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Sage Manifolds: Asymptotically de Sitter Spacetime in Fefferman-Graham Gauge

asked 2 years ago

Loreno Heer gravatar image

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!nρnqab|ρ=0.

I am supposed to get

q(1)ab=0 q(2)ab=˚Rab14˚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.

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answered 2 years ago

eric_g gravatar image

updated 2 years ago

Computations of the Ricci tensor in Sage are explicit, in given coordinate charts, and you cannot keep an undefined (symbolic) metric qab. What you can do though is to have some components of a metric expressed in terms of (unspecified) symbolic functions. An example is provided by the Friedmann-Robertson-Walker-Lemaître metric, the components of which contain the symbolic function a(t), see this notebook, which introduces as well the cosmological constant Λ as a symbolic variable.

Regarding tensor expansions with respect to a parameter ρ, they are implemented in Sage, see the methods series_expansion, truncate and set_calc_order and this notebook for an example of use.

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Asked: 2 years ago

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Last updated: Aug 16 '22