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一般相対論で使うあれこれ

物理 相対論 数学

 

添字がややこしくて混乱しそうなのでお勉強するときにそばに置いておくために作った。定義とかは全部Winbergの"Gravitation and Cosmology"で書いてたやつ。{ \displaystyle \xi ^{mu} }は自由落下系、または局所慣性系で、{ \displaystyle x^{\mu} }は一般の座標系。[tex:{ \displaystyle \eta _{\alpha \beta} }]はミンコフスキー計量。

 

f:id:inarizusi24:20151114150339j:plain

f:id:inarizusi24:20151114150357j:plain

f:id:inarizusi24:20151114150359j:plain

はてなで書いたらなんか上手く出力できなくて飽きたのでtexで書いたの貼り付け。証明はそこら辺の相対論の本参照で。

 

  • アフィン接続、クリストッフェル記号(The affine connection, Christoffel symbol)

{ \displaystyle \Gamma ^{\lambda}_{\mu \nu} \equiv \frac{\partial x^{\lambda}}{\partial \xi ^{\alpha}}\frac{\partial ^2 \xi ^{\alpha}}{\partial x^{\mu} \partial x^{\nu}} }

{ \displaystyle g_{\mu \nu} \equiv \frac{\partial \xi^{\alpha}}{\partial x^{\mu}}\frac{\partial \xi ^{\beta}}{\partial x^{\nu}} \eta _{\alpha \beta} }

{ \displaystyle \frac{\partial g_{\mu \nu}}{\partial x^{\lambda}} = \Gamma ^{\rho} _{\lambda \nu} g_{\rho \nu} + \Gamma ^{\rho}_{\lambda \nu} g_{ro \mu} }

  • アフィン接続を計量テンソルを用いて表す。

{ \displaystyle \Gamma ^{\sigma}_{\lambda \mu} = \frac{1}{2} g^{\nu \sigma}\{ \frac{\partial g_{\mu \nu}}{\partial x^{\lambda}} + \frac{\partial g_{\lambda \nu}}{\partial x^{\mu}} - \frac{\partial g_{\mu \lambda}}{\partial x^{\nu}}\} }

  • 測地線(Geodesics)

{ \displaystyle \frac{d^2 x^{\nu}}{d \tau ^2} + \Gamma ^{\nu}_{\mu \sigma}\frac{dx^{\mu}}{d\tau}\frac{dx^{\sigma}}{d\tau} =0 }

{ \displaystyle \frac{\partial {x'}^{\rho}}{\partial x^{\mu}}\frac{\partial {x'}^{\sigma}}{\partial x^{\nu}}\frac{\partial {x'}^{\eta}}{\partial x^{\lambda}}\frac{\partial {x'}^{\xi}}{\partial x^{\kappa}} \epsilon ^{\mu \nu \lambda \kappa} = \begin{vmatrix}\frac{\partial x'}{\partial x}\end{vmatrix}\epsilon ^{\rho \sigma \eta \xi} }

  • アフィン接続のたまに使う変換

{ \displaystyle {\Gamma '} ^{\lambda} _{\mu \nu} = \frac{\partial {x'}^{\lambda}}{\partial x^{\rho}}\frac{\partial x^{\tau}}{\partial {x'}^{\mu}}\frac{\partial x^{\sigma}}{\partial {x'}^{\nu}}\Gamma ^{\rho} _{\tau \sigma} - \frac{\partial x^{\rho}}{\partial {x'}^{\nu}}\frac{\partial x^{\sigma}}{\partial {x'}^{\mu}}\frac{\partial ^2 {x'}^{\lambda}}{\partial x^{\rho} \partial x^{\sigma}} }

  • 反変ベクトル(Contravariant vector)の共変微分(Covariant derivative)

{ \displaystyle V^{\mu} _{\ ; \lambda} = \frac{\partial V^{\mu}}{\partial x^{\nu}} + \Gamma ^{\mu} _{\lambda \kappa} V^{\kappa} }

{ \displaystyle V_{\mu ; \nu} = \frac{\partial V_{\mu}}{\partial x^{\nu}} - \Gamma ^{\lambda} _{\mu \nu} V^{\lambda} }

[tex:{ \displaystyle T^{\mu \sigma} _{\ \ \lambda ;\rho} = \frac{\partial}{\partial x^{\rho}} T^{\mu \sigma} _{\ \ \rho} + \Gamma ^{\mu} _{\rho \nu}

T^{\nu \sigma} _{\ \ \lambda} + \Gamma ^{\sigma} _{\rho \nu} T^{\mu \nu} _{\ \ \lambda} - \Gamma ^{\kappa} _{\lambda \rho} T^{\mu \sigma} _{\ \ \kappa} }]

 

{ \displaystyle \mathscr{P} ^{\mu} _{\ \lambda ;\rho} = \frac{\partial}{\partial x^{\rho}} \mathscr{P} ^{\mu} _{\ \lambda} + \Gamma ^{\mu} _ {\rho \nu} \mathscr{P} ^{\nu} _{\ \lambda} - \Gamma ^{\kappa} _ {\lambda \rho} \mathscr{P} ^{\mu} _{\ \kappa} + \frac{W}{2g}\frac{\partial g}{\partial x^{\rho}} \mathscr{P} ^{\mu} _{\ \lambda} }

[tex:{ \displaystyle S_{;\mu} = \frac{\partial

S}{\partial x^{\mu}} }]

  • Covariant rotation

{ \displaystyle V_{\mu ;\nu} - V_{\nu ;\mu} = \frac{\partial V_{\mu}}{\partial x^{\nu}} - \frac{\partial V_{\nu}}{\partial x^{\mu}} }

  • Covariant divergence

{ \displaystyle \Gamma ^{\mu} _{\mu \lambda} = \frac{1}{2} \frac{\partial}{\partial x^{\lambda}}\log g = \frac{1}{\sqrt{g}}\frac{\partial}{\partial x^{\lambda}}\sqrt{g} }

を使って、

{ \displaystyle V^{\mu}_{\ ;\mu} &= \frac{\partial V^{\mu}}{\partial x^{\mu}} + \Gamma ^{\mu} _{\mu \lambda}V^{\lambda} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial x^{\mu}}\sqrt{g} }

[tex:{ \displaystyle \int d^4x\sq

rt{g}V^{\mu}_{\ ;\mu} = 0 }]

{ \displaystyle A^{\mu \nu}_{;\mu} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial x^{\mu}}(\sqrt{g} A^{\mu \nu}) }

{ \displaystyle A_{\mu \nu ;\lambda} +A_{\lambda \mu ;\nu} + A_{\nu \lambda ;\mu} = \frac{\partial A_{\mu \nu}}{\partial x^{\lambda}} + \frac{\partial A_{\lambda \mu}}{\partial x^{\nu}} + \frac{\partial A_{\nu \lambda}}{\partial x^{\mu}} }

  • 曲線に沿った共変微分(共変ベクトルについては第二項にマイナスがつくだけ)

{ \displaystyle \frac{DA^{\mu}}{D\tau} = \frac{dA^{\mu}}{d\tau} + \Gamma ^{\mu}_{\nu \lambda}\frac{dx^{\lambda}}{d\tau}A^{\nu} }

  • さっきの{ \displaystyle A^{\mu} }が曲線にそって動いても変わらない時、平行移動の方程式(parallel transport)

{ \displaystyle \frac{dA^{\mu}}{d\tau} = - \Gamma ^{\mu}_{\nu \lambda}\frac{dx^{\lambda}}{d\tau}A^{\nu} }

[tex:{ \displaystyle \frac{\partial}{\partial x^{\mu}}\sqrt{g} F^{\mu \nu} = -\sqrt{g}J^{\nu}

\frac{\partial}{\partial x^{\alpha}}F_{\beta \gamma} + \frac{\partial}{\partial x^{\beta}}F_{\gamma \alpha} + \frac{\partial}{\partial x^{\gamma}}F_{\alpha \beta} }]

{ \displaystyle T^{\mu \nu} = g^{-\frac{1}{2}}\sum_{n}m_n\int \frac{dx_{n}^{\alpha}}{d\tau}dx_{n}^{\alpha}\delta ^4(x - x_n) }

{ \displaystyle R^{\lambda}_{\ \mu \nu \kappa} \equiv \frac{\partial \Gamma ^{\lambda}_{\mu \nu}}{\partial x^{\kappa}} - \frac{\partial \Gamma ^{\lambda}_{\mu \kappa}}{\partial x^{\nu}} + \Gamma ^{\eta}_{\mu \nu}\Gamma ^{\lambda}_{\kappa \eta} - \Gamma ^{\eta}_{\mu \kappa}\Gamma ^{\lambda}_{\nu \eta} }

  • なんか役立ちそう

[tex:{ \displaystyle V_{\mu ;\nu ;\kappa} - V_{\mu ;\kappa ;\nu} = -V_{\sigma}R^{\sigma}_{\ \mu \nu \kappa}

\end{equation} }]

{ \displaystyle V^{\lambda ;\nu ;\kappa} - V_{\lambda ;\kappa ;\nu} = V^{\sigma}R^{\lambda}_{\ \sigma \nu \kappa} }

{ \displaystyle R_{\mu \nu} = R^{\lambda}_{\ \mu \lambda \kappa} }

  • Curvature scalar

{ \displaystyle R = g^{\mu \kappa} R_{\mu \kappa} }

[tex:{ \displaystyle

R_{\lambda \mu \nu \kappa} = \frac{1}{2}\Bigl\{ \frac{\partial ^2 g_{\lambda \nu}}{\partial x^{\kappa} \partial x^{\mu}} - \frac{\partial ^2 g_{\mu \nu}}{\partial x^{\kappa} \partial x^{\lambda}} - \frac{\partial ^2 g_{\lambda \kappa}}{\partial x^{\nu} \partial x^{\mu}} + \frac{\partial ^2 g_{\mu \kappa}}{\partial x^{\nu} \partial x^{\lambda}} \Bigr\} + g_{\eta \sigma} (\Gamma ^{\eta}_{\nu \lambda}\Gamma ^{\sigma}_{\mu \lambda} - \Gamma ^{\eta}_{\kappa \lambda}\Gamma ^{\sigma}_{\mu \nu}) }]

[tex:{ \displaystyle

R_{\lambda \mu \nu \kappa} = R_{\nu \kappa \lambda \mu}

R_{\lambda \mu \nu \kappa} = -R_{\mu \lambda \nu \kappa} = -R_{\lambda \mu \kappa \nu} = R_{\mu \lambda \kappa \nu}

R_{\lambda \mu \nu \kappa} + R_{\lambda \kappa \mu \nu} + R_{\lambda \nu \kappa \mu} = 0 }]

{ \displaystyle R_{\lambda \mu \nu \kappa ;\eta} + R_{\lambda \mu \eta \nu ;\kappa} + R_{\lambda \mu \kappa \eta ;\nu} = 0 }

{ \displaystyle G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2}g_{\mu \nu}R = -8\pi GT_{\mu \nu} }

  • 宇宙定数を入れたやつ

{ \displaystyle G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2}g_{\mu \nu}R -\lambda g_{\mu \nu} = -8\pi GT_{\mu \nu} }

{ \displaystyle G^{\mu \nu}_{\ \ ;\mu} = 0 }

  • the harmonic coordinate conditions

{ \displaystyle \Gamma ^{\lambda} \equiv g^{\mu \nu}\Gamma ^{\lambda}_{\mu \nu} = 0 }