Conformally flat metric

A metric is conformally flat when it is locally a positive scalar multiple of a flat metric. Multiplication by the conformal factor preserves null cones and unparametrized null geodesics.

A conformally flat metric is locally a positive scalar multiple of a flat metric. It makes its domain a conformally flat manifold.

For $\lambda >0$, this four-dimensional Euclidean metric has two asymptotically flat ends. Inversion $\rho =\lambda /r$ puts the metric near the $r=0$ end into the same form with $\rho \to \infty$. On $r>0$, its Ricci scalar vanishes because $\Delta (r^{-2})=0$, but its Ricci tensor is $R_{ij}=4\lambda ({\delta }_{ij}-4x_i{x}_j/r^2)/(r^2+\lambda {)}^2$. It therefore does not satisfy the Vacuum Einstein equations unless $\lambda =0$. For negative $\lambda$ the conformal factor vanishes at a finite radius, so that metric is not a smooth wormhole across that sphere.

For a positive conformal factor in dimension $n\ge 3$, set $S_{ij}=-u_{ij}+u_i{u}_j-\frac{1}{2}|\nabla u{|}^2{\delta }_{ij}$. These are the coordinate components of the Schouten tensor. In the convention $R_{ijkl}=g(e_i,R(e_k,e_l)e_j)$, the Riemann curvature tensor is $R_{ijkl}=e^{2u}({\delta }_{ik}{S}_{jl}+{\delta }_{jl}{S}_{ik}-{\delta }_{il}{S}_{jk}-{\delta }_{jk}{S}_{il})$. The orthonormal coframe $e^{u}d{x}^i$ has connection 1-forms ${\omega }^i{}_j=u_{j}d{x}^i-u_{i}d{x}^j$; substituting into Cartan's second structure equation proves this expression directly. Its contractions give $R_{ij}=(n-2)S_{ij}+{\delta }_{ij}{\sum }_k{S}_{kk}$ and $R=e^{-2u}[-2(n-1)\Delta u-(n-1)(n-2)|\nabla u{|}^2]$.

A conformal factor is the positive scalar field multiplying a metric under a conformal rescaling. In numerical relativity one often writes a spatial metric as ${\gamma }_{ij}={\psi }^4{\tilde{\gamma }}_{ij}$.

The difference of the two Levi-Civita connections is ${\tilde{\Gamma }}^a{}_{bc}-{\Gamma }^a{}_{bc}={\delta }_b^a{\partial }_c\log \Omega +{\delta }_c^a{\partial }_b\log \Omega -g_{bc}{g}^{ad}{\partial }_d\log \Omega$. Along a null geodesic with tangent $k$, the last term vanishes and the remaining acceleration is $2k(\log \Omega )k$. It is parallel to the tangent, so changing the affine parameter removes it. Thus a positive conformal factor preserves unparametrized null geodesics, although it generally does not preserve timelike or spacelike geodesics.

If $\tilde{g}={\Lambda }^{2}g$, a null geodesic with affine parameter $\tau$ for $g$ has affine parameter $\tilde{\tau }$ for $\tilde{g}$ when $d\tilde{\tau }/d\tau =C{\Lambda }^2$ along the curve. Indeed the conformal change of the Levi-Civita connection gives ${\tilde{\nabla }}_{V}V=2V(\log |\Lambda |)V$; rescaling $V$ by $d\tau /d\tilde{\tau }$ cancels this tangential acceleration. Constant $C>0$ preserves the direction of parametrization. Agreement of unparametrized null geodesics does not imply agreement of their affine parameters or completeness.