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When we put the BPS condition and the extremality condition together on the most general black hole solution in $AdS_5$ (with minimally gauged supergravity), we get that the relation between the horizon radius and a parameter $a$ (that can be varied from $0$ to $1$) is the following: $$r_+^2=a(a+2)\tag{1}$$ I work in the units where the $AdS$ radius is set to unity - this sets the $g$ of the linked paper to unity. Also, I should mention that I have taken both the angular momenta to be identical - reducing two parameters $a,b$ of the linked paper to a single parameter $a=b$.

The equation $(1)$ clearly suggests that the radius of a SUSY extremal black hole can not be larger than $1$ by orders of magnitude. On the other hand, both the large extremal and large SUSY black holes individually seem to enjoy the status of valid solutions as one can easily verify from the general expressions in the linked (or any other similar) paper.

Mathematically, I find no inconsistency with these results (nor do I hope/wish to find) but I can't understand in a qualitative sense why the large black holes can't accommodate SUSY and extremality at the same time. In other words, no matter what combination of mass, charge and angular momenta you provide a large SUSY black hole with (that are consistent with the BPS condition), its temperature will always be non-zero. Is there any heuristic explanation for this?

PS: Moreover, as explained in the linked paper, the only way for a SUSY black hole to be free of naked CTCs is to be extremal (or to be a topological soliton). So, in other words, all the large SUSY black holes in $AdS_5$ will always suffer from naked CTCs as they can't be extremal.

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