曲率テンソル (3回目)
2023/8/5(土)
曲率テンソル (3回目)
(Curvature tensor)
ビアンキの恒等式の導出
■ 定義
▼ 曲率テンソル
[∇α,∇β]em = Rnm,αβen と定義する
gknRnm,αβ = Rkm,αβ = -gknRnm,βα = -Rkm,βα
Rnm,αβ
= (∂/∂xα)Γnβm - (∂/∂xβ)Γnαm
+ ΓnαkΓkβm - ΓnβkΓkαm
■ 導出
▼ ビアンキの恒等式
Γkij = gkaΓaij
= (1/2)gka{(∂gja/∂Xi)+(∂gai/∂Xj)-(∂gij/∂Xa)}
導出は
https://ulprojectmail.blogspot.com/2023/07/christoffel-3.html
クリストッフェル記号 (3回目)
x':局所慣性系
以後、局所慣性系
クリストッフェル記号 = 0、gkn = gkn とする
Rkm,αβ = gknRnm,αβ
= gkn{(∂/∂x'α)Γnβm - (∂/∂x'β)Γnαm
+ ΓnαiΓiβm - ΓnβiΓiαm)
= gkn{(∂/∂x'α)Γnβm - (∂/∂x'β)Γnαm }
= (1/2)gkn
[(∂/∂x'α)gni{(∂gmi/∂x'β)+(∂giβ/∂x'm)-(∂gβm/∂x'i)}
-(∂/∂x'β)gni{(∂gmi/∂x'α)+(∂giα/∂x'm)-(∂gαm/∂x'i)}]
= (1/2)gkngni
[(∂/∂x'α){(∂giβ/∂x'm)-(∂gβm/∂x'i)}
-(∂/∂x'β){(∂giα/∂x'm)-(∂gαm/∂x'i)}]
= (1/2)δki
{(∂2giβ/∂x'α∂x'm)-(∂2gβm/∂x'α∂x'i)
-(∂2giα/∂x'β∂x'm)+(∂2gαm/∂x'β∂x'i)}
= (1/2){(∂2giβ/∂x'α∂x'm)-(∂2gβm/∂x'α∂x'i)
-(∂2giα/∂x'β∂x'm)+(∂2gαm/∂x'β∂x'i)}
Rkm,αβ = (1/2){(∂2giβ/∂x'α∂x'm)-(∂2gβm/∂x'α∂x'i)
-(∂2giα/∂x'β∂x'm)+(∂2gαm/∂x'β∂x'i)}
Rkα,βm = (1/2){(∂2gim/∂x'β∂x'α)-(∂2gmα/∂x'β∂x'i)
-(∂2giβ/∂x'm∂x'α)+(∂2gβα/∂x'm∂x'i)}
Rkβ,mα = (1/2){(∂2giα/∂x'm∂x'β)-(∂2gαβ/∂x'm∂x'i)
-(∂2gim/∂x'α∂x'β)+(∂2gmβ/∂x'α∂x'i)}
2(Rkm,αβ + Rkα,βm + Rkβ,mα)
=(∂2giβ/∂x'α∂x'm)-(∂2gβm/∂x'α∂x'i)
-(∂2giα/∂x'β∂x'm)+(∂2gαm/∂x'β∂x'i)
+(∂2gim/∂x'β∂x'α)-(∂2gmα/∂x'β∂x'i)
-(∂2giβ/∂x'm∂x'α)+(∂2gβα/∂x'm∂x'i)
+(∂2giα/∂x'm∂x'β)-(∂2gαβ/∂x'm∂x'i)
-(∂2gim/∂x'α∂x'β)+(∂2gmβ/∂x'α∂x'i)
=(∂2giβ/∂x'α∂x'm)
-(∂2giβ/∂x'm∂x'α)
-(∂2gβm/∂x'α∂x'i)
+(∂2gmβ/∂x'α∂x'i)
+(∂2gim/∂x'β∂x'α)
-(∂2gim/∂x'α∂x'β)
-(∂2giα/∂x'β∂x'm)
+(∂2giα/∂x'm∂x'β)
+(∂2gαm/∂x'β∂x'i)
-(∂2gmα/∂x'β∂x'i)
+(∂2gβα/∂x'm∂x'i)
-(∂2gαβ/∂x'm∂x'i)
= 0
Rnm,αβ + Rnα,βm + Rnβ,mα
= gnk(Rkm,αβ + Rkα,βm + Rkβ,mα) = 0
Rkm,αβ + Rkα,βm + Rkβ,mα = 0
Rnm,αβ + Rnα,βm + Rnβ,mα = 0
▼ ビアンキの恒等式(微分形)
Rnm,αβ
= (∂/∂xα)Γnβm - (∂/∂xβ)Γnαm
+ ΓnαkΓkβm - ΓnβkΓkαm
x':局所慣性系
以後、局所慣性系
クリストッフェル記号 = 0、gkn = gkn とする
∇k(ΓnαkΓkβm) = ∇kΓnαkΓkβm + Γnαk∇kΓkβm = 0
より
∇k(ΓnαkΓkβm - ΓnβkΓkαm) = 0
∇γRnm,αβ
= (∂/∂xγ)(∂/∂xα)Γnβm - (∂/∂xγ)(∂/∂xβ)Γnαm
∇αRnm,βγ
= (∂/∂xα)(∂/∂xβ)Γnγm - (∂/∂xα)(∂/∂xγ)Γnβm
∇βRnm,γα
= (∂/∂xβ)(∂/∂xγ)Γnαm - (∂/∂xβ)(∂/∂xα)Γnγm
∇γRnm,αβ + ∇αRnm,βγ + ∇βRnm,γα
= (∂/∂xγ)(∂/∂xα)Γnβm - (∂/∂xγ)(∂/∂xβ)Γnαm
+ (∂/∂xα)(∂/∂xβ)Γnγm - (∂/∂xα)(∂/∂xγ)Γnβm
+ (∂/∂xβ)(∂/∂xγ)Γnαm - (∂/∂xβ)(∂/∂xα)Γnγm
= (∂/∂xγ)(∂/∂xα)Γnβm
- (∂/∂xα)(∂/∂xγ)Γnβm
- (∂/∂xγ)(∂/∂xβ)Γnαm
+ (∂/∂xβ)(∂/∂xγ)Γnαm
+ (∂/∂xα)(∂/∂xβ)Γnγm
- (∂/∂xβ)(∂/∂xα)Γnγm
= 0
gkn(∇γRnm,αβ + ∇αRnm,βγ + ∇βRnm,γα )
∇γRkm,αβ + ∇αRkm,βγ + ∇βRkm,γα = 0
■ 結果
▼ ビアンキの恒等式
Rkm,αβ + Rkα,βm + Rkβ,mα = 0
Rnm,αβ + Rnα,βm + Rnβ,mα = 0
∇γRkm,αβ + ∇αRkm,βγ + ∇βRkm,γα = 0
∇γRnm,αβ + ∇αRnm,βγ + ∇βRnm,γα = 0
曲率テンソル (3回目)
(Curvature tensor)
ビアンキの恒等式の導出
■ 定義
▼ 曲率テンソル
[∇α,∇β]em = Rnm,αβen と定義する
gknRnm,αβ = Rkm,αβ = -gknRnm,βα = -Rkm,βα
Rnm,αβ
= (∂/∂xα)Γnβm - (∂/∂xβ)Γnαm
+ ΓnαkΓkβm - ΓnβkΓkαm
■ 導出
▼ ビアンキの恒等式
Γkij = gkaΓaij
= (1/2)gka{(∂gja/∂Xi)+(∂gai/∂Xj)-(∂gij/∂Xa)}
導出は
https://ulprojectmail.blogspot.com/2023/07/christoffel-3.html
クリストッフェル記号 (3回目)
x':局所慣性系
以後、局所慣性系
クリストッフェル記号 = 0、gkn = gkn とする
Rkm,αβ = gknRnm,αβ
= gkn{(∂/∂x'α)Γnβm - (∂/∂x'β)Γnαm
+ ΓnαiΓiβm - ΓnβiΓiαm)
= gkn{(∂/∂x'α)Γnβm - (∂/∂x'β)Γnαm }
= (1/2)gkn
[(∂/∂x'α)gni{(∂gmi/∂x'β)+(∂giβ/∂x'm)-(∂gβm/∂x'i)}
-(∂/∂x'β)gni{(∂gmi/∂x'α)+(∂giα/∂x'm)-(∂gαm/∂x'i)}]
= (1/2)gkngni
[(∂/∂x'α){(∂giβ/∂x'm)-(∂gβm/∂x'i)}
-(∂/∂x'β){(∂giα/∂x'm)-(∂gαm/∂x'i)}]
= (1/2)δki
{(∂2giβ/∂x'α∂x'm)-(∂2gβm/∂x'α∂x'i)
-(∂2giα/∂x'β∂x'm)+(∂2gαm/∂x'β∂x'i)}
= (1/2){(∂2giβ/∂x'α∂x'm)-(∂2gβm/∂x'α∂x'i)
-(∂2giα/∂x'β∂x'm)+(∂2gαm/∂x'β∂x'i)}
Rkm,αβ = (1/2){(∂2giβ/∂x'α∂x'm)-(∂2gβm/∂x'α∂x'i)
-(∂2giα/∂x'β∂x'm)+(∂2gαm/∂x'β∂x'i)}
Rkα,βm = (1/2){(∂2gim/∂x'β∂x'α)-(∂2gmα/∂x'β∂x'i)
-(∂2giβ/∂x'm∂x'α)+(∂2gβα/∂x'm∂x'i)}
Rkβ,mα = (1/2){(∂2giα/∂x'm∂x'β)-(∂2gαβ/∂x'm∂x'i)
-(∂2gim/∂x'α∂x'β)+(∂2gmβ/∂x'α∂x'i)}
2(Rkm,αβ + Rkα,βm + Rkβ,mα)
=(∂2giβ/∂x'α∂x'm)-(∂2gβm/∂x'α∂x'i)
-(∂2giα/∂x'β∂x'm)+(∂2gαm/∂x'β∂x'i)
+(∂2gim/∂x'β∂x'α)-(∂2gmα/∂x'β∂x'i)
-(∂2giβ/∂x'm∂x'α)+(∂2gβα/∂x'm∂x'i)
+(∂2giα/∂x'm∂x'β)-(∂2gαβ/∂x'm∂x'i)
-(∂2gim/∂x'α∂x'β)+(∂2gmβ/∂x'α∂x'i)
=(∂2giβ/∂x'α∂x'm)
-(∂2giβ/∂x'm∂x'α)
-(∂2gβm/∂x'α∂x'i)
+(∂2gmβ/∂x'α∂x'i)
+(∂2gim/∂x'β∂x'α)
-(∂2gim/∂x'α∂x'β)
-(∂2giα/∂x'β∂x'm)
+(∂2giα/∂x'm∂x'β)
+(∂2gαm/∂x'β∂x'i)
-(∂2gmα/∂x'β∂x'i)
+(∂2gβα/∂x'm∂x'i)
-(∂2gαβ/∂x'm∂x'i)
= 0
Rnm,αβ + Rnα,βm + Rnβ,mα
= gnk(Rkm,αβ + Rkα,βm + Rkβ,mα) = 0
Rkm,αβ + Rkα,βm + Rkβ,mα = 0
Rnm,αβ + Rnα,βm + Rnβ,mα = 0
▼ ビアンキの恒等式(微分形)
Rnm,αβ
= (∂/∂xα)Γnβm - (∂/∂xβ)Γnαm
+ ΓnαkΓkβm - ΓnβkΓkαm
x':局所慣性系
以後、局所慣性系
クリストッフェル記号 = 0、gkn = gkn とする
∇k(ΓnαkΓkβm) = ∇kΓnαkΓkβm + Γnαk∇kΓkβm = 0
より
∇k(ΓnαkΓkβm - ΓnβkΓkαm) = 0
∇γRnm,αβ
= (∂/∂xγ)(∂/∂xα)Γnβm - (∂/∂xγ)(∂/∂xβ)Γnαm
∇αRnm,βγ
= (∂/∂xα)(∂/∂xβ)Γnγm - (∂/∂xα)(∂/∂xγ)Γnβm
∇βRnm,γα
= (∂/∂xβ)(∂/∂xγ)Γnαm - (∂/∂xβ)(∂/∂xα)Γnγm
∇γRnm,αβ + ∇αRnm,βγ + ∇βRnm,γα
= (∂/∂xγ)(∂/∂xα)Γnβm - (∂/∂xγ)(∂/∂xβ)Γnαm
+ (∂/∂xα)(∂/∂xβ)Γnγm - (∂/∂xα)(∂/∂xγ)Γnβm
+ (∂/∂xβ)(∂/∂xγ)Γnαm - (∂/∂xβ)(∂/∂xα)Γnγm
= (∂/∂xγ)(∂/∂xα)Γnβm
- (∂/∂xα)(∂/∂xγ)Γnβm
- (∂/∂xγ)(∂/∂xβ)Γnαm
+ (∂/∂xβ)(∂/∂xγ)Γnαm
+ (∂/∂xα)(∂/∂xβ)Γnγm
- (∂/∂xβ)(∂/∂xα)Γnγm
= 0
gkn(∇γRnm,αβ + ∇αRnm,βγ + ∇βRnm,γα )
∇γRkm,αβ + ∇αRkm,βγ + ∇βRkm,γα = 0
■ 結果
▼ ビアンキの恒等式
Rkm,αβ + Rkα,βm + Rkβ,mα = 0
Rnm,αβ + Rnα,βm + Rnβ,mα = 0
∇γRkm,αβ + ∇αRkm,βγ + ∇βRkm,γα = 0
∇γRnm,αβ + ∇αRnm,βγ + ∇βRnm,γα = 0