Quantum gravity in JNW spacetime

In this paper we study the behavior of a scalar field coupled to gravitons on the Janis-Newman-Winicour background, which somewhat interpolates between Minkowski and Schwarzschild space-times. The most important physical effect we find is that there is a 17-dimensional position-dependent mass matrix YABpxq which happens to be non-diagonal in the basis in which the kinetic energy term is diagonal. There is a different basis with a mixing between the scalar field and the graviton trace in which the mass matrix is diagonal, but this basis fails to diagonalize the kinetic energy piece. This is at variance with what happens in the Standard Model with the quark mixing, and is of course due to the fact that the mass matrix here is position dependent and thus it does not commute with the kinetic energy operator, so that both operators cannot be diagonalized simultaneously.