Island Paradigm and Information Recovery from Radiation

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 $\mathbb{U}_{bath}$ 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 $\mathbb{U}_{bath}$ 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.