Black hole transitions, AdS spacetime, and the distance conjecture

In this work, we investigate the connection between black hole instabilities and Swampland constraints, presenting new insights into the anti–de Sitter (AdS) distance conjecture. By examining the scale at which horizon instabilities of Schwarzschild-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>AdS</a:mi><a:mi>d</a:mi></a:msub></a:math> black holes take place—<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi mathvariant="normal">Λ</c:mi><c:mrow><c:mi>BH</c:mi></c:mrow></c:msub></c:math>—we uncover a universal scaling relation, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msub><f:mi mathvariant="normal">Λ</f:mi><f:mrow><f:mi>BH</f:mi></f:mrow></f:msub><f:mo>∼</f:mo><f:mo stretchy="false">|</f:mo><f:msub><f:mi mathvariant="normal">Λ</f:mi><f:mi>AdS</f:mi></f:msub><f:msup><f:mo stretchy="false">|</f:mo><f:mi>α</f:mi></f:msup></f:math>, with <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mfrac><l:mn>1</l:mn><l:mi>d</l:mi></l:mfrac><l:mo>≤</l:mo><l:mi>α</l:mi><l:mo>≤</l:mo><l:mfrac><l:mn>1</l:mn><l:mn>2</l:mn></l:mfrac></l:math>, linking the emergence of towers of states directly to instability scales as <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msub><n:mi mathvariant="normal">Λ</n:mi><n:mi>AdS</n:mi></n:msub><n:mo stretchy="false">→</n:mo><n:mn>0</n:mn></n:math>. 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 <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mrow><r:msub><r:mrow><r:mi>AdS</r:mi></r:mrow><r:mrow><r:mn>5</r:mn></r:mrow></r:msub><r:mo>×</r:mo><r:msup><r:mi>S</r:mi><r:mn>5</r:mn></r:msup></r:mrow></r:math>, we illustrate how higher-derivative corrections may alter these bounds, potentially extending their applicability toward the interior of moduli space. Using also general results about gravitational collapse in AdS, our analysis points toward a possible breakdown of the conjecture in <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>d</t:mi><t:mo>></t:mo><t:mn>10</t:mn></t:math>, suggesting an intriguing upper limit on the number of noncompact spacetime dimensions. Finally, we briefly discuss parallel considerations and implications for the dS case.