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。因埃尔温·布鲁诺·克里斯托费尔 (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,多数作者用和乐 (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。
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 (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)}
.
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