How to calculate backreaction in AdS space?

Question

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.

This post imported from StackExchange Physics at 2015-08-27 07:53 (UTC), posted by SE-user pinu

Answer 1

If you are ok with a computation of the self-force to linear order, it is best to use linear perturbations of the AdS background. I do not know whether this has been done for AdS in any non-horrible way (it may be non-obviously buried in some paper), but I certainly liked the treatment of linearized waves on dS by Bini, Cappozziello and Esposito.

The obvious problem with AdS are the boundary conditions at conformal infinity, but if you care about some immediate self-force, i.e. not something such as a particle accelerating throughout the whole cosmic history, this should be possible to hand-waive away (but maybe you are even interested in the interaction with boundary conditions because of AdS/CFT, those are unknown waters for me). I don't think AdS₃ will be any simpler, but one should expect the solutions to be very different in every such low dimension.

Comments

@Void Thank you for answering. Actually I was thinking about an ansatz based method may be very useful here. Suppose I start with the AdS metric and by imposing some symmetries (asymptotically the metric should remain AdS, etc) if I could write down an ansatz for the metric with few new parameters. Then by demanding that it should satisfy Einstein equations I can fix those parameters. Is it a right way of thinking about this problem? May be it won't give me very general result for the metric.. I don't know. 

---

@pinu By this method, you will obtain the field of a specific source which you do not have under control. Generically, the source will include negative matter and all sorts of things you do not really like. If you just want to see how self-force works for "some" source, I would instead recommend to read it off from some exact solution such as the C-Metric, see http://arxiv.org/pdf/gr-qc/0510101v1.pdf