Noncommutative (sometimes called "fuzzy") black holes are solutions of Einstein's equations obtained with a previous basic assumption of noncommutativity of the coordinates x^{\mu}. This assumption is considered helpful in the sense that removes the (otherwise irremovable) singularity in r=0. See the link at the bottom for a review, even though there have been more recent developments.

My question regards the possibility of extending the black hole solution to a time-dependent one.

http://arxiv.org/pdf/0807.1939.pdf