I am currently working through Chapter 1 on Chandrasekhar's book Mathematical Theory of Black holes (where he lays out the Einstein equations in terms of the Newman-Penrose formalism). I am presently trying to relate components of the Riemann tensor to the Newman Penrose scalars using
\begin{equation} R_{abcd}=W_{abcd}+\frac{1}{2}\left(\eta_{ac}R_{bd}-\eta_{bc}R_{ad}-\eta_{ad}R_{bc}+\eta_{bd}R_{ac}\right)-\frac{1}{6}\left(\eta_{ac}\eta_{bd}-\eta_{ad}\eta_{bc}\right)R , \end{equation} where $R_{abcd}$ is the Riemann tensor, $W_{abcd}$ the Weyl tensor, $R_{ab}$ the Ricci tensor, and $R$ the Ricci scalar.
I use $\eta_{ab}$ above as the above equation is written in terms of the Newman-Penrose formalism, so that $\eta_{ab}$ is in fact the "tetrad metric" and equals \begin{equation} \eta_{ab} = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{pmatrix} \end{equation} (but the Riemann tensor, etc are not zero). The indices $a$, etc run over $1,2,3,4$. In his book (Eq. (293) in Chapter 1), Chandrasekhar (using the above two formulas) derives formulas such as \begin{equation} R_{1314}=\frac{1}{2}R_{11} . \end{equation} (using arguments I won't get into here, he shows $C_{1314}=0$). The problem I have is: I get the opposite sign for all the Ricci tensor/scalar terms in my equations. For example I instead get \begin{equation} R_{1314}=-\frac{1}{2}R_{11} , \end{equation} as $\eta_{34}=-1$. Does anyone know what I could be doing wrong (and then give a correct derivation of the above equation), or if there is a series of typos in Chandrasekhar's book about the signs of the Ricci tensor/scalar terms?