Back Reaction of 4D Conformal Fields on Static Geometry

In classical mechanics, a black hole is described by a vacuum solution with the horizon of the Einstein equation. For spherical case, it is the Schwarzschild metric, and the location of the horizon is given by the Schwarzschild radius. (Note that the Schwarzschild radius can also be defined even for a star without horizon.) In quantum mechanics, a black hole evaporates and information inside it seems to be lost, which is contradict to the principle of quantum mechanics. An effective way to address this problem is to consider again “What is the black hole in quantum mechanics?” In this paper, we examined how robust the use of the Schwarzschild metric to represent a black hole is in quantum mechanics. We consider conformal matters (e.g. electromagnetic field) and introduce the quantum effect (4D conformal anomaly) into the Einstein equation, which necessarily makes the equation non-vacuum. We start from the Schwarzschild metric, add the quantum effect perturbatively, and solve the Einstein equation in a self-consistent manner. Then, we showed that the quantum effect can play a crucial role in shaping the static geometry near the Schwarzschild radius. The geometry depends on a parameter corresponding to a boundary condition, and the existence of the horizon requires the fine-tuning. Therefore, in quantum mechanics, a typical static spherical solution does not have a horizon.