  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.
