Probing hidden topology with quantum detectors

We consider the transition rate of a static Unruh-DeWitt detector in two (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mn>2</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:math>)-dimensional black hole spacetimes that are isometric to the static Bañados-Teitelboim-Zanelli black hole outside the horizon but have no asymptotically locally anti–de Sitter exterior behind the horizon. The spacetimes are the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi mathvariant="double-struck">R</c:mi><c:msup><c:mi mathvariant="normal">P</c:mi><c:mn>2</c:mn></c:msup></c:math> geon, with spatial topology <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi mathvariant="double-struck">R</g:mi><g:msup><g:mi mathvariant="normal">P</g:mi><g:mn>2</g:mn></g:msup><g:mo>\</g:mo><g:mo stretchy="false">{</g:mo><g:mtext>point at infinity</g:mtext><g:mo stretchy="false">}</g:mo></g:math>, and the Swedish geon of Åminneborg ., with spatial topology <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msup><m:mi>T</m:mi><m:mn>2</m:mn></m:msup><m:mo>\</m:mo><m:mo stretchy="false">{</m:mo><m:mtext>point at infinity</m:mtext><m:mo stretchy="false">}</m:mo></m:math>. For a conformal scalar field, prepared in the Hartle-Hawking-type state that is induced from the global vacuum on the anti–de Sitter covering space, we show numerically that the detector’s transition rate distinguishes the two spacetimes, particularly at late exterior times, and we trace this phenomenon to the differences in the isometries that are broken by the quotient construction from the universal covering space. Our results provide an example in which information about the interior topology of a black hole is accessible to a quantum observer outside the black hole.