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I need to compute the linearised Einstein Equations around a fixed metric $g_{μν}=\text{Minkowski metric} + h_{μν}$, which is not the flat metric.

Does anyone know a Mathematica package or a review paper that can help me?

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You can start there: Ricci tensor and Christoffel symbols in Mathematica –  Kuba May 21 at 12:15

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