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We know that we can obtain $AdS_{2}$ x $S_{2}$ spacetime as the near-horizon geometry of an extremal/ near extremal RN (or Kerr) Black Hole in asymptotically flat spacetime. It is also known that we can also obtain $AdS_{3}$, from the near horizon limit of the extremal 6d Black string solution.

My question is as follows: Are there possible black hole solutions to Einstein Field equations, whose near -horizon geometry is a de-Sitter space (may or may not arise together with a compact space), instead of an AdS geometry? If yes, can someone please point me to the correct sources (research articles, review notes) to read about this.

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  • $\begingroup$ I am not an expert in the field, I have found an article by Andrew Strominger arxiv.org/abs/hep-th/0106113 where he proposes the duality between quantum gravity on $dS_D$ and CFT on $S^{D-1}$, however, without a black hole $\endgroup$ – spiridon_the_sun_rotator May 16 at 19:33
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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 $𝐻$. If $𝐻$ is simply connected this statement is global. In this case if $𝐻$ is compact and the strong energy conditions holds it must be the $AdS_2$ case or the direct product $\mathbb{R}^{1,1}×𝐻$.

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.

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Is not possible to obtain a dS geometry from the near horizon of a supersymmetric black hole because, by definition, the microscopic description of a supersymmetric black hole as a D-brane bound state preserves some vacuum supersymmetries, then its near-horizon geometry must preserve the same amount of supersymmetries.

The problem is that no supersymmetry can be unitarily realized on $dS$-spaces (see my answer to this question), it follows that a near horizon geometry of the sort $dS \times$$($Something$)$ cann't be the near horizon geometry of any supersymmetric black hole.

Nevertheless, if you relax some assumptions and allow non-unitary realizations of SUSY or branes with supergroups as gauge groups for their worldvolume theory, then exotic objects whose near-horizon geometry have $dS$ factors are possible.

References:

-Negative Branes, Supergroups and the Signature of Spacetime .

-Talk: Robbert Dijkgraaf - Negative Branes, Supergroups and the Signature of Spacetime.

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