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Non-commutative (sometimes called "fuzzy") black holes are solutions of Einstein's equations obtained with a previous basic assumption of non-commutativity of the coordinates $[x^{\mu},x^{\nu}]=i\, \theta^{\mu\nu}$. This assumption is considered helpful in the sense that removes the singularity in $r=0$. See this link for a review, even though there have been more recent developments.

My question regards the possibility of extending this black hole solution to a time-dependent one, in order to describe an evaporating or accreting case.

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