# de-Sitter space as near-horizon limit of Black Holes

For 4D spacetimes with positive cosmological constant, de Sitter factor can emerge as a near-horizon limit in maximal Schwarzschild–de Sitter and Kerr–de Sitter black holes, when the event horizon of black hole approaches the cosmological horizon of de Sitter universe. The corresponding limiting solution is the Nariai metric (or its rotating generalization) which is the direct (or fibered for rotating case) product of two–dimensional de Sitter space and a two–sphere. Explicit forms of the metrics could be found for example in this paper, with links to earlier works.

There are various avenues for generalizations of these solutions: additional fields (Maxwell, dilaton etc.), replacing cosmological constant with some form of matter, generalizations for higher dimensional black holes.

Here is a general theorem that might help (theorem is proven here, see also this review):

Theorem. Any static near-horizon geometry is locally a warped product of $$AdS_2$$, $$dS_2$$ or $$\mathbb{R}^{1,1}$$ and . If  is simply connected this statement is global. In this case if  is compact and the strong energy conditions holds it must be the $$AdS_2$$ case or the direct product $$\mathbb{R}^{1,1}×$$.

So a rather general conclusion is that in order to have a de Sitter space as a factor in a near-horizon geometry we must have matter violating strong energy condition such as quintessence or positive cosmological constant and then the negative pressures must be large enough in comparison with black hole curvature, so that in the case of positive $$\Lambda$$ the size of black hole is comparable with de Sitter length scale.