- #1
ft_c
- 34
- 0
Hi all
I'm having trouble understanding what I'm missing here. Basically, if I write the Ricci scalar as the contracted Ricci tensor, then take the covariant derivative, I get something that disagrees with the Bianchi identity:
\begin{align*}
R&=g^{\mu\nu}R_{\mu\nu}\\
\Rightarrow \nabla R&=\nabla(g^{\mu\nu}R_{\mu\nu})\\
&=\nabla(g^{\mu\nu})R_{\mu\nu}+g^{\mu\nu}\nabla(R_{\mu\nu})\\
&=g^{\mu\nu}\nabla(R_{\mu\nu})\\
g_{\mu\nu}\nabla R&=g_{\mu\nu}g^{\mu\nu}\nabla(R_{\mu\nu})\\
&=4\nabla(R_{\mu\nu})\\
\Rightarrow \nabla(R_{\mu\nu}-\tfrac{1}{4}g_{\mu\nu} R)&=0
\end{align*}
whereas the contracted Bianchi identity is
$$\nabla(R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R)=0$$
If anyone could let me know what's going wrong here that would be much appreciated! Thanks very much in advance
I'm having trouble understanding what I'm missing here. Basically, if I write the Ricci scalar as the contracted Ricci tensor, then take the covariant derivative, I get something that disagrees with the Bianchi identity:
\begin{align*}
R&=g^{\mu\nu}R_{\mu\nu}\\
\Rightarrow \nabla R&=\nabla(g^{\mu\nu}R_{\mu\nu})\\
&=\nabla(g^{\mu\nu})R_{\mu\nu}+g^{\mu\nu}\nabla(R_{\mu\nu})\\
&=g^{\mu\nu}\nabla(R_{\mu\nu})\\
g_{\mu\nu}\nabla R&=g_{\mu\nu}g^{\mu\nu}\nabla(R_{\mu\nu})\\
&=4\nabla(R_{\mu\nu})\\
\Rightarrow \nabla(R_{\mu\nu}-\tfrac{1}{4}g_{\mu\nu} R)&=0
\end{align*}
whereas the contracted Bianchi identity is
$$\nabla(R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R)=0$$
If anyone could let me know what's going wrong here that would be much appreciated! Thanks very much in advance