Appendix D: Manasse–Misner index order and the quadratic algebra

The original Manasse–Misner paper uses an index placement that can make its printed signs appear opposite to modern all-lowered conventions. The following translation keeps the mixed tensor fixed and shows where the sign enters when the first index pair is lowered and reordered.

Manasse and Misner write

\[ R_\mu{}^\nu{}_{\alpha\beta}, \tag{D.1}\]

whereas these notes write the same mixed tensor as

\[ R^\nu{}_{\mu\alpha\beta}. \tag{D.2}\]

After lowering the contravariant index, their ordering gives

\[ R^{\mathrm{MM}}_{\mu\nu\alpha\beta} =g_{\nu\rho}R^\rho{}_{\mu\alpha\beta} =R_{\nu\mu\alpha\beta} =-R_{\mu\nu\alpha\beta}. \tag{D.3}\]

Thus their printed expansion

\[ \begin{aligned} g_{00}&=-1+R^{\mathrm{MM}}_{0i0j}X^iX^j+\cdots, \\ g_{0i}&=\frac23R^{\mathrm{MM}}_{0jik}X^jX^k+\cdots, \\ g_{ij}&=\delta_{ij}+\frac13R^{\mathrm{MM}}_{ikjl}X^kX^l+\cdots \end{aligned} \tag{D.4}\]

becomes Eq. 4.48, Eq. 4.49, and Eq. 4.50.

For completeness, the mixed component algebra in Eq. 4.46 is

\[ \begin{aligned} g_{\hat 0k,ij} &=-\Gamma^{\hat 0}_{ki,j}+\Gamma^k_{\hat 0i,j}\\ &=-\frac23(R_{\hat 0ikj}+R_{\hat 0jki}), \end{aligned} \tag{D.5}\]

where Eq. 4.39 and Eq. 4.42, pair exchange, and the first Bianchi identity have been used. Likewise,

\[ \begin{aligned} g_{\ell m,ij} &=\Gamma^\ell_{mi,j}+\Gamma^m_{\ell i,j}\\ &=-\frac13(R_{\ell i m j}+R_{\ell j m i}). \end{aligned} \tag{D.6}\]

Because \(X^iX^j\) is symmetric, these second-derivative expressions reduce to the compact metric coefficients in the main text.