- #1
direct99
- 9
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Hello,
I was reading a book about general relativity and I came across these two equations
$$ \begin{align}
\mathrm{g}^{\mu\nu}_{,\rho}+
\mathrm{g}^{\sigma\nu}{\Gamma}^{\mu}_{\sigma\rho}+
\mathrm{g}^{\mu\sigma}{\Gamma}^{\nu}_{\rho\sigma}
-\frac{1}{2}(
{\Gamma}^{\sigma}_{\sigma}+{{\Gamma}^{\sigma}_{\sigma\rho}}
)
\mathrm{g}^{\mu\nu}
&=0,
\tag1
\\
\mathrm{g}^{[\mu\nu]}_{,\nu}
-\frac{1}{2}(
{\Gamma}^{\rho}_{\rho\nu}+{\Gamma}^{\rho}_{\nu\rho}
)
\mathrm{g}^{(\mu\nu)}
&=0,
\tag2
\end{align}
$$
and it says that by contracting equation (1) once with respect to ($\mu,\rho$), then with respect to ($\nu,\rho$), then subtracting the resulting equations we can get equation (2), however I can't see how that's possible.
Also, This connection is not symmetric with respect to the lower Indices
I was reading a book about general relativity and I came across these two equations
$$ \begin{align}
\mathrm{g}^{\mu\nu}_{,\rho}+
\mathrm{g}^{\sigma\nu}{\Gamma}^{\mu}_{\sigma\rho}+
\mathrm{g}^{\mu\sigma}{\Gamma}^{\nu}_{\rho\sigma}
-\frac{1}{2}(
{\Gamma}^{\sigma}_{\sigma}+{{\Gamma}^{\sigma}_{\sigma\rho}}
)
\mathrm{g}^{\mu\nu}
&=0,
\tag1
\\
\mathrm{g}^{[\mu\nu]}_{,\nu}
-\frac{1}{2}(
{\Gamma}^{\rho}_{\rho\nu}+{\Gamma}^{\rho}_{\nu\rho}
)
\mathrm{g}^{(\mu\nu)}
&=0,
\tag2
\end{align}
$$
and it says that by contracting equation (1) once with respect to ($\mu,\rho$), then with respect to ($\nu,\rho$), then subtracting the resulting equations we can get equation (2), however I can't see how that's possible.
Also, This connection is not symmetric with respect to the lower Indices
Last edited: