Island paradigm and information recovery from radiation

A bstract The island paradigm for an AdS 2 eternal black hole in the Hartle-Hawking state coupled to a finite temperature non-gravitating bath asserts that after the Page time the operators in the black hole interior can be reconstructed using the bath degrees of freedom. We demonstrate this assertion by consideing a special operator $$ {\mathbbm{U}}_{\textrm{bath}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mtext>bath</mml:mtext> </mml:msub> </mml:math> that has nontrivial action only on the bath degrees of freedom. We show via the gravitational Euclidean path integral analysis that though before the Page time $$ {\mathbbm{U}}_{\textrm{bath}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mtext>bath</mml:mtext> </mml:msub> </mml:math> does not translate the operators in the bath to the black hole interior, after the Page time it takes them to the black hole interior.