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I would appreciate if someone can confirm or correct my understanding of extrinsic-curvature (as in the ADM 3+1 decomposition of spacetime) when dealing with a conformally-flat spacetime. (I updated the question's title and body after comments were rightfully made)

I followed mathematical derivations in literature (such as "3+1 Formalism and Bases of Numerical Relativity" ch. 8 by Gourgoulhon (2007)) and I read the answer in this thread. I assume that a conformally-flat spacetime is also asymptotically-flat if the factor function ($\Psi$ usually) is bound. Perhaps this is even trivial to mention, but I'll try to be precise.

Is it correct to infer that in a conformally-flat spacetime that is asymptotically-flat and stretches out to infinity - the extrinsic curvature tensor $K_{mn}$ vanishes (all of its components) at infinity?

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  • $\begingroup$ It would seem a confusing usage of the term if "asymptotically flat" spacetime did not tend to zero curvature at infinity. There may, of course, be some additional technical conditions. $\endgroup$
    – gandalf61
    Commented Jul 5 at 16:17
  • $\begingroup$ The title of the question mentions conformally-flat but the body speaks only about asymptotically flat. $\endgroup$
    – A.V.S.
    Commented 2 days ago
  • $\begingroup$ Thank you for the comments, I updated the question. $\endgroup$
    – AmnonJW
    Commented 2 days ago

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