リーマン幾何学 (Riemannian Geometry)

計量テンソル (Metric tensor)

ds2=gij(x)dxidxj=g′ij(x′)dx′idx′jgikgkj=δijgij∂gij∂xk=−gij∂gij∂xk∂gij∂xk=−gilgjm∂gjm∂xk∂g(x)∂xi=g(x)gjm∂gjm∂xi1−g∂−g∂xi=12gjk∂gjk∂xi\begin{aligned} {ds}^2 & = g_{ij}(x) {dx}^{i} {dx}^{j} = {g'}_{ij}(x') {dx'}^{i} {dx'}^{j} \\ g^{ik} g_{kj} & = {{\delta }^i}_j \\ g_{ij} \frac{\partial g^{ij}}{\partial x^k} & = -g^{ij} \frac{\partial g_{ij}}{\partial x^k} \\ \frac{\partial g^{ij}}{\partial x^k} & = - g^{il} g^{jm} \frac{\partial g_{jm}}{\partial x^k} \\ \frac{\partial g(x)}{\partial x^i} & = g(x) g^{jm} \frac{\partial g_{jm}}{\partial x^i} \\ \frac{1}{\sqrt[]{ -g}} \frac{\partial \sqrt[]{ -g}}{\partial x^i} & = \frac{1}{2} g^{jk} \frac{\partial g_{jk}}{\partial x^i} \end{aligned}

Examples

Flat spacetime with coordinates $ (t, x, y, z) $

ds2=−dt2+dx2+dy2+dz2 ds^2 = -dt^2 +dx^2 + dy^2 + dz^2

In spherical coordinates $ (t, r, \theta, \phi) $

ds2=−dt2+dr2+r2dΩ2 ds^2 = -dt^2 + dr^2 + r^2d\Omega^2

where

dΩ2=dθ2+sin2θdϕ2 d\Omega^2 = d\theta^2 + \sin^2 \theta d\phi^2

is the standard metric on the 2-sphere.

The round metric on a sphere

ds2=dθ2+sin2θdϕ2 ds^2 = d{\theta }^2 + sin^2\theta d\phi^2

Christoffel symbols（クリストッフェル記号）

Γkli=12gim(∂gmk∂xl+∂gml∂xk−∂gkl∂xm)=12gim(gmk,l+gml,k−gkl,m)Γjki=Γkji\begin{aligned} {\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) \\ {\Gamma }^{i}_{jk} & = {\Gamma }^{i}_{kj} \end{aligned}

Covariant derivative（共変微分）

Examples

∇aϕ=∂aϕ∇bVa=∂bVa+ΓcbaVc∇bVa=∂bVa−ΓabcVc∇cTab=∂cTab+ΓdcaTdb+ΓdcbTad∇cTab=∂cTab−ΓacdTdb−ΓbcdTad\begin{aligned} {\nabla }_a \phi & = {\partial }_a \phi \\ {\nabla }_b V^a & = {\partial }_b V^a + {\Gamma }^{a}_{cb} V^c \\ {\nabla }_b V_a & = {\partial }_b V_a - {\Gamma }^{c}_{ab} V_c \\ {\nabla }_c T^{ab} & = {\partial }_c T^{ab} + {\Gamma }^{a}_{dc} T^{db} + {\Gamma }^{b}_{dc} T^{ad} \\ {\nabla }_c T_{ab} & = {\partial }_c T_{ab} - {\Gamma }^{d}_{ac} T_{db} - {\Gamma }^{d}_{bc} T_{ad} \end{aligned}

リーマンの曲率テンソル (Riemann curvature tensor)

Am[;i;j]≡Am;i;j−Am;j;i {A_m}_{ [ ;i ;j ] } \equiv A_{m ;i ;j } - A_{m ;j ;i }
Am[;i;j]=RbmijAb {A_m}_{ [ ;i ;j ] } = {R^b}_{ mij } A_b

Symmetries and identities

Skew symmetry

[ R_{abcd} = -R_{bacd} = -R_{abdc} ]

Interchange symmetry

[ R_{abcd} = R_{cdab} ]

First Bianchi identity

[ R_{abcd} + R_{acdb} + R_{adbc} = 0 ] [ R_{a \left[ bcd \right] } = 0 ]

Second Bianchi identity

[ R_{abcd;e} + R_{abde;c} + R_{abec;d} = \nabla_e R_{abcd} + \nabla_c R_{abde} + \nabla_d R_{abec} = 0 ] [ R_{ab \left[ cd;e \right] } = 0 ]

アインシュタインテンソル (Einstein tensor)

Gμν=Rμν−12gμνR G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R

The Einstein tensor is symmetric

Gμν=Gνμ G_{\mu \nu } = G_{\nu \mu }

and, like the stress-energy tensor, divergenceless

Gμν;ν=∇νGμν=0 {G^{\mu \nu }}_{;\nu } = {\nabla }_{\nu } G^{\mu \nu } = 0