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This might be a very straight forward and basic question in GR. I am interested in calculating backreaction due to certain matter field (say, scalar) in AdS space.

  1. Should I put the energy-momentum tensor ($T_{\mu \nu}$) for the scalar field into the Einstein equation with negative cosmological constant and try to solve it subject to some symmetries? (I feel this can be a difficult task!)

  2. Can I somehow use the original $AdS$ metric and see how the matter field (perturbatively) change it. I think this is the original idea of backreaction on some particular background. But don't have any idea how to implement it practically.

  3. Is there any subtlety involved due to non-trivial asymptotic behavior of $AdS$ space compared to Minkowski space.

  4. Any further simplification to this computation if I work specifically in $AdS_3$?

Any good references (books or original articles) on this topic where explicit calculations have been done will be highly appreciated.

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