Abstract
Here we perform the Kaluza–Klein dimensional reduction from D+1 to D dimensions of massless Lagrangians described by a symmetric rank-2 tensor and invariant under transverse differmorphisms (TDiff). They include the linearized Einstein–Hilbert theory, linearized unimodular gravity and scalar tensor models. We obtain simple expressions in terms of gauge invariant field combinations and show that unitarity is preserved in all cases. After fixing a gauge, the reduced model becomes a massive scalar tensor theory. We show that the diffeomorphism (Diff) symmetry, instead of TDiff, is a general feature of the massless sector of consistent massive scalar tensor models. We discuss some subtleties when eliminating Stückelberg fields directly at action level as gauge conditions. A non local connection between the massless sector of the scalar tensor theory and the pure tensor TDiff model leads to a parametrization of the non conserved source which naturally separates spin-0 and spin-2 contributions in the pure tensor theory. The case of curved backgrounds is also investigated. If we truncate the non minimal couplings to linear terms in the curvature, vector and scalar constraints require Einstein spaces as in the Diff and WTDiff (Weyl plus Diff) cases. We prove that our linearized massive scalar tensor models admit those curved background extensions.
Highlights
In order to figure out why the graviton is eventually massless we must, start with some non zero mass and search for its consequences
We show that the diffeomorphism (Diff) symmetry, instead of transverse differmorphisms (TDiff), is a general feature of the massless sector of consistent massive scalar tensor models
We have lost information since the gauge φ = 0 does not completely determine the gauge parameters ψμ due to the residual symmetry δψμ = ∂μγ with γ = 0. This is similar to the fact that h = 0 at action level in the LEH action make us lose the trace of the linearized Einstein–Hilbert equation which becomes an integration constant related to the cosmological constant in the WTDiff model
Summary
In order to figure out why the graviton is eventually massless we must, start with some non zero mass and search for its consequences. C (2021) 81:547 sive Fierz–Pauli (FP) model [16] is the paradigmatic starting point It is a free theory for massive spin-2 particles where the linearized diffeomorphism (Diff) symmetry of the massless sector, linearized Einstein–Hilbert (LEH), is broken by the mass terms. The authors of [18,19] have concluded that, one could add a consistent mass term for the spin-0 sector without problems, there is no mass term in the spin-2 sector that might avoid the presence of ghosts They have shown that such negative result holds in the special case of the linearized unimodular gravity where the TDiff symmetry is enhanced to WTDiff (Weyl plus TDiff). We show, for the reader’s convenience, some technical details about spin projection and transition operators used to write down propagators
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