= Linearized curvature of a two-potential weak-field metric
{title2=$ds^2=-(1+2\Phi/c^2)(dx^0)^2+(1-2\Psi/c^2)d\mathbf x^2$}
For <metric signature> $(-,+,+,+)$, $x^0=ct$ and small dimensionless potentials, the <metric perturbation> is $h_{00}=-2\Phi/c^2$, $h_{ij}=-2\Psi\delta_{ij}/c^2$. Substitution into the <linearized Ricci tensor and scalar> formula gives
$$
R_{00}=\frac{\nabla^2\Phi+3\partial_0^2\Psi}{c^2},\quad
R_{0i}=\frac{2\partial_0\partial_i\Psi}{c^2},\quad
R_{ij}=\frac{\partial_i\partial_j(\Psi-\Phi)+\delta_{ij}(\nabla^2\Psi-\partial_0^2\Psi)}{c^2}.
$$
The contraction is $R=(4\nabla^2\Psi-2\nabla^2\Phi-6\partial_0^2\Psi)/c^2$, so $G^0{}_0=-2\nabla^2\Psi/c^2$. These signs use $R^\mu{}_{\nu\alpha\beta}V^\nu=[\nabla_\alpha,\nabla_\beta]V^\mu$. Static zero-anisotropic-stress boundary conditions give $\Phi=\Psi$ and the <Newtonian limit>; without those assumptions the two potentials and their time derivatives must be retained.
Back to article page
