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Perturbation[expr,n] | returns the nth order perturbation of expr |

ExpandPerturbation[expr] | expand any unexpanded Perturbation[tensor] terms in expr |

DefTensorPerturbation[pert,tens,M] | defines the perturbation pert for the tensor tens on the manifold M |

DeMetricPerturbation[g,pert,par] | defines the perturbation pert of the metric g and the perturbations of the associated curvature tensors, with expansion parameter par |

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This gives the unexpanded perturbations of the input expressions, unless the input is a tensor for which a perturbation is defined. *xTras* automatically defines a metric perturbation (see the tutorial Metric Variations), hence the perturbation of the metric returned the perturbation tensor .

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This expanded the expressions in terms of any defined tensor perturbations, in this case . The covariant derivatives and the curvature tensors are now those of the background. They can easily be replaced with particular background values with the help of *xTras* functions.

SymmetricSpaceRules[cd,K] | produces replacement rules for the curvature tensors of the covariant derivative cd on a symmetric space of constant curvature K |

ToBackground[expr] | sends unperturbed (curvature) tensors to background values |

ExpandBackground[expr] | returns the perbutation expansion of expr on an arbitrary background |

The main functions for doing AdS perturbations.

In this section we will set up things to do perturbations around AdS. First, begin with defining a constant symbol for the cosmological constant.

Next, compute the equations of motion for the Einstein-Hilbert term plus a cosmological constant with VarL.

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We now set this background to the default background solution of ToBackground. It then also becomes the default background for PerturbBackground and ExpandBackground.

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Checking that the background solves the equations of motion can now be done with ToBackground:

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We can now expand the equations of motion on this background with ExpandBackground, in order to obtain the linearized Einstein equations:

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The tensor has a first index that labels the perturbation order. Often, it is convenient to remove this label. To that end, we define a new tensor that we will use as the metric perturbation.

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This process of replacing with can be automated by setting another option of ToBackground. The second rule below is for eliminating all higher order perturbations.

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Instead of perturbing the equations of motion, we can also perturb the action. Of course, the first non-trivial order here is not the first, but the second.

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When we vary the linear action with respect to , we should obtain the linearized equations of motion:

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