# 線性化重力

## 方法

${\displaystyle g\,=\eta +h}$

## 線性化重力下的愛因斯坦重力場方程式[1]

${\displaystyle \Gamma _{\beta \gamma }^{\alpha }={\frac {1}{2}}\eta ^{\alpha \delta }(\partial _{\beta }h_{\delta \gamma }+\partial _{\gamma }h_{\beta \delta }-\partial _{\delta }h_{\beta \gamma })+{\mathcal {O}}(|h|)}$

${\displaystyle R_{bcd}^{a}=\partial _{c}\Gamma _{bd}^{a}-\partial _{d}\Gamma _{bc}^{a}+\Gamma _{cm}^{a}\Gamma _{bd}^{m}-\Gamma _{dm}^{a}\Gamma _{bc}^{m}}$

{\displaystyle {\begin{aligned}R_{\mu \nu }&=\partial _{\alpha }\Gamma _{\mu \nu }^{\alpha }-\partial _{\nu }\Gamma _{\mu \alpha }^{\alpha }\\&={\frac {1}{2}}\{h_{\ \nu ,\mu \alpha }^{\alpha }+h_{\ \mu ,\nu \alpha }^{\alpha }-h_{\mu \nu ,\alpha }^{\ \ \ \ \ \alpha }-h_{\ \alpha ,\mu \nu }^{\alpha }-h_{\ \mu ,\alpha \nu }^{\alpha }+h_{\mu \alpha ,\nu }^{\ \ \ \ \ \alpha }\}\end{aligned}}}

{\displaystyle {\begin{aligned}G_{\mu \nu }&=R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }R\\&={\frac {1}{2}}\{\partial _{\mu }\partial _{\alpha }h_{\ \nu }^{\alpha }+\partial _{\nu }\partial _{\alpha }h_{\ \mu }^{\alpha }-\partial _{\alpha }\partial ^{\alpha }h_{\mu \nu }-\partial _{\mu }\partial _{\nu }h\}-{\frac {1}{2}}\eta _{\mu \nu }(\partial _{\alpha }\partial _{\beta }h^{\alpha \beta }-\partial _{\alpha }\partial ^{\alpha }h)\\&=8\pi T_{\mu \nu }\end{aligned}}}

1. ^ Misner, Charles; Thorne, Kip; Wheeler, John. Gravitation. Princeton University Press. 2017. ISBN 9780691177793.