シュワルツシルト半径 (1回目)
2023/8/14(月)
シュワルツシルト半径 (1回目)
(Schwarzschild radius)
極座標の微小不変量dsを求める
■ 導出
▼ 極座標の微小不変量ds
https://ulprojectmail.blogspot.com/2023/07/polar-4.html
極座標 (4回目)
より
gαβ = diag(1, r2, r2sin2θ)
dw = cdtとして
ds2 = -dw2 + dx2 + dy2 + dz2
を極座標で求めると
時間軸を追加して
gαβ = diag(-1, 1, r2, r2sin2θ)
drα = dw, dr,dθ,dφ
として
ds2 = gαβdxαdxβ = gααdxαdxα
= -dw2 + dr2 + r2dθ2 + r2sin2θdφ2
■ 結果
▼ 極座標の微小不変量
ds2 = -dw2 + dr2 + r2dθ2 + r2sin2θdφ2
■ 別解
▼ 極座標の全微分
https://ulprojectmail.blogspot.com/2023/06/polar-1.html
極座標 (1回目)
x = rsinθcosφ
y = rsinθsinφ
z = rcosθ
xα = x,y,z、rβ = r,θ,φ
全微分
dxα = (∂xα/∂rβ)drβ
dx = sinθcosφdr+rcosθcosφdθ-rsinθsinφdφ
dy = sinθsinφdr+rcosθsinφdθ+rsinθcosφdφ
dz = cosθdr-rsinθdθ
(a + b + c)2 = (a+b)2 + 2(a+b)c + c2
= a2 + b2 + c2 + 2(ab + bc + ca)
dx2 = (sinθcosφdr)2+(rcosθcosφdθ)2+(rsinθsinφdφ)2
+ 2(sinθcosφdr)(rcosθcosφdθ)
- 2(rcosθcosφdθ)(rsinθsinφdφ)
- 2(rsinθsinφdφ)(sinθcosφdr)
dy2 = (sinθsinφdr)2+(rcosθsinφdθ)2+(rsinθcosφdφ)2
+ 2(sinθsinφdr)(rcosθsinφdθ)
+ 2(rcosθsinφdθ)(rsinθcosφdφ)
+ 2(rsinθcosφdφ)(sinθsinφdr)
dz2 = (cosθdr)2+(rsinθdθ)2-2(cosθdr)(rsinθdθ)
dx2 + dy2 + dz2
= (sinθcosφdr)2+(rcosθcosφdθ)2+(rsinθsinφdφ)2
+ 2(sinθcosφdr)(rcosθcosφdθ)
- 2(rcosθcosφdθ)(rsinθsinφdφ)
- 2(rsinθsinφdφ)(sinθcosφdr)
+ (sinθsinφdr)2+(rcosθsinφdθ)2+(rsinθcosφdφ)2
+ 2(sinθsinφdr)(rcosθsinφdθ)
+ 2(rcosθsinφdθ)(rsinθcosφdφ)
+ 2(rsinθcosφdφ)(sinθsinφdr)
+ (cosθdr)2+(rsinθdθ)2-2(cosθdr)(rsinθdθ)
= (sinθcosφdr)2+(rcosθcosφdθ)2+(rsinθsinφdφ)2
+ (sinθsinφdr)2+(rcosθsinφdθ)2+(rsinθcosφdφ)2
+ 2(sinθcosφdr)(rcosθcosφdθ)
+ 2(sinθsinφdr)(rcosθsinφdθ)
- 2(rcosθcosφdθ)(rsinθsinφdφ)
+ 2(rcosθsinφdθ)(rsinθcosφdφ)
- 2(rsinθsinφdφ)(sinθcosφdr)
+ 2(rsinθcosφdφ)(sinθsinφdr)
+ (cosθdr)2+(rsinθdθ)2 - 2(cosθdr)(rsinθdθ)
= (sinθdr)2+(rcosθdθ)2+(rsinθdφ)2
+ 2(sinθdr)(rcosθdθ)
+ (cosθdr)2+(rsinθdθ)2 - 2(cosθdr)(rsinθdθ)
= (sinθdr)2+(rcosθdθ)2+(rsinθdφ)2
+ (cosθdr)2+(rsinθdθ)2
= dr2 + r2dθ2 + r2sin2θdφ2
dx2 + dy2 + dz2 = dr2 + r2dθ2 + r2sin2θdφ2
結果
ds2 = -dw2 + dx2 + dy2 + dz2
= -dw2 + dr2 + r2dθ2 + r2sin2θdφ2
シュワルツシルト半径 (1回目)
(Schwarzschild radius)
極座標の微小不変量dsを求める
■ 導出
▼ 極座標の微小不変量ds
https://ulprojectmail.blogspot.com/2023/07/polar-4.html
極座標 (4回目)
より
gαβ = diag(1, r2, r2sin2θ)
dw = cdtとして
ds2 = -dw2 + dx2 + dy2 + dz2
を極座標で求めると
時間軸を追加して
gαβ = diag(-1, 1, r2, r2sin2θ)
drα = dw, dr,dθ,dφ
として
ds2 = gαβdxαdxβ = gααdxαdxα
= -dw2 + dr2 + r2dθ2 + r2sin2θdφ2
■ 結果
▼ 極座標の微小不変量
ds2 = -dw2 + dr2 + r2dθ2 + r2sin2θdφ2
■ 別解
▼ 極座標の全微分
https://ulprojectmail.blogspot.com/2023/06/polar-1.html
極座標 (1回目)
x = rsinθcosφ
y = rsinθsinφ
z = rcosθ
xα = x,y,z、rβ = r,θ,φ
全微分
dxα = (∂xα/∂rβ)drβ
dx = sinθcosφdr+rcosθcosφdθ-rsinθsinφdφ
dy = sinθsinφdr+rcosθsinφdθ+rsinθcosφdφ
dz = cosθdr-rsinθdθ
(a + b + c)2 = (a+b)2 + 2(a+b)c + c2
= a2 + b2 + c2 + 2(ab + bc + ca)
dx2 = (sinθcosφdr)2+(rcosθcosφdθ)2+(rsinθsinφdφ)2
+ 2(sinθcosφdr)(rcosθcosφdθ)
- 2(rcosθcosφdθ)(rsinθsinφdφ)
- 2(rsinθsinφdφ)(sinθcosφdr)
dy2 = (sinθsinφdr)2+(rcosθsinφdθ)2+(rsinθcosφdφ)2
+ 2(sinθsinφdr)(rcosθsinφdθ)
+ 2(rcosθsinφdθ)(rsinθcosφdφ)
+ 2(rsinθcosφdφ)(sinθsinφdr)
dz2 = (cosθdr)2+(rsinθdθ)2-2(cosθdr)(rsinθdθ)
dx2 + dy2 + dz2
= (sinθcosφdr)2+(rcosθcosφdθ)2+(rsinθsinφdφ)2
+ 2(sinθcosφdr)(rcosθcosφdθ)
- 2(rcosθcosφdθ)(rsinθsinφdφ)
- 2(rsinθsinφdφ)(sinθcosφdr)
+ (sinθsinφdr)2+(rcosθsinφdθ)2+(rsinθcosφdφ)2
+ 2(sinθsinφdr)(rcosθsinφdθ)
+ 2(rcosθsinφdθ)(rsinθcosφdφ)
+ 2(rsinθcosφdφ)(sinθsinφdr)
+ (cosθdr)2+(rsinθdθ)2-2(cosθdr)(rsinθdθ)
= (sinθcosφdr)2+(rcosθcosφdθ)2+(rsinθsinφdφ)2
+ (sinθsinφdr)2+(rcosθsinφdθ)2+(rsinθcosφdφ)2
+ 2(sinθcosφdr)(rcosθcosφdθ)
+ 2(sinθsinφdr)(rcosθsinφdθ)
- 2(rcosθcosφdθ)(rsinθsinφdφ)
+ 2(rcosθsinφdθ)(rsinθcosφdφ)
- 2(rsinθsinφdφ)(sinθcosφdr)
+ 2(rsinθcosφdφ)(sinθsinφdr)
+ (cosθdr)2+(rsinθdθ)2 - 2(cosθdr)(rsinθdθ)
= (sinθdr)2+(rcosθdθ)2+(rsinθdφ)2
+ 2(sinθdr)(rcosθdθ)
+ (cosθdr)2+(rsinθdθ)2 - 2(cosθdr)(rsinθdθ)
= (sinθdr)2+(rcosθdθ)2+(rsinθdφ)2
+ (cosθdr)2+(rsinθdθ)2
= dr2 + r2dθ2 + r2sin2θdφ2
dx2 + dy2 + dz2 = dr2 + r2dθ2 + r2sin2θdφ2
結果
ds2 = -dw2 + dx2 + dy2 + dz2
= -dw2 + dr2 + r2dθ2 + r2sin2θdφ2
