Sage Manifolds: Asymptotically de Sitter Spacetime in Fefferman-Graham Gauge
I am new to the Sage Manifolds package and I am trying to model an asymptotically de Sitter spacetime in the Fefferman-Graham gauge (also called Ambient Metric). In the article I am currently reading this is defined as such:
g=−3dρ2Λρ2+3qabdxadxbΛρ2,
where qab=qab(ρ,xc) smooth, and Λ>0 is the cosmological constant.
I want to input the above data into Sage and compute the Ricci curvature tensor and then solve the (vacuum) Einstein equation (Rαβ=Λgαβ) using the expansion of q=q(0)+ρq(1)+ρ2q(2)+… where q(n)ab=1n!∂n∂ρnqab|ρ=0.
I am supposed to get
q(1)ab=0 q(2)ab=˚Rab−14˚Rq(0)ab q(0)abq(3)ab=0 Daq(3)ab=0
where ˚Rab and ˚R are the Ricci tensor and scalar of q(0) and D its covariant derivative. I looked at the sage manifolds tutorial. And it is somewhat clear how to define a manifold and metric. However what I can not figure out is how to have a metric with symbolic functions (eg. q above) and the cosmological constant which I prefer to just keep as a symbolic variable.