特殊相対性理論 (Special Relativity)

Zur Elektrodynamik bewegter Körper

ローレンツ変換

t′=t−(v/c2)x1−(v/c)2x′=x−vt1−(v/c)2y′=yz′=z\begin{aligned} t' & = \frac{t - (v/c^2)x}{\sqrt[]{ 1 - {(v/c)}^2}} \\ x' & = \frac{x - vt}{\sqrt[]{ 1 - {(v/c)}^2}} \\ y' & = y \\ z' & = z \end{aligned}
[ct′x′y′z′]=[γ−γvc00−γvcγ0000000000][ctxyz]\left[ \begin{array}{c} ct' \\ x' \\ y' \\ z' \\ \end{array} \right] = \left[ \begin{array}{cccc} \gamma & -\gamma \frac{v}{c} & 0 & 0 \\ -\gamma \frac{v}{c} & \gamma & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right] \left[ \begin{array}{c} ct \\ x \\ y \\ z \\ \end{array} \right]

相対論的電磁気学 (Electromagnetism in 4D)

4次元ポテンシャルベクトル (4-vector potential)

Aμ=(ϕ/c,A)Aμ=ημνAν=(−ϕ/c,A)\begin{aligned} A^{\mu } & = \left( \phi /c , \mathbf{A} \right)\\ A_{\mu } & = {\eta }_{\mu \nu } A^{\nu } = \left( - \phi /c , \mathbf{A} \right) \end{aligned}

electromagnetic field tensor

Fμν=∂μAν−∂νAμ=(0−Ex/c−Ey/c−Ez/cEx/c0Bz−ByEy/c−Bz0BxEz/cBy−Bx0)\begin{aligned} F_{\mu \nu } & = {\partial}_{\mu } A_{\nu } - {\partial }_{\nu } A_{\mu } \\ & = \left( \begin{array}{cccc} 0 & -E_x / c & -E_y / c & -E_z / c \\ E_x / c & 0 & B_z & -B_y \\ E_y / c & -B_z & 0 & B_x \\ E_z / c & B_y & -B_x & 0 \\ \end{array} \right) \end{aligned}
Fμν=∂μAν−∂νAμ=(0−Ex/c−Ey/c−Ez/cEx/c0Bz−ByEy/c−Bz0BxEz/cBy−Bx0)\begin{aligned} F_{\mu \nu } & = {\partial}_{\mu } A_{\nu } - {\partial }_{\nu } A_{\mu } \\ & = \left( \begin{array}{cccc} 0 & -E_x / c & -E_y / c & -E_z / c \\ E_x / c & 0 & B_z & -B_y \\ E_y / c & -B_z & 0 & B_x \\ E_z / c & B_y & -B_x & 0 \\ \end{array} \right) \end{aligned}

Properties

Fμν=−FνμF_{\mu \nu} = - F_{\nu \mu}

4 元電流 (Four-current)

jμ=(cρ,j)j^\mu = \left( c\rho, \mathbf{j} \right)

Continuity equation

∂μjμ=0\partial_\mu j^\mu = 0

マクスウェルの方程式 (Maxwell's equations)

Ampère's circuital law (with Maxwell's correction)

∇×B=μ0(J+ε0∂B∂t)∇⋅E=ρε0\begin{aligned} {\mathbf \nabla} \times {\mathbf B} & = \mu_0 \left( {\mathbf J} + \varepsilon_0 \frac{\partial {\mathbf B}}{\partial t} \right) \\ {\mathbf \nabla} \cdot {\mathbf E} & = \frac{\rho}{\varepsilon_0} \end{aligned}
∂μFμν=−μ0jμ□Aμ−∂μ∂νAν=−μ0jμ\begin{aligned} \partial_\mu F^{\mu \nu} & = -\mu_0 j^\mu \\ \Box A^\mu - \partial^\mu \partial_\nu A^\nu & = -\mu_0 j^\mu \end{aligned}

Maxwell–Faraday equation (Faraday's law of induction)

∇×E=−∂B∂t∇⋅B=0\begin{aligned} {\mathbf \nabla} \times {\mathbf E} & = - \frac{\partial {\mathbf B}}{\partial t} \\ {\mathbf \nabla} \cdot {\mathbf B} & = 0 \end{aligned}
∂λFμν+∂μFνλ+∂νFλμ=0\partial_\lambda F_{\mu \nu} + \partial_\mu F_{\nu \lambda} + \partial_\nu F_{\lambda \mu} = 0

エネルギー運動量テンソル (Stress-energy tensor)

Stress-energy in special situations

Stress-energy of a fluid in equilibrium

Tαβ=(ρ+pc2)uαuβ+pgαβgαβ=(−c−2000010000100001)Tαβ=(ρ0000p0000p0000p)\begin{aligned} T^{\alpha \beta } & = \left( \rho + \frac{p}{c^2} \right) u^{\alpha } u^{\beta } + p g^{\alpha \beta } \\ g^{\alpha \beta } & = \left( \begin{array}{cccc} -{c}^{-2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right) \\ T^{\alpha \beta } & = \left( \begin{array}{cccc} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \\ \end{array} \right) \end{aligned}

Electromagnetic stress-energy tensor

Tμν=1μ0(FμαgαβFνβ−14gμνFδγFδγ)T^{\mu \nu } = \frac{1}{{\mu }_0} \left( F^{\mu \alpha } g_{\alpha \beta } F^{\nu \beta } - \frac{1}{4} g^{\mu \nu } F_{\delta \gamma } F^{\delta \gamma } \right)