General tensor Lagrangians from the gravitational Higgs mechanism

The gravitational Higgs mechanism proposed by 't Hooft in arXiv:0708.3184 involves the spacetime metric gμν as well as the induced metric where ϕa (a = 0, …, 3), as we call it, break all four diffeomorphisms spontaneously via the vacuum expectation values ⟨ϕa⟩ ∝ xa. In this framework, we construct and analyze the most general action density in terms of various invariants involving the curvature tensors, connexion coefficients, and the contractions and the determinants of the two metric fields. We show that this action admits a consistent expansion about the flat background such that the resulting Lagrangian possesses several novel features not found in the linearized Einstein–Hilbert Lagrangian with Fierz–Pauli mass term (LEHL-FP): (i) its kinetic part generalizes that of LELHL-FP by weighing the corresponding structures with certain coefficients generated by invariants, (ii) the entire Lagrangian is ghost– and tachyon–free for mass terms not necessarily in the Fierz–Pauli form, and, (iii) a consistent mass term is generated with no apparent need to higher derivative couplings.