Massless limit of gravitational Higgs mechanism

In gravitational Higgs mechanism graviton components acquire mass via spontaneous diffeomorphism breaking by scalar vacuum expectation values. We point out that in the massless limit the resulting theory is not Einstein-Hilbert gravity (EHG) but constrained gravity (CG). Consequently, massive solutions in the massless limit must be compared to those in CG (as opposed to EHG). We discuss spherically symmetric solutions in this context. The Schwarzschild solution in EHG can be coordinate-transformed such that it is also a solution in CG. The resulting solutions are non-perturbative in the asymptotic regime, and are reproduced in the massless limit of asymptotic massive solutions, hence no van Dam-Veltman-Zakharov discontinuity. We point out that higher curvature terms must be included to obtain non-singular spherically symmetric massive solutions and discuss a suitable framework.