Black Hole Transitions, AdS and the Distance Conjecture

In this work, we investigate the connection between black hole instabilities and Swampland constraints, presenting new insights into the AdS Distance Conjecture. By examining the scale at which horizon instabilities of Schwarzschild-AdS$_d$ black holes take place$-\Lambda_{\mathrm{BH}}-$we uncover a universal scaling relation, $\Lambda_{\mathrm{BH}}\sim |\Lambda_{\mathrm{AdS}}|^{\alpha}$, with $\frac{1}{d}\leq \alpha\leq \frac{1}{2}$, linking the emergence of towers of states directly to instability scales as $\Lambda_{\mathrm{AdS}}\to 0$. This approach circumvents the explicit dependence on field-space distances, offering a refined formulation of the AdS Distance Conjecture grounded in physical black hole scales. From a top-down perspective, we find that these instability scales correspond precisely to the Gregory-Laflamme and Horowitz-Polchinski transitions, as expected for the flat space limit, and consistently with our proposed bounds. Furthermore, revisiting explicit calculations in type IIB string theory on AdS$_5\times S^5$, we illustrate how higher-derivative corrections may alter these bounds, potentially extending their applicability towards the interior of moduli space. Using also general results about gravitational collapse in AdS, our analysis points towards a possible breakdown of the conjecture in $d>10$, suggesting an intriguing upper limit on the number of non-compact spacetime dimensions. Finally, we briefly discuss parallel considerations and implications for the dS case.