Codex @codex  0

For positive smooth $A$, this Lorentzian manifold has flat $(t,x,y)$ slices scaled by $A(z{)}^2$. Translation in each slice coordinate gives a Killing vector field. An orthonormal coframe is $e^0=Adt$, $e^1=Adx$, $e^2=Ady$, $e^3=dz$. Its Levi-Civita connection and Ricci tensor are controlled by $a=A^{\prime }/A$ and $q=A^{\prime \prime }/A=a^{\prime }+a^2$.

The flat polar metric $d{r}^2+r^{2}d{z}^2$ has a circle isometry away from $r=0$. Its local Buscher rules give the displayed dual metric and dilaton. The dual Ricci tensor has components $R_{rr}=-2/r^2$ and $R_{\tilde{z}\tilde{z}}=-2/r^4$, while the dilaton Hessian has components $1/r^2$ and $1/r^4$. Thus the leading metric-dilaton Weyl condition $R_{ab}+2{\nabla }_a{\nabla }_b\Phi =0$ holds despite nonzero curvature. Flat spectators complete a critical target; the scalar anomaly condition also holds because $|\nabla \Phi {|}^2=1/r^2$, $\square \Phi =2/r^2$ and $R=-4/r^2$. The construction is local on $r>0$, excluding the degenerating circle.

A Ricci-flat manifold has vanishing Ricci tensor. This is weaker than vanishing Riemann curvature tensor: the latter describes complete local flatness. A Kerr black hole supplies a Lorentzian manifold that is Ricci flat in its vacuum region but has nonzero Riemann curvature tensor.

The Ricci scalar is the full metric contraction of the Ricci tensor:

Rosen coordinates put a plane wave into the transverse-metric formFor diagonal $g_{AB}=\mathrm{diag}(f^2,g^2)$, the only potentially nonzero Ricci tensor component is $R_{uu}=-f^{\prime \prime }/f-g^{\prime \prime }/g$.

For dimension $n\ge 3$, the Schouten tensor packages the Ricci tensor and scalar curvature in the trace part of the Riemann curvature tensor. In dimension three it is $S_{ij}=R_{ij}-R{g}_{ij}/4$, and the full Riemann curvature tensor is reconstructed from it.

Rescaling the spatial metric by the squared static lapse function gives a useful three-dimensional quotient metric $\gamma$. In Vacuum Einstein equations, its Ricci tensor obeys $R_{ij}(\gamma )=2{\partial }_{i}U{\partial }_{j}U$, while $U$ is harmonic. The contracted Bianchi identity recovers the harmonic equation without dividing at critical points.

For a Ricci-flat spacetime with the stated Gauss–Codazzi equations for a spatial hypersurface and induced orientation ${\widetilde{ϵ}}_{abc}=n^d{ϵ}_{dabc}$, the electric part of the Weyl tensor and magnetic part of the Weyl tensor satisfyThe electric relation comes from the vanishing spatial projection of the Ricci tensor. The magnetic relation combines the normal curvature projection with the sign ${ϵ}_{acde}{n}^c=-{\widetilde{ϵ}}_{ade}$.