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# 物理代写|广义相对论代写General Relativity代考|The C Symbols

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## 物理代写|广义相对论代写General Relativity代考|The C Symbols

The $\mathrm{C}$ symbols involve $g^{\mu \nu}$. They are also obtained as expansions in $\bar{v}^n$, that satisfy Eq. (8.21). From, $g^{\mu \xi} g_{\nu \xi}=\delta_\nu^\mu$,
\begin{aligned} 1 & =g^{0 \xi} g_{0 \xi}=g^{00} g_{00}+g^{0 i} g_{0 i}=\left(\eta^{00}+{ }^2 h^{00}\right)\left(\eta_{00}+{ }^2 h_{00}\right) \ & =1-{ }^2 h^{00}-{ }^2 h_{00} \ { }^2 h^{00} & =-{ }^2 h_{00}, \ 0 & =g^{0 \xi} g_{i \xi}=g^{00} g_{i 0}+g^{0 j} g_{i j}={ }^3 h_{i 0} \eta^{00}+{ }^3 h^{0 j} \eta_{i j}, \ { }^3 h^{i 0} & ={ }^3 h_{i 0}, \ \delta_j^i & =g^{i \xi} g_{j \xi}=g^{i 0} g_{j 0}+g^{i k} g_{j k}=\left(\eta^{i k}+{ }^2 h^{i k}\right)\left(\eta_{j k}+{ }^2 h_{j k}\right) \ & =\eta^{i k} \eta_{j k}+{ }^2 h_{j k} \eta^{i k}+{ }^2 h^{i k} \eta_{j k}=\delta^i{ }j+{ }^2 h{j k} \eta^{i k}+{ }^2 h^{i k} \eta_{j k}, \ { }^2 h^{i j} & =-{ }^2 h_{j i} . \end{aligned}

The $\mathrm{C}$ symbols involve terms with a partial derivative with respect to time. It’s important to note that as for powers of $\bar{v}$,
$$\frac{\partial}{\partial t} \propto \bar{v} / r$$
This result and Eqs. (8.18)-(8.20) are required to make sure Eq. (8.21) is satisfied. The $\mathrm{C}$ symbols are obtained from
$$\Gamma_{\mu \nu}^{\xi}=g^{\xi \chi}\left(g_{\mu \chi}, \nu+g_{\nu \chi}, \mu-g_{\mu \nu}, \chi\right) / 2 .$$
Those with contravariant index 0 are as follows:
\begin{aligned} \Gamma_{00}^0 & =g^{0 \chi}\left(g_{0 \chi, 0}+g_{0 \chi}, 0-g_{00, \chi}\right) / 2 \ & =g^{00}\left(g_{00,0}\right) / 2+g^{0 i}\left(2 g_{0 i}, 0-g_{00, i}\right) / 2, \ { }^3 \Gamma_{00}^0 & =\eta^{00}{ }^2 h_{00,0} / 2=-{ }^2 h_{00,0} / 2, \ \Gamma_{0 i}^0 & =g^{0 \chi}\left(g_{i \chi, 0}+g_{0 \chi, i}-g_{i 0, \chi}\right) / 2 \ & =g^{00}\left(g_{i 0,0}+g_{00, i}-g_{i 0,0}\right) / 2+g^{0 j}\left(g_{i j, 0}+g_{0 j, i}-g_{0 i, j}\right) / 2, \ { }^2 \Gamma_{0 i}^0 & =\eta^{00}\left({ }^2 h_{00, i}\right) / 2=-{ }^2 h_{00, i} / 2 \ \Gamma_{i j}^0 & =g^{0 \chi}\left(g_{i \chi, j}+g_{j \chi, i}-g_{i j}, \chi\right) / 2 \ & =g^{00}\left(g_{i 0, j}+g_{j 0, i}-g_{i j, 0}\right) / 2+g^{0 k}\left(g_{i k, j}+g_{j k, i}-g_{i j, k}\right) / 2, \ { }^3 \Gamma_{i j}^0 & =-\left({ }^3 h_{i 0, j}+{ }^3 h_{j 0, i}-{ }^2 h_{i j, 0}\right) / 2 . \end{aligned}

## 物理代写|广义相对论代写General Relativity代考|Ricci Tensor and Einstein Field Equations

In order to express the various expansion terms of a metric, as potentials in terms of the energy momentum tensor, just like ${ }^2 h_{00}=-2 \Psi_G$, the Einstein field equations are needed,
\begin{aligned} G_{\mu \xi} & =R_{\mu \xi}-g_{\mu \xi} R / 2=8 \pi T_{\mu \xi}, \ g^{\mu \xi}\left(R_{\mu \xi}-g_{\mu \xi} R / 2\right) & =8 \pi g^{\mu \xi} T_{\mu \xi}, \ R-2 R & =-R=8 \pi g^{\mu \xi} T_{\mu \xi}, \ R_{\mu \nu} & =8 \pi\left(T_{\mu \nu}-g_{\mu \nu} g^{\chi \xi} T_{\chi \xi} / 2\right) . \end{aligned}
The nonzero Ricci tensor elements $R_{\mu \nu}$ will be evaluated so that Eq. (8.21) is obeyed. The reader should note that $R_{\mu \nu}$ in this text has the opposite sign of $R_{\mu \nu}$ in Weinberg’s text. In some of the equations below, $\eta^{i i}$ is used to remind the reader to sum over $i$.
$$R_{\mu \nu}=R_{\mu \xi \nu}^{\xi}=\Gamma_{\mu \nu}^\chi \Gamma_{\xi \chi}^{\xi}-\Gamma_{\xi \mu}^\chi \Gamma_{\nu \chi}^{\xi}+\Gamma_{\mu \nu, \xi}^{\xi}-\Gamma_{\xi \mu}^{\xi}, \nu$$
However, the product $\Gamma \Gamma \propto \bar{v}^{>3}$ and may be neglected. Moreover,
\begin{aligned} R_{00} & =\Gamma_{00, \xi}^{\xi}-\Gamma_{\xi 0,0}^{\xi}=\Gamma_{00, i}^i-\Gamma_{i 0,0}^i, \ { }^2 R_{00} & ={ }^2 \Gamma_{00, i}^i=-\eta^{i i}{ }^i h_{00, i},{ }i / 2=-\nabla^2{ }^2 h{00} / 2 . \end{aligned}

## 物理代写|广义相对论代写General Relativity代考|The C Symbols

$\mathrm{C}$符号包含$g^{\mu \nu}$。它们也可以在$\bar{v}^n$中展开，满足式(8.21)。来自，$g^{\mu \xi} g_{\nu \xi}=\delta_\nu^\mu$;
\begin{aligned} 1 & =g^{0 \xi} g_{0 \xi}=g^{00} g_{00}+g^{0 i} g_{0 i}=\left(\eta^{00}+{ }^2 h^{00}\right)\left(\eta_{00}+{ }^2 h_{00}\right) \ & =1-{ }^2 h^{00}-{ }^2 h_{00} \ { }^2 h^{00} & =-{ }^2 h_{00}, \ 0 & =g^{0 \xi} g_{i \xi}=g^{00} g_{i 0}+g^{0 j} g_{i j}={ }^3 h_{i 0} \eta^{00}+{ }^3 h^{0 j} \eta_{i j}, \ { }^3 h^{i 0} & ={ }^3 h_{i 0}, \ \delta_j^i & =g^{i \xi} g_{j \xi}=g^{i 0} g_{j 0}+g^{i k} g_{j k}=\left(\eta^{i k}+{ }^2 h^{i k}\right)\left(\eta_{j k}+{ }^2 h_{j k}\right) \ & =\eta^{i k} \eta_{j k}+{ }^2 h_{j k} \eta^{i k}+{ }^2 h^{i k} \eta_{j k}=\delta^i{ }j+{ }^2 h{j k} \eta^{i k}+{ }^2 h^{i k} \eta_{j k}, \ { }^2 h^{i j} & =-{ }^2 h_{j i} . \end{aligned}

$\mathrm{C}$符号涉及对时间有偏导数的项。需要注意的是，对于$\bar{v}$的幂，
$$\frac{\partial}{\partial t} \propto \bar{v} / r$$

$$\Gamma_{\mu \nu}^{\xi}=g^{\xi \chi}\left(g_{\mu \chi}, \nu+g_{\nu \chi}, \mu-g_{\mu \nu}, \chi\right) / 2 .$$

\begin{aligned} \Gamma_{00}^0 & =g^{0 \chi}\left(g_{0 \chi, 0}+g_{0 \chi}, 0-g_{00, \chi}\right) / 2 \ & =g^{00}\left(g_{00,0}\right) / 2+g^{0 i}\left(2 g_{0 i}, 0-g_{00, i}\right) / 2, \ { }^3 \Gamma_{00}^0 & =\eta^{00}{ }^2 h_{00,0} / 2=-{ }^2 h_{00,0} / 2, \ \Gamma_{0 i}^0 & =g^{0 \chi}\left(g_{i \chi, 0}+g_{0 \chi, i}-g_{i 0, \chi}\right) / 2 \ & =g^{00}\left(g_{i 0,0}+g_{00, i}-g_{i 0,0}\right) / 2+g^{0 j}\left(g_{i j, 0}+g_{0 j, i}-g_{0 i, j}\right) / 2, \ { }^2 \Gamma_{0 i}^0 & =\eta^{00}\left({ }^2 h_{00, i}\right) / 2=-{ }^2 h_{00, i} / 2 \ \Gamma_{i j}^0 & =g^{0 \chi}\left(g_{i \chi, j}+g_{j \chi, i}-g_{i j}, \chi\right) / 2 \ & =g^{00}\left(g_{i 0, j}+g_{j 0, i}-g_{i j, 0}\right) / 2+g^{0 k}\left(g_{i k, j}+g_{j k, i}-g_{i j, k}\right) / 2, \ { }^3 \Gamma_{i j}^0 & =-\left({ }^3 h_{i 0, j}+{ }^3 h_{j 0, i}-{ }^2 h_{i j, 0}\right) / 2 . \end{aligned}

## 物理代写|广义相对论代写General Relativity代考|Ricci Tensor and Einstein Field Equations

\begin{aligned} G_{\mu \xi} & =R_{\mu \xi}-g_{\mu \xi} R / 2=8 \pi T_{\mu \xi}, \ g^{\mu \xi}\left(R_{\mu \xi}-g_{\mu \xi} R / 2\right) & =8 \pi g^{\mu \xi} T_{\mu \xi}, \ R-2 R & =-R=8 \pi g^{\mu \xi} T_{\mu \xi}, \ R_{\mu \nu} & =8 \pi\left(T_{\mu \nu}-g_{\mu \nu} g^{\chi \xi} T_{\chi \xi} / 2\right) . \end{aligned}

$$R_{\mu \nu}=R_{\mu \xi \nu}^{\xi}=\Gamma_{\mu \nu}^\chi \Gamma_{\xi \chi}^{\xi}-\Gamma_{\xi \mu}^\chi \Gamma_{\nu \chi}^{\xi}+\Gamma_{\mu \nu, \xi}^{\xi}-\Gamma_{\xi \mu}^{\xi}, \nu$$

\begin{aligned} R_{00} & =\Gamma_{00, \xi}^{\xi}-\Gamma_{\xi 0,0}^{\xi}=\Gamma_{00, i}^i-\Gamma_{i 0,0}^i, \ { }^2 R_{00} & ={ }^2 \Gamma_{00, i}^i=-\eta^{i i}{ }^i h_{00, i},{ }i / 2=-\nabla^2{ }^2 h{00} / 2 . \end{aligned}

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