克氏符号 ,全称克里斯托费尔符号 (Christoffel symbols ),在数学 和物理 中,是从度量张量 导出的列维-奇维塔联络 (Levi-Civita connection )的坐标表达式。因埃爾溫·布魯諾·克里斯托費爾 (1829年-1900年)命名。克氏符号在每当进行涉及到几何的实用演算时都会被用到,因为他们使得非常复杂的演算不被搞混。不幸的是,它们写起来较繁琐,并要求对细节的仔细关注。相反,无下标的形式化的列维-奇维塔联络的概念是相当漂亮,并允许定理用典雅的方式表达,但是在实用演算中没有什么用处。
克氏符号可以从度量张量 g i k {\displaystyle g_{ik}} 的共变导数 为0这一事实来导出:
D l g i k = ∂ g i k ∂ x l − g m k Γ i l m − g i m Γ k l m = 0 {\displaystyle D_{l}g_{ik}={\frac {\partial g_{ik}}{\partial x^{l}}}-g_{mk}\Gamma _{il}^{m}-g_{im}\Gamma _{kl}^{m}=0} 。通过交换指标(index ),和求和,可以解出联络:
Γ k l i = 1 2 g i m ( ∂ g m k ∂ x l + ∂ g m l ∂ x k − ∂ g k l ∂ x m ) {\displaystyle \Gamma _{kl}^{i}={\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)} ,注意虽然记号有三个指标,他们不 是张量。它们不像张量那样变换。它们是二阶切丛上的物体的分量,是一个喷射 ,参看jet丛 。克氏符号在坐标变换下的变换性质见下面。
注意,多数作者用和樂 (或称完全,holonomic)的坐标系,我们也用这样的常规做法。在非和乐的坐标中,克氏符号有更复杂的形式
Γ k l i = 1 2 g i m ( ∂ g m k ∂ x l + ∂ g m l ∂ x k − ∂ g k l ∂ x m + c m k l + c m l k − c k l m ) {\displaystyle \Gamma _{kl}^{i}={\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}}}+c_{mkl}+c_{mlk}-c_{klm}\right)} 其中c k l m = g m p c k l p {\displaystyle c_{klm}=g_{mp}{c_{kl}}^{p}} 是该基的交换系数 ;也就是
[ e k , e l ] = c k l m e m {\displaystyle [e_{k},e_{l}]={c_{kl}}^{m}e_{m}} 其中e k 是向量的基而[ , ] {\displaystyle [,]} 是李括号 。
以下的表达式除作特殊说明外都是在和乐坐标基中。
和无指标符号的关系
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把指标缩并起来,就得到
Γ k i i = 1 2 g i m ∂ g i m ∂ x k = 1 2 g ∂ g ∂ x k = ∂ ln | g | ∂ x k {\displaystyle \Gamma _{ki}^{i}={\frac {1}{2}}g^{im}{\frac {\partial g_{im}}{\partial x_{k}}}={\frac {1}{2g}}{\frac {\partial g}{\partial x_{k}}}={\frac {\partial \ln {\sqrt {|g|}}}{\partial x_{k}}}} 其中|g |是度量张量g i k {\displaystyle g_{ik}} 的行列式 的绝对值。
类似的,
g k l Γ k l i = − 1 | g | ∂ | g | g i k ∂ x k . {\displaystyle g^{kl}\Gamma _{kl}^{i}={\frac {-1}{\sqrt {|g|}}}\;{\frac {\partial {\sqrt {|g|}}\,g^{ik}}{\partial x^{k}}}.} 向量场 V m {\displaystyle V^{m}} 的共变导数(covariant derivative) 是
D l V m = ∂ V m ∂ x l + Γ k l m V k . {\displaystyle D_{l}V^{m}={\frac {\partial V^{m}}{\partial x^{l}}}+\Gamma _{kl}^{m}V^{k}.} 共变散度(covariant divergence) 是
D m V m = ∂ V m ∂ x m + V k ∂ log | g | ∂ x k = 1 | g | ∂ ( V m | g | ) ∂ x m {\displaystyle D_{m}V^{m}={\frac {\partial V^{m}}{\partial x^{m}}}+V^{k}{\frac {\partial \log {\sqrt {|g|}}}{\partial x^{k}}}={\frac {1}{\sqrt {|g|}}}{\frac {\partial (V^{m}{\sqrt {|g|}})}{\partial x^{m}}}} .张量 A i k {\displaystyle A^{ik}} 的共变导数是
D l A i k = ∂ A i k ∂ x l + Γ m l i A m k + Γ m l k A i m {\displaystyle D_{l}A^{ik}={\frac {\partial A^{ik}}{\partial x^{l}}}+\Gamma _{ml}^{i}A^{mk}+\Gamma _{ml}^{k}A^{im}} .若张量是反对称 的,则其散度简化为
D k A i k = 1 | g | ∂ ( A i k | g | ) ∂ x k {\displaystyle D_{k}A^{ik}={\frac {1}{\sqrt {|g|}}}{\frac {\partial (A^{ik}{\sqrt {|g|}})}{\partial x^{k}}}} .标量场ϕ {\displaystyle \phi } 的反变导数称为ϕ {\displaystyle \phi } 的梯度 。也就是说,梯度就是把微分的指标升到上面:
D i ϕ = g i k ∂ ϕ ∂ x k . {\displaystyle D^{i}\phi =g^{ik}{\frac {\partial \phi }{\partial x^{k}}}.} 标量势的拉普拉斯算子 Laplacian 是
Δ ϕ = 1 | g | ∂ ∂ x i ( g i k | g | ∂ ϕ ∂ x k ) {\displaystyle \Delta \phi ={\frac {1}{\sqrt {|g|}}}{\frac {\partial }{\partial x^{i}}}\left(g^{ik}{\sqrt {|g|}}{\frac {\partial \phi }{\partial x^{k}}}\right)} .拉普拉斯也就是梯度的共变散度(对于标量场来讲)
Δ ϕ = D i D i ϕ {\displaystyle \Delta \phi =D_{i}D^{i}\phi } .
黎曼曲率
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黎曼曲率张量 是
R i k l m = 1 2 ( ∂ 2 g i m ∂ x k ∂ x l + ∂ 2 g k l ∂ x i ∂ x m − ∂ 2 g i l ∂ x k ∂ x m − ∂ 2 g k m ∂ x i ∂ x l ) + g n p ( Γ k l n Γ i m p − Γ k m n Γ i l p ) {\displaystyle R_{iklm}={\frac {1}{2}}\left({\frac {\partial ^{2}g_{im}}{\partial x^{k}\partial x^{l}}}+{\frac {\partial ^{2}g_{kl}}{\partial x^{i}\partial x^{m}}}-{\frac {\partial ^{2}g_{il}}{\partial x^{k}\partial x^{m}}}-{\frac {\partial ^{2}g_{km}}{\partial x^{i}\partial x^{l}}}\right)+g_{np}\left(\Gamma _{kl}^{n}\Gamma _{im}^{p}-\Gamma _{km}^{n}\Gamma _{il}^{p}\right)} .该张量的对称性有
R i k l m = R l m i k {\displaystyle R_{iklm}=R_{lmik}} 和R i k l m = − R k i l m = − R i k m l {\displaystyle R_{iklm}=-R_{kilm}=-R_{ikml}} .也就是交换前后两对指标是对称的,交换其中一对是反对称的。
循环替换的和是
R i k l m + R i m k l + R i l m k = 0. {\displaystyle R_{iklm}+R_{imkl}+R_{ilmk}=0.} 比安基恒等式 是
D m R i k l n + D l R i m k n + D k R i l m n = 0. {\displaystyle D_{m}R_{ikl}^{n}+D_{l}R_{imk}^{n}+D_{k}R_{ilm}^{n}=0.} Ricci曲率
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Ricci张量 由下式给出
R i k = ∂ Γ i k l ∂ x l − ∂ Γ i l l ∂ x k + Γ i k l Γ l m m − Γ i l m Γ k m l . {\displaystyle R_{ik}={\frac {\partial \Gamma _{ik}^{l}}{\partial x^{l}}}-{\frac {\partial \Gamma _{il}^{l}}{\partial x^{k}}}+\Gamma _{ik}^{l}\Gamma _{lm}^{m}-\Gamma _{il}^{m}\Gamma _{km}^{l}.} 该张量是对称的:R i k = R k i {\displaystyle R_{ik}=R_{ki}} .它可以通过收缩黎曼张量的指标得到:
R i k = g l m R l i m k . {\displaystyle R_{ik}=g^{lm}R_{limk}.} 标量曲率 由下式给出
R = g i k R i k {\displaystyle R=g^{ik}R_{ik}} .标量的共变导数可以从Bianchi等式推出:
D l R m l = 1 2 ∂ R ∂ x m {\displaystyle D_{l}R_{m}^{l}={\frac {1}{2}}{\frac {\partial R}{\partial x^{m}}}} . 外尔张量
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外尔张量 (Weyl tensor) 是
C i k l m = R i k l m + 1 2 ( − R i l g k m + R i m g k l + R k l g i m − R k m g i l ) + 1 6 R ( g i l g k m − g i m g k l ) {\displaystyle C_{iklm}=R_{iklm}+{\frac {1}{2}}\left(-R_{il}g_{km}+R_{im}g_{kl}+R_{kl}g_{im}-R_{km}g_{il}\right)+{\frac {1}{6}}R\left(g_{il}g_{km}-g_{im}g_{kl}\right)} . 坐标变换
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在从( x 1 , . . . , x n ) {\displaystyle (x^{1},...,x^{n})} 到( y 1 , . . . , y n ) {\displaystyle (y^{1},...,y^{n})} 的坐标变换下,向量的变换为
∂ ∂ y i = ∂ x k ∂ y i ∂ ∂ x k {\displaystyle {\frac {\partial }{\partial y^{i}}}={\frac {\partial x^{k}}{\partial y^{i}}}{\frac {\partial }{\partial x^{k}}}} 所以
Γ i j k ¯ = ∂ x p ∂ y i ∂ x q ∂ y j Γ p q r ∂ y k ∂ x r + ∂ y k ∂ x m ∂ 2 x m ∂ y i ∂ y j {\displaystyle {\overline {\Gamma _{ij}^{k}}}={\frac {\partial x^{p}}{\partial y^{i}}}\,{\frac {\partial x^{q}}{\partial y^{j}}}\,\Gamma _{pq}^{r}\,{\frac {\partial y^{k}}{\partial x^{r}}}+{\frac {\partial y^{k}}{\partial x^{m}}}\,{\frac {\partial ^{2}x^{m}}{\partial y^{i}\partial y^{j}}}} 其中上划线表示y 坐标系中的克氏符号。注意克氏符号不 像张量那样变换,而是像jet丛 中的对象那样。
Lev Davidovich Landau and Evgeny Mikhailovich Lifshitz , The Classical Theory of Fields, Fourth Revised English Edition, Course of Theoretical Physics, Volume 2 , (1951) Pergamon Press, Oxford; ISBN 0-08-025072-6 . See chapter 10, paragraphs 85,86 and 87.
Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics , (1978) Benjamin/Cummings Publishing, London; ISBN 0-8053-0102-X . See chapter 2, paragraph 2.7.1
Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation , (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0 . See chapter 8, paragraph 8.5