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

asked 2022-08-15 10:40:24 +0200

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:

$$g = - \frac{3 d\rho^2}{\Lambda \rho^2} + \frac{3 q_{ab}dx^adx^b}{\Lambda \rho^2},$$

where $q_{ab} = q_{ab}(\rho,x^c)$ smooth, and $\Lambda> 0$ 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 ($R_{\alpha \beta} = \Lambda g_{\alpha\beta}$) using the expansion of $q = q^{(0)} + \rho q^{(1)} + \rho^2 q^{(2)} + \ldots$ where $q_{ab}^{(n)} = \frac{1}{n!}\frac{\partial^n}{\partial \rho^n}q_{ab}|_{\rho = 0}$.

I am supposed to get

$$q_{ab}^{(1)} = 0 $$ $$q_{ab}^{(2)} = \mathring{R}_{ab} - \frac{1}{4} \mathring{R} q_{ab}^{(0)} $$ $$q^{(0)ab}q_{ab}^{(3)} = 0 $$ $$D^aq_{ab}^{(3)} = 0 $$

where $\mathring{R}_{ab}$ and $\mathring{R}$ 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 2022-08-16 19:15:41 +0200

eric_g gravatar image

updated 2022-08-16 19:32:45 +0200

Computations of the Ricci tensor in Sage are explicit, in given coordinate charts, and you cannot keep an undefined (symbolic) metric $q_{ab}$. 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 $\Lambda$ as a symbolic variable.

Regarding tensor expansions with respect to a parameter $\rho$, 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: 2022-08-15 10:40:24 +0200

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Last updated: Aug 16 '22