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In the poincare co-ordinate representation metric of the ads space has a flat part that corresponds to the minkowski metric. Can one show that this flat part can be replaced by other metric that satisfies the condition of zero curvature and still the whole ads metric satisfies vaccum Einstein's equation with negative cosmological constant? Please help me with a proof.

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  • $\begingroup$ If your metric has zero curvature, locally you can choose normal coordinates in which it will just be the Minkowski metric, so you don't need to prove anything. $\endgroup$ – Holographer Feb 13 '15 at 15:45
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You can easily convince yourself that if you replace the Minkowski metric with another flat, maximally symmetric 4D metric, it has no impact on the Einstein equations.

Writing this up in a formal manner is left for the OP as an excercise :P

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