Direct derivation of the dual generalized second law in the Unruh vacuum at null infinity

Establish the dual generalized second law for quantum fields in the Unruh vacuum directly from the asymptotic subalgebras of observables A_i^ at future null infinity I^+. Specifically, compute the one-sided modular Hamiltonian of the Unruh vacuum on A_i^ and prove the corresponding relative-entropy monotonicity that yields the dual generalized second law, without relying on effective modular Hamiltonians built from transmission or greybody factors.

Background

The paper extends results on dual generalized second laws at future null infinity by analyzing the Unruh vacuum in Schwarzschild and Kerr backgrounds. While formal expressions for modular Hamiltonians are motivated using symmetry properties on the past horizon and transmission through potential barriers, a fully algebraic proof at null infinity analogous to prior work for other vacua is not provided.

The authors emphasize that a satisfactory proof requires computing the modular Hamiltonian of the Unruh vacuum directly on the asymptotic subalgebras A_i^ at I^+, and then applying the monotonicity of relative entropy to obtain a dual generalized second law, paralleling their previous rigorous treatments for Hartle–Hawking and its regularizations.