ds^{2} = |
┌ 1 0 0 ┐ │ 0 1 0 │ └ 0 0 1 ┘ |
┌ x ┐ │ y │ └ z ┘ |
┌ x ┐ │ y │ └ z ┘ |
= |
┌ 1x+0y+0z ┐ │ 0x+1y+0z │ └ 0x+0y+1z ┘ |
┌ x ┐ │ y │ └ z ┘ |
= |
┌ x ┐ │ y │ └ z ┘ |
┌ x ┐ │ y │ └ z ┘ |
= | x^{2} + y^{2} + z^{2} | (equation 4) | |
g_{μν} = |
┌ 1 0 0 ┐ │ 0 1 0 │ └ 0 0 1 ┘ |
(equation 5) |
g_{μν} = |
┌ ∂x/∂x ∂x/∂y ∂x/∂z ┐ │ ∂y/∂x ∂y/∂y ∂y/∂z │ └ ∂z/∂x ∂z/∂y ∂z/∂z ┘ |
(equation 6) |
A = |
┌ a_{11} a_{12} a_{13} ┐ │ a_{21} a_{22} a_{23} │ └ a_{31} a_{32} a_{33} ┘ |
(equation 7) |
V = |
┌ v_{1} ┐ │ v_{2} │ │ v_{3} │ └ v_{4} ┘ |
a_{ij} | = |
∂x_{i} ---- (equation 12) ∂x_{j} |
x'_{i} | = | Σ |
∂x'_{i} ---- x_{j} (equation 13) ∂x_{j} |
┌ cos(ϕ) -sin(ϕ) ┐ └ sin(ϕ) cos(ϕ.) ┘ |
┌ x_{1}' ┐ └ x_{2}' ┘ |
= |
┌ cos(ϕ) -sin(ϕ) ┐ └ sin(ϕ) cos(ϕ.) ┘ |
┌ x_{1} ┐ └ x_{2} ┘ |
A'^{i} | = | Σ |
∂x'_{i} ---- A^{j} (equation 16) ∂x_{j} |
A'_{i} | = | Σ |
∂x_{j} ---- A_{j} (equation 17) ∂x'_{i} |
┌ 1 0 0 ┐ │ 0 1 0 │ └ 0 0 1 ┘ |
┌ -xy -y^{2} ┐ └ x^{2} xy ┘ |
A'^{ij} | = | Σ Σ |
∂x'_{i} ∂x'_{j} ----------- A^{kl} (equation 18) ∂x_{k}∂x_{l} |
C'_{ij} | = | Σ Σ |
∂x_{k} ∂x_{l} ----------- C_{kl} (equation 19) ∂x'_{i}∂x'_{j} |
ds^{2} = |
┌ -1 0 0 0 ┐ │ 0 1 0 0 │ │ 0 0 1 0 │ └ 0 0 0 1 ┘ |
┌ ct ┐ │ x │ │ y │ └ z ┘ |
┌ ct ┐ │ x │ │ y │ └ z ┘ |
= |
┌ -ct+0x+0y+0z ┐ │ 0ct+1x+0y+0z │ │ 0ct+0x+1y+0z │ └ 0ct+0x+0y+1z ┘ |
┌ ct ┐ │ x │ │ y │ └ z ┘ |
= |
┌ -ct┐ │ x │ │ y │ └ z ┘ |
┌ ct ┐ │ x │ │ y │ └ z ┘ |
= | -c^{2} t^{2} + x^{2} + y^{2} + z^{2} | ||
c^{2} | = |
1 ---- ε_{0} μ_{0} |
x' | = | x-vt ------------- √(1-v^{2}/c^{2}) |
y' | = | y |
z' | = | z |
t' | = | t - (v/c^{2}).x ------------- √(1-v^{2}/c^{2}) |
γ | = | 1 ------------- √(1-v^{2}/c^{2}) |
x' | = | γ | (x-vt) |
y' | = | y |
z' | = | z |
t' | = | γ | (t - (v/c^{2}).x) |
ds^{2} = |
┌ -1 0 0 0 ┐ │ 0 1 0 0 │ │ 0 0 1 0 │ └ 0 0 0 1 ┘ |
┌ ct ┐ │ x │ │ y │ └ z ┘ |
┌ ct ┐ │ x │ │ y │ └ z ┘ |
= |
┌ -ct+0x+0y+0z ┐ │ 0ct+1x+0y+0z │ │ 0ct+0x+1y+0z │ └ 0ct+0x+0y+1z ┘ |
┌ ct ┐ │ x │ │ y │ └ z ┘ |
= |
┌ -ct┐ │ x │ │ y │ └ z ┘ |
┌ ct ┐ │ x │ │ y │ └ z ┘ |
= | -c^{2} t^{2} + x^{2} + y^{2} + z^{2} | ||