Cold Black Holes in the Einstein-Scalar Field System

We study Einstein gravity coupled to a massless scalar field in a static spherically symmetric space-time in four dimensions. Black hole solutions exist when the kinetic energy of the scalar field is negative, that is, for a phantom field. These “scalar black holes” have an infinite horizon area and zero temperature. They are related through a conformal transformation with similar objects in the Jordan frames of scalar-tensor theories of gravity. The thermodynamical properties of these solutions are discussed. It is proved that any static, spherically symmetric black holes with an infinite horizon area have zero Hawking temperature.