}}g^{im}\left(g_{mk,l}+g_{ml,k}-g_{kl,m}\right),}

where(gjk) is the inverse of thematrix(gjk), defined as (using theKronecker delta, andEinstein notation for summation)gjigik =δ jk. Although the Christoffel symbols are written in the same notation astensors with index notation, they do not transform like tensors undera change of coordinates.

Contraction of indices

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Contracting the upper index with either of the lower indices (those being symmetric) leads toΓiki=∂∂xkln⁡|g|{\displaystyle {\Gamma ^{i}}_{ki}={\frac {\partial }{\partial x^{k}}}\ln {\sqrt {|g|}}}whereg=detgik{\displaystyle g=\det g_{ik}} is the determinant of the metric tensor. This identity can be used to evaluate the divergence of vectors and the covariant derivatives oftensor densities. Also

Γiki=Γiik=12(gmigmk,i+gmigmi,k−gimgki,m)=12gmigmi,k{\displaystyle {\Gamma ^{i}}_{ki}={\Gamma ^{i}}_{ik}={\tfrac {1}{2}}\left(g^{mi}g_{mk,i}+g^{mi}g_{mi,k}-g^{im}g_{ki,m}\right)={\tfrac {1}{2}}g^{mi}g_{mi,k}} .

Christoffel symbols of the first kind

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The Christoffel symbols of the first kind can be derived either from the Christoffel symbols of the second kind and the metric,[11]Γcab=gcdΓdab,{\displaystyle \Gamma _{cab}=g_{cd}{\Gamma ^{d}}_{ab}\,,}

or from the metric alone,[11]Γcab=12(∂gca∂xb+∂gcb∂xa−∂gab∂xc)=12(gca,b+gcb,a−gab,c)=12(∂bgca+∂agcb−∂cgab).{\displaystyle {\begin{aligned}\Gamma _{cab}&={\frac {1}{2}}\left({\frac {\partial g_{ca}}{\partial x^{b}}}+{\frac {\partial g_{cb}}{\partial x^{a}}}-{\frac {\partial g_{ab}}{\partial x^{c}}}\right)\\&={\frac {1}{2}}\,\left(g_{ca,b}+g_{cb,a}-g_{ab,c}\right)\\&={\frac {1}{2}}\,\left(\partial _{b}g_{ca}+\partial _{a}g_{cb}-\partial _{c}g_{ab}\right)\,.\\\end{aligned}}}

As an alternative notation one also finds[7][12][13]

Γcab=[ab,c].{\displaystyle \Gamma _{cab}=[ab,c].}It is worth noting that[ab,c] = [ba,c].[10]

Connection coefficients in a nonholonomic basis

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The Christoffel symbols are most typically defined in a coordinate basis, which is the convention followed here. In other words, the nameChristoffel symbols is reserved only for coordinate (i.e.,holonomic) frames. However, the connection coefficients can also be defined in an arbitrary (i.e., nonholonomic) basis of tangent vectorsui by∇uiuj=ωkijuk.{\displaystyle \nabla _{\mathbf {u} _{i}}\mathbf {u} _{j}={\omega ^{k}}_{ij}\mathbf {u} _{k}.}

Explicitly, in terms of the metric tensor, this is[9]ωikl=12gim(gmk,l+gml,k−gkl,m+cmkl+cmlk−cklm),{\displaystyle {\omega ^{i}}_{kl}={\frac {1}{2}}g^{im}\left(g_{mk,l}+g_{ml,k}-g_{kl,m}+c_{mkl}+c_{mlk}-c_{klm}\right),}

wherecklm =gmpcklp are thecommutation coefficients of the basis; that is,[uk,ul]=cklmum{\displaystyle [\mathbf {u} _{k},\,\mathbf {u} _{l}]={c_{kl}}^{m}\mathbf {u} _{m}}

whereuk are the basisvectors and[ , ] is theLie bracket. The standard unit vectors inspherical and cylindrical coordinates furnish an example of a basis with non-vanishing commutation coefficients. The difference between the connection in such a frame, and the Levi-Civita connection is known as thecontorsion tensor.

Ricci rotation coefficients (asymmetric definition)

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When we choose the basisXi ≡ui orthonormal:gab ≡ηab = ⟨Xa,Xb⟩ thengmk,l ≡ηmk,l = 0. This implies thatωikl=12ηim(cmkl+cmlk−cklm){\displaystyle {\omega ^{i}}_{kl}={\frac {1}{2}}\eta ^{im}\left(c_{mkl}+c_{mlk}-c_{klm}\right)}and the connection coefficients become antisymmetric in the first two indices:ωabc=−ωbac,{\displaystyle \omega _{abc}=-\omega _{bac}\,,}whereωabc=ηadωdbc.{\displaystyle \omega _{abc}=\eta _{ad}{\omega ^{d}}_{bc}\,.}

In this case, the connection coefficientsωabc are called theRicci rotation coefficients.[14][15]

Equivalently, one can define Ricci rotation coefficients as follows:[9]ωkij:=uk⋅(∇jui),{\displaystyle {\omega ^{k}}_{ij}:=\mathbf {u} ^{k}\cdot \left(\nabla _{j}\mathbf {u} _{i}\right)\,,}whereui is an orthonormal nonholonomic basis anduk =ηklul itsco-basis.

Transformation law under change of variable

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Under a change of variable from(x1,…,xn){\displaystyle \left(x^{1},\,\ldots ,\,x^{n}\right)} to(x¯1,…,x¯n){\displaystyle \left({\bar {x}}^{1},\,\ldots ,\,{\bar {x}}^{n}\right)}, Christoffel symbols transform as

Γ¯ikl=∂x¯i∂xm∂xn∂x¯k∂xp∂x¯lΓmnp+∂2xm∂x¯k∂x¯l∂x¯i∂xm{\displaystyle {{\bar {\Gamma }}^{i}}_{kl}={\frac {\partial {\bar {x}}^{i}}{\partial x^{m}}}\,{\frac {\partial x^{n}}{\partial {\bar {x}}^{k}}}\,{\frac {\partial x^{p}}{\partial {\bar {x}}^{l}}}\,{\Gamma ^{m}}_{np}+{\frac {\partial ^{2}x^{m}}{\partial {\bar {x}}^{k}\partial {\bar {x}}^{l}}}\,{\frac {\partial {\bar {x}}^{i}}{\partial x^{m}}}}

where the overline denotes the Christoffel symbols in thex¯i{\displaystyle {\bar {x}}^{i}} coordinate system. The Christoffel symbol doesnot transform as a tensor, but rather as an object in thejet bundle. More precisely, the Christoffel symbols can be considered as functions on the jet bundle of the frame bundle ofM, independent of any local coordinate system. Choosing a local coordinate system determines a local section of this bundle, which can then be used to pull back the Christoffel symbols to functions onM, though of course these functions then depend on the choice of local coordinate system.

For each point, there exist coordinate systems in which the Christoffel symbols vanish at the point.[16] These are called (geodesic)normal coordinates, and are often used inRiemannian geometry.

There are some interesting properties which can be derived directly from the transformation law.

Relationship to parallel transport and derivation of Christoffel symbols in Riemannian space

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If a vectorξi{\displaystyle \xi ^{i}} is transported parallel on a curve parametrized by some parameters{\displaystyle s} on aRiemannian manifold, the rate of change of the components of the vector is given bydξids=−Γimjdxmdsξj.{\displaystyle {\frac {d\xi ^{i}}{ds}}=-{\Gamma ^{i}}_{mj}{\frac {dx^{m}}{ds}}\xi ^{j}.}

Now just by using the condition that the scalar productgikξiηk{\displaystyle g_{ik}\xi ^{i}\eta ^{k}} formed by two arbitrary vectorsξi{\displaystyle \xi ^{i}} andηk{\displaystyle \eta ^{k}} is unchanged is enough to derive the Christoffel symbols. The condition isdds(gikξiηk)=0{\displaystyle {\frac {d}{ds}}\left(g_{ik}\xi ^{i}\eta ^{k}\right)=0}which by the product rule expands to∂gik∂xldxldsξiηk+gikdξidsηk+gikξidηkds=0.{\displaystyle {\frac {\partial g_{ik}}{\partial x^{l}}}{\frac {dx^{l}}{ds}}\xi ^{i}\eta ^{k}+g_{ik}{\frac {d\xi ^{i}}{ds}}\eta ^{k}+g_{ik}\xi ^{i}{\frac {d\eta ^{k}}{ds}}=0.}

Applying the parallel transport rule for the two arbitrary vectors and relabelling dummy indices and collecting the coefficients ofξiηkdxl{\displaystyle \xi ^{i}\eta ^{k}dx^{l}} (arbitrary), we obtain

∂gik∂xl=grkΓril+girΓrlk.{\displaystyle {\frac {\partial g_{ik}}{\partial x^{l}}}=g_{rk}{\Gamma ^{r}}_{il}+g_{ir}{\Gamma ^{r}}_{lk}.}

This is same as the equation obtained by requiring the covariant derivative of the metric tensor to vanish in the General definition section. The derivation from here is simple. By cyclically permuting the indicesikl{\displaystyle ikl} in above equation, we can obtain two more equations and then linearly combining these three equations, we can expressΓijk{\displaystyle {\Gamma ^{i}}_{jk}} in terms of the metric tensor.

Relationship to index-free notation

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LetX andY bevector fields with componentsXi andYk. Then thekth component of the covariant derivative ofY with respect toX is given by(∇XY)k=Xi(∇iY)k=Xi(∂Yk∂xi+ΓkimYm).{\displaystyle \left(\nabla _{X}Y\right)^{k}=X^{i}(\nabla _{i}Y)^{k}=X^{i}\left({\frac {\partial Y^{k}}{\partial x^{i}}}+{\Gamma ^{k}}_{im}Y^{m}\right).}

Here, theEinstein notation is used, so repeated indices indicate summation over indices and contraction with the metric tensor serves to raise and lower indices:g(X,Y)=XiYi=gikXiYk=gikXiYk.{\displaystyle g(X,Y)=X^{i}Y_{i}=g_{ik}X^{i}Y^{k}=g^{ik}X_{i}Y_{k}.}

Keep in mind thatgik ≠gik and thatgik =δ ik, theKronecker delta. The convention is that the metric tensor is the one with the lower indices; the correct way to obtaingik fromgik is to solve the linear equationsgijgjk =δ ik.

The statement that the connection istorsion-free, namely that∇XY−∇YX=[X,Y]{\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,\,Y]}

is equivalent to the statement that—in a coordinate basis—the Christoffel symbol is symmetric in the lower two indices:Γijk=Γikj.{\displaystyle {\Gamma ^{i}}_{jk}={\Gamma ^{i}}_{kj}.}

The index-less transformation properties of a tensor are given bypullbacks for covariant indices, andpushforwards for contravariant indices. The article oncovariant derivatives provides additional discussion of the correspondence between index-free notation and indexed notation.

Covariant derivatives of tensors

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Thecovariant derivative of a vector field with componentsVm is∇lVm=∂Vm∂xl+ΓmklVk.{\displaystyle \nabla _{l}V^{m}={\frac {\partial V^{m}}{\partial x^{l}}}+{\Gamma ^{m}}_{kl}V^{k}.}

By corollary, divergence of a vector can be obtained as∇iVi=1−g∂(−gVi)∂xi.{\displaystyle \nabla _{i}V^{i}={\frac {1}{\sqrt {-g}}}{\frac {\partial \left({\sqrt {-g}}\,V^{i}\right)}{\partial x^{i}}}.}

The covariant derivative of acovector fieldωm is∇lωm=∂ωm∂xl−Γkmlωk.{\displaystyle \nabla _{l}\omega _{m}={\frac {\partial \omega _{m}}{\partial x^{l}}}-{\Gamma ^{k}}_{ml}\omega _{k}.}

The symmetry of the Christoffel symbol now implies∇i∇jφ=∇j∇iφ{\displaystyle \nabla _{i}\nabla _{j}\varphi =\nabla _{j}\nabla _{i}\varphi }for any scalar field, but in general the covariant derivatives of higher order tensor fields do not commute (seecurvature tensor).

The covariant derivative of a type(2, 0)tensor fieldAik is∇lAik=∂Aik∂xl+ΓimlAmk+ΓkmlAim,{\displaystyle \nabla _{l}A^{ik}={\frac {\partial A^{ik}}{\partial x^{l}}}+{\Gamma ^{i}}_{ml}A^{mk}+{\Gamma ^{k}}_{ml}A^{im},}that is,Aik;l=Aik,l+AmkΓiml+AimΓkml.{\displaystyle {A^{ik}}_{;l}={A^{ik}}_{,l}+A^{mk}{\Gamma ^{i}}_{ml}+A^{im}{\Gamma ^{k}}_{ml}.}

If the tensor field ismixed then its covariant derivative isAik;l=Aik,l+AmkΓiml−AimΓmkl,{\displaystyle {A^{i}}_{k;l}={A^{i}}_{k,l}+{A^{m}}_{k}{\Gamma ^{i}}_{ml}-{A^{i}}_{m}{\Gamma ^{m}}_{kl},}and if the tensor field is of type(0, 2) then its covariant derivative isAik;l=Aik,l−AmkΓmil−AimΓmkl.{\displaystyle A_{ik;l}=A_{ik,l}-A_{mk}{\Gamma ^{m}}_{il}-A_{im}{\Gamma ^{m}}_{kl}.}

Contravariant derivatives of tensors

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To find the contravariant derivative of a vector field, we must first transform it into a covariant derivative using the metric tensor∇lVm=gil∇iVm=gil∂iVm+gilΓkimVk=∂lVm+gilΓkimVk{\displaystyle \nabla ^{l}V^{m}=g^{il}\nabla _{i}V^{m}=g^{il}\partial _{i}V^{m}+g^{il}\Gamma _{ki}^{m}V^{k}=\partial ^{l}V^{m}+g^{il}\Gamma _{ki}^{m}V^{k}}

Applications

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In general relativity

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The Christoffel symbols find frequent use in Einstein's theory ofgeneral relativity, wherespacetime is represented by a curved 4-dimensionalLorentz manifold with aLevi-Civita connection. TheEinstein field equations—which determine the geometry of spacetime in the presence of matter—contain theRicci tensor, and so calculating the Christoffel symbols is essential. Once the geometry is determined, the paths of particles and light beams are calculated by solving thegeodesic equations in which the Christoffel symbols explicitly appear.

In classical (non-relativistic) mechanics

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Letxi{\displaystyle x^{i}} be the generalized coordinates andx˙i{\displaystyle {\dot {x}}^{i}} be the generalized velocities, then the kinetic energy for a unit mass is given byT=12gikx˙ix˙k{\displaystyle T={\tfrac {1}{2}}g_{ik}{\dot {x}}^{i}{\dot {x}}^{k}}, wheregik{\displaystyle g_{ik}} is themetric tensor. IfV(xi){\displaystyle V\left(x^{i}\right)}, the potential function, exists then the contravariant components of the generalized force per unit mass areFi=∂V/∂xi{\displaystyle F_{i}=\partial V/\partial x^{i}}. The metric (here in a purely spatial domain) can be obtained from the line elementds2=2Tdt2{\displaystyle ds^{2}=2Tdt^{2}}. Substituting the LagrangianL=T−V{\displaystyle L=T-V} into theEuler-Lagrange equation, we get[19]

gikx¨k+12(∂gik∂xl+∂gil∂xk−∂glk∂xi)x˙lx˙k=Fi.{\displaystyle g_{ik}{\ddot {x}}^{k}+{\frac {1}{2}}\left({\frac {\partial g_{ik}}{\partial x^{l}}}+{\frac {\partial g_{il}}{\partial x^{k}}}-{\frac {\partial g_{lk}}{\partial x^{i}}}\right){\dot {x}}^{l}{\dot {x}}^{k}=F_{i}.}

Now multiplying bygij{\displaystyle g^{ij}}, we getx¨j+Γjlkx˙lx˙k=Fj.{\displaystyle {\ddot {x}}^{j}+{\Gamma ^{j}}_{lk}{\dot {x}}^{l}{\dot {x}}^{k}=F^{j}.}

When Cartesian coordinates can be adopted (as in inertial frames of reference), we have an Euclidean metrics, the Christoffel symbol vanishes, and the equation reduces toNewton's second law of motion. In curvilinear coordinates[20] (forcedly in non-inertial frames, where the metrics is non-Euclidean and not flat), fictitious forces like theCentrifugal force andCoriolis force originate from the Christoffel symbols, so from the purely spatial curvilinear coordinates.

In Earth surface coordinates

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Given aspherical coordinate system, which describes points on the Earth surface (approximated as an ideal sphere).

x(R,θ,φ)=(Rcos⁡θcos⁡φRcos⁡θsin⁡φRsin⁡θ){\displaystyle {\begin{aligned}x(R,\theta ,\varphi )&={\begin{pmatrix}R\cos \theta \cos \varphi &R\cos \theta \sin \varphi &R\sin \theta \end{pmatrix}}\\\end{aligned}}}

For a point x,R is the distance to the Earth core (usually approximately theEarth radius).θ andφ are thelatitude andlongitude. Positiveθ is the northern hemisphere. To simplify the derivatives, the angles are given inradians (where d sin(x)/dx = cos(x), the degree values introduce an additional factor of 360 / 2 pi).

At any location, the tangent directions areeR{\displaystyle e_{R}} (up),eθ{\displaystyle e_{\theta }} (north) andeφ{\displaystyle e_{\varphi }} (east) - you can also use indices 1,2,3.

eR=(cos⁡θcos⁡φcos⁡θsin⁡φsin⁡θ)eθ=R⋅(−sin⁡θcos⁡φ−sin⁡θsin⁡φcos⁡θ)eφ=Rcos⁡θ⋅(−sin⁡φcos⁡φ0){\displaystyle {\begin{aligned}e_{R}&={\begin{pmatrix}\cos \theta \cos \varphi &\cos \theta \sin \varphi &\sin \theta \end{pmatrix}}\\e_{\theta }&=R\cdot {\begin{pmatrix}-\sin \theta \cos \varphi &-\sin \theta \sin \varphi &\cos \theta \end{pmatrix}}\\e_{\varphi }&=R\cos \theta \cdot {\begin{pmatrix}-\sin \varphi &\cos \varphi &0\end{pmatrix}}\\\end{aligned}}}

The relatedmetric tensor has only diagonal elements (the squared vector lengths). This is an advantage of the coordinate system and not generally true.

[21]gRR=1gθθ=R2gφφ=R2cos2⁡θgij=0elsegRR=1gθθ=1/R2gφφ=1/(R2cos2⁡θ)gij=0else{\displaystyle {\begin{aligned}g_{RR}=1\qquad &g_{\theta \theta }=R^{2}\qquad &g_{\varphi \varphi }=R^{2}\cos ^{2}\theta \qquad &g_{ij}=0\quad \mathrm {else} \\g^{RR}=1\qquad &g^{\theta \theta }=1/R^{2}\qquad &g^{\varphi \varphi }=1/(R^{2}\cos ^{2}\theta )\qquad &g^{ij}=0\quad \mathrm {else} \\\end{aligned}}}

Now the necessary quantities can be calculated. Examples:

eR=eRgRR=1⋅eR=(cos⁡θcos⁡φcos⁡θsin⁡φsin⁡θ)ΓRφφ=eR⋅∂∂φeφ=eR⋅(−Rcos⁡θcos⁡φ−Rcos⁡θsin⁡φ0)=−Rcos2⁡θ{\displaystyle {\begin{aligned}e^{R}=e_{R}g^{RR}=1\cdot e_{R}&={\begin{pmatrix}\cos \theta \cos \varphi &\cos \theta \sin \varphi &\sin \theta \end{pmatrix}}\\{\Gamma ^{R}}_{\varphi \varphi }=e^{R}\cdot {\frac {\partial }{\partial \varphi }}e_{\varphi }&=e^{R}\cdot {\begin{pmatrix}-R\cos \theta \cos \varphi &-R\cos \theta \sin \varphi &0\end{pmatrix}}=-R\cos ^{2}\theta \\\end{aligned}}}

The resulting Christoffel symbols of the second kindΓkji=ek⋅∂ej∂xi{\displaystyle {\Gamma ^{k}}_{ji}=e^{k}\cdot {\frac {\partial e_{j}}{\partial x^{i}}}} then are (organized by the "derivative" indexi in a matrix):

(ΓRRRΓRθRΓRφRΓθRRΓθθRΓθφRΓφRRΓφθRΓφφR)=(00001/R0001/R)(ΓRRθΓRθθΓRφθΓθRθΓθθθΓθφθΓφRθΓφθθΓφφθ)=(0−R01/R0000−tan⁡θ)(ΓRRφΓRθφΓRφφΓθRφΓθθφΓθφφΓφRφΓφθφΓφφφ)=(00−Rcos2⁡θ00cos⁡θsin⁡θ1/R−tan⁡θ0){\displaystyle {\begin{aligned}{\begin{pmatrix}{\Gamma ^{R}}_{RR}&{\Gamma ^{R}}_{\theta R}&{\Gamma ^{R}}_{\varphi R}\\{\Gamma ^{\theta }}_{RR}&{\Gamma ^{\theta }}_{\theta R}&{\Gamma ^{\theta }}_{\varphi R}\\{\Gamma ^{\varphi }}_{RR}&{\Gamma ^{\varphi }}_{\theta R}&{\Gamma ^{\varphi }}_{\varphi R}\\\end{pmatrix}}&=\quad {\begin{pmatrix}0&0&0\\0&1/R&0\\0&0&1/R\end{pmatrix}}\\{\begin{pmatrix}{\Gamma ^{R}}_{R\theta }&{\Gamma ^{R}}_{\theta \theta }&{\Gamma ^{R}}_{\varphi \theta }\\{\Gamma ^{\theta }}_{R\theta }&{\Gamma ^{\theta }}_{\theta \theta }&{\Gamma ^{\theta }}_{\varphi \theta }\\{\Gamma ^{\varphi }}_{R\theta }&{\Gamma ^{\varphi }}_{\theta \theta }&{\Gamma ^{\varphi }}_{\varphi \theta }\\\end{pmatrix}}\quad &={\begin{pmatrix}0&-R&0\\1/R&0&0\\0&0&-\tan \theta \end{pmatrix}}\\{\begin{pmatrix}{\Gamma ^{R}}_{R\varphi }&{\Gamma ^{R}}_{\theta \varphi }&{\Gamma ^{R}}_{\varphi \varphi }\\{\Gamma ^{\theta }}_{R\varphi }&{\Gamma ^{\theta }}_{\theta \varphi }&{\Gamma ^{\theta }}_{\varphi \varphi }\\{\Gamma ^{\varphi }}_{R\varphi }&{\Gamma ^{\varphi }}_{\theta \varphi }&{\Gamma ^{\varphi }}_{\varphi \varphi }\\\end{pmatrix}}&=\quad {\begin{pmatrix}0&0&-R\cos ^{2}\theta \\0&0&\cos \theta \sin \theta \\1/R&-\tan \theta &0\end{pmatrix}}\\\end{aligned}}}

These values show how the tangent directions (columns:eR{\displaystyle e_{R}},eθ{\displaystyle e_{\theta }},eφ{\displaystyle e_{\varphi }}) change, seen from an outside perspective (e.g. from space), but given in the tangent directions of the actual location (rows:R,θ,φ).

As an example, take the nonzero derivatives byθ inΓkj θ{\displaystyle {\Gamma ^{k}}_{j\ \theta }}, which corresponds to a movement towards north (positive dθ):

These effects are maybe not apparent during the movement, because they are the adjustments that keep the measurements in the coordinatesR,θ,φ. Nevertheless, it can affect distances, physics equations, etc. So if e.g. you need the exact change of amagnetic field pointing approximately "south", it can be necessary to alsocorrect your measurement by the change of the north direction using the Christoffel symbols to get the "true" (tensor) value.

The Christoffel symbols of the first kindΓlji=glkΓkji{\displaystyle {\Gamma _{l}}_{ji}=g_{lk}{\Gamma ^{k}}_{ji}} show the same change using metric-corrected coordinates, e.g. for derivative byφ:

(ΓRRφΓRθφΓRφφΓθRφΓθθφΓθφφΓφRφΓφθφΓφφφ)=Rcos⁡θ(00−cos⁡θ00Rsin⁡θcos⁡θ−Rsin⁡θ0){\displaystyle {\begin{aligned}{\begin{pmatrix}{\Gamma _{R}}_{R\varphi }&{\Gamma _{R}}_{\theta \varphi }&{\Gamma _{R}}_{\varphi \varphi }\\{\Gamma _{\theta }}_{R\varphi }&{\Gamma _{\theta }}_{\theta \varphi }&{\Gamma _{\theta }}_{\varphi \varphi }\\{\Gamma _{\varphi }}_{R\varphi }&{\Gamma _{\varphi }}_{\theta \varphi }&{\Gamma _{\varphi }}_{\varphi \varphi }\\\end{pmatrix}}&=R\cos \theta {\begin{pmatrix}0&0&-\cos \theta \\0&0&R\sin \theta \\\cos \theta &-R\sin \theta &0\end{pmatrix}}\\\end{aligned}}}

Lagrangian approach at finding a solution

In cylindrical coordinates, Cartesian and cylindrical polar coordinates exist as:

{x=rcos⁡φy=rsin⁡φz=h{\textstyle {\begin{cases}x=r\cos \varphi \\y=r\sin \varphi \\z=h\end{cases}}} and{r=x2+y2φ=arctan⁡(yx)h=z{\displaystyle {\begin{cases}r={\sqrt {x^{2}+y^{2}}}\\\varphi =\arctan \left({\frac {y}{x}}\right)\\h=z\end{cases}}}

Cartesian points exist and Christoffel Symbols vanish as time passes, therefore, in cylindrical coordinates:

Γrrr=Γφrr=∂2x∂r2∂r∂x+∂2y∂r2∂r∂y+∂2z∂r2∂r∂z=0{\displaystyle \Gamma _{rr}^{r}=\Gamma _{\varphi r}^{r}={\frac {\partial ^{2}x}{\partial r^{2}}}{\frac {\partial r}{\partial x}}+{\frac {\partial ^{2}y}{\partial r^{2}}}{\frac {\partial r}{\partial y}}+{\frac {\partial ^{2}z}{\partial r^{2}}}{\frac {\partial r}{\partial z}}=0}

Γrφr=Γφrr=∂2x∂r∂φ∂r∂x+∂2y∂r∂φ∂r∂y+∂2z∂r∂φ∂r∂z=−sin⁡φcos⁡φ+sin⁡φcos⁡φ=0{\displaystyle \Gamma _{r\varphi }^{r}=\Gamma _{\varphi r}^{r}={\frac {\partial ^{2}x}{\partial r\partial \varphi }}{\frac {\partial r}{\partial x}}+{\frac {\partial ^{2}y}{\partial r\partial \varphi }}{\frac {\partial r}{\partial y}}+{\frac {\partial ^{2}z}{\partial r\partial \varphi }}{\frac {\partial r}{\partial z}}=-\sin \varphi \cos \varphi +\sin \varphi \cos \varphi =0}

Γφφr=∂2x∂φ2∂r∂x+∂2y∂φ2∂r∂y+∂2z∂φ2∂r∂z=−xr−yr=−r{\displaystyle \Gamma _{\varphi \varphi }^{r}={\frac {\partial ^{2}x}{\partial \varphi ^{2}}}{\frac {\partial r}{\partial x}}+{\frac {\partial ^{2}y}{\partial \varphi ^{2}}}{\frac {\partial r}{\partial y}}+{\frac {\partial ^{2}z}{\partial \varphi ^{2}}}{\frac {\partial r}{\partial z}}=-{\frac {x}{r}}-{\frac {y}{r}}=-r}

Γrrφ=Γφrφ=∂2x∂r2∂φ∂x+∂2y∂r2∂φ∂y+∂2z∂r2∂φ∂z=0{\displaystyle \Gamma _{rr}^{\varphi }=\Gamma _{\varphi r}^{\varphi }={\frac {\partial ^{2}x}{\partial r^{2}}}{\frac {\partial \varphi }{\partial x}}+{\frac {\partial ^{2}y}{\partial r^{2}}}{\frac {\partial \varphi }{\partial y}}+{\frac {\partial ^{2}z}{\partial r^{2}}}{\frac {\partial \varphi }{\partial z}}=0}

Γrφφ=Γφrφ=∂2x∂r∂φ∂φ∂x+∂2y∂r∂φ∂φ∂y+∂2z∂r∂φ∂φ∂z=−yr2+cos⁡φxr2=1r{\displaystyle \Gamma _{r\varphi }^{\varphi }=\Gamma _{\varphi r}^{\varphi }={\frac {\partial ^{2}x}{\partial r\partial \varphi }}{\frac {\partial \varphi }{\partial x}}+{\frac {\partial ^{2}y}{\partial r\partial \varphi }}{\frac {\partial \varphi }{\partial y}}+{\frac {\partial ^{2}z}{\partial r\partial \varphi }}{\frac {\partial \varphi }{\partial z}}=-{\frac {y}{r^{2}}}+\cos \varphi {\frac {x}{r^{2}}}={\frac {1}{r}}}

Γφφφ=∂2x∂φ2∂φ∂x+∂2y∂φ2∂φ∂y+∂2z∂φ2∂φ∂z=−xr2−yr2=0{\displaystyle \Gamma _{\varphi \varphi }^{\varphi }={\frac {\partial ^{2}x}{\partial \varphi ^{2}}}{\frac {\partial \varphi }{\partial x}}+{\frac {\partial ^{2}y}{\partial \varphi ^{2}}}{\frac {\partial \varphi }{\partial y}}+{\frac {\partial ^{2}z}{\partial \varphi ^{2}}}{\frac {\partial \varphi }{\partial z}}=-{\frac {x}{r^{2}}}-{\frac {y}{r^{2}}}=0}

Spherical coordinates (using Lagrangian 2x2x2)

ds2=dθ2+sin2⁡θdϕ2{\displaystyle ds^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}

The Lagrangian can be evaluated as:

L=θ˙2+sin2⁡θϕ˙2{\displaystyle L={\dot {\theta }}^{2}+\sin ^{2}\theta {\dot {\phi }}^{2}}

Hence,

{ϕ¨+2cos⁡θsin⁡θθ˙ϕ˙=0θ¨−sin⁡θcos⁡θϕ˙2=0d2xkdλ2+Γijkdxidλdxjdλ=0∂L∂θ¨=0{\displaystyle {\begin{cases}{\ddot {\phi }}+2{\frac {\cos \theta }{\sin \theta }}{\dot {\theta }}{\dot {\phi }}=0\\{\ddot {\theta }}-\sin \theta \cos \theta {\dot {\phi }}^{2}=0\\{\frac {d^{2}x^{k}}{d\lambda ^{2}}}+\Gamma _{ij}^{k}{\frac {dx^{i}}{d\lambda }}{\frac {dx^{j}}{d\lambda }}=0\\{\frac {\partial L}{\partial {\ddot {\theta }}}}=0\end{cases}}} can be rearranged to{ϕ¨+2cos⁡θsin⁡θθ˙ϕ˙=0θ¨−sin⁡θcos⁡θϕ˙2=0{\displaystyle {\begin{cases}{\ddot {\phi }}+2{\frac {\cos \theta }{\sin \theta }}{\dot {\theta }}{\dot {\phi }}=0\\{\ddot {\theta }}-\sin \theta \cos \theta {\dot {\phi }}^{2}=0\end{cases}}}

By using the following geodesic equation:

d2xkdλ2+Γijkdxidλdxjdλ=0{\displaystyle {\frac {d^{2}x^{k}}{d\lambda ^{2}}}+\Gamma _{ij}^{k}{\frac {dx^{i}}{d\lambda }}{\frac {dx^{j}}{d\lambda }}=0}

The following can be obtained:

Γ221=−sin⁡θcos⁡θ(Γ122)=Γ212cos⁡θsin⁡θ{\displaystyle \Gamma _{22}^{1}=-\sin \theta \cos \theta (\Gamma _{12}^{2})=\Gamma _{21}^{2}{\frac {\cos \theta }{\sin \theta }}}

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Lagrangian mechanics in geodesics (principles of least action in Christoffel symbols)

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IncorporatingLagrangian mechanics and using theEuler–Lagrange equation, Christoffel symbols can be substituted into the Lagrangian to account for the geometry of the manifold. Christoffel symbols being calculated from themetric tensor, the equations can be derived and expressed from the principle of least action. When applying the Euler-Lagrange equation to a system of equations, the Lagrangian will include terms involving the Christoffel symbols, allowing the equation to act for the curvature which can determine the correct equations of motion for objects moving along geodesics.

Using the principle of least action from the Euler-Lagrange equation

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The Euler-Lagrange equation is applied to a functional related to the path of an object in a spherical coordinate system,

GivenL∈C2(R3){\displaystyle L\in C^{2}(\mathbb {R} ^{3})} andy∈C1[a,b]{\displaystyle y\in C^{1}[a,b]} such thaty(a)=C{\displaystyle y(a)=C} andey(b)=d{\displaystyle ey(b)=d}

if

{∫abL(y(x))dx∫abL(y′(x))dx∫abL(x)dx{\displaystyle {\begin{cases}\int _{a}^{b}L(y(x))dx\\\int _{a}^{b}L(y'(x))dx\\\int _{a}^{b}L(x)dx\end{cases}}}

Reaches its minimummin≡y0∈C{\displaystyle min\equiv y_{0}\in C} , wherey0{\displaystyle y_{0}} is a solution that can be found by solving the differential equation:

ddx(∂L∂y′(y(x),y′(x)))−∂L∂y(y(x),y′(x))=0{\displaystyle {\frac {d}{dx}}\left({\frac {\partial L}{\partial y'}}(y(x),y'(x))\right)-{\frac {\partial L}{\partial y}}(y(x),y'(x))=0}

The differential equation provides the mathematical conditions that must be satisfied for this optimal path.

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See also

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Notes

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  1. ^See, for instance, (Spivak 1999) and (Choquet-Bruhat & DeWitt-Morette 1977)
  2. ^Ronald Adler, Maurice Bazin, Menahem Schiffer,Introduction to General Relativity (1965) McGraw-Hill Book CompanyISBN 0-07-000423-4 (See section 2.1)
  3. ^Charles W. Misner, Kip S. Thorne, John Archibald Wheeler,Gravitation (1973) W. H. FreemanISBN 0-7167-0334-3 (See chapters 8-11)
  4. ^Misner, Thorne, Wheeler,op. cit. (See chapter 13)
  5. ^Jurgen Jost,Riemannian Geometry and Geometric Analysis, (2002) Springer-VerlagISBN 3-540-42627-2
  6. ^David Bleeker,Gauge Theory and Variational Principles (1991) Addison-Wesely Publishing CompanyISBN 0-201-10096-7
  7. ^abcChristoffel, E.B. (1869),"Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades",Journal für die reine und angewandte Mathematik,70:46–70
  8. ^abcChatterjee, U.; Chatterjee, N. (2010).Vector & Tensor Analysis. p. 480.
  9. ^abc"Christoffel Symbol of the Second Kind -- from Wolfram MathWorld".mathworld.wolfram.com. Archived fromthe original on 2009-01-23.
  10. ^abBishop, R.L.; Goldberg (1968),Tensor Analysis on Manifolds, p. 241
  11. ^abLudvigsen, Malcolm (1999),General Relativity: A Geometrical Approach, p. 88
  12. ^Chatterjee, U.; Chatterjee, N. (2010).Vector and Tensor Analysis. p. 480.
  13. ^Struik, D.J. (1961).Lectures on Classical Differential Geometry (first published in 1988 Dover ed.). p. 114.
  14. ^G. Ricci-Curbastro (1896). "Dei sistemi di congruenze ortogonali in una varietà qualunque".Mem. Acc. Lincei.2 (5):276–322.
  15. ^H. Levy (1925)."Ricci's coefficients of rotation".Bull. Amer. Math. Soc.31 (3–4):142–145.doi:10.1090/s0002-9904-1925-03996-8.
  16. ^This is assuming that the connection is symmetric (e.g., the Levi-Civita connection). If the connection hastorsion, then only the symmetric part of the Christoffel symbol can be made to vanish.
  17. ^Einstein, Albert (2005)."The Meaning of Relativity (1956, 5th Edition)". Princeton University Press (2005).
  18. ^Schrödinger, E. (1950). Space-time structure. Cambridge University Press.
  19. ^Adler, R., Bazin, M., & Schiffer, M. Introduction to General Relativity (New York, 1965).
  20. ^David, Kay,Tensor Calculus (1988) McGraw-Hill Book CompanyISBN 0-07-033484-6 (See section 11.4)
  21. ^abcd"Alexander J. Sesslar".sites.google.com. Retrieved2024-10-22.

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