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Fierz–Pauli Massive Gravity

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A relativistic massive spin-2 field has five polarizations in four dimensions, but a generic quadratic mass term propagates a sixth scalar with the wrong-sign kinetic term. The Fierz–Pauli tuning is the unique Lorentz-invariant quadratic tuning that removes this scalar. The result is linear: it identifies the correct weak-field starting point, not by itself a consistent nonlinear theory.

Required background. Variational and Symplectic Geometry supplies the constraint language used in the degree-of-freedom count, and Weak-Field Gravity supplies linearized diffeomorphisms and the spin-2 kinetic operator.

Helpful background. Einstein Equations fixes the field-equation normalization; Gravitational Actions and Boundary Terms supplies the action principle; Formulations of General Relativity supplies the ADM comparison; and Gravity Beyond General Relativity places the mass deformation among modified-gravity theories.

The quadratic spin-2 theory

Section titled “The quadratic spin-2 theory”

Let hμν=hνμh_{\mu\nu}=h_{\nu\mu} be a field on Minkowski spacetime, with h=ημνhμνh=\eta^{\mu\nu}h_{\mu\nu}. Up to integrations by parts and an overall normalization, ghost-free Lorentz-invariant two-derivative kinetics are the quadratic Einstein–Hilbert term. We write

LA=−14hμνEμνρσhρσ−m28(hμνhμν−Ah2),\mathcal L_A=-\frac14 h^{\mu\nu}\mathcal E_{\mu\nu}{}^{\rho\sigma}h_{\rho\sigma} -\frac{m^2}{8}\left(h_{\mu\nu}h^{\mu\nu}-A h^2\right),

where

Eμνρσhρσ=−12[□hμν−2∂(μ∂ρhν)ρ+∂μ∂νh−ημν(□h−∂ρ∂σhρσ)].\mathcal E_{\mu\nu}{}^{\rho\sigma}h_{\rho\sigma} =-\frac12\left[ \Box h_{\mu\nu}-2\partial_{(\mu}\partial^\rho h_{\nu)\rho} +\partial_\mu\partial_\nu h -\eta_{\mu\nu}\left(\Box h-\partial^\rho\partial^\sigma h_{\rho\sigma}\right) \right].

The kinetic term is invariant under hμν↦hμν+∂(μξν)h_{\mu\nu}\mapsto h_{\mu\nu}+\partial_{(\mu}\xi_{\nu)}; the mass term breaks that gauge symmetry. The derivation from the most general quadratic kinetic operator and the normalization used here appear in de Rham 2014, § 2.2.1, eqs. (2.32)–(2.38).

The dimensionless number AA is a tuning parameter, not a field. The Fierz–Pauli point is A=1A=1. This distinction matters below, where the vector Stückelberg field is denoted by BμB_\mu.

Why a generic tuning carries a sixth mode

Section titled “Why a generic tuning carries a sixth mode”

Varying the action and taking a divergence uses the linearized Bianchi identity, ∂μEμν=0\partial^\mu\mathcal E_{\mu\nu}=0. In vacuum it gives four relations,

∂μhμν=A ∂νh.\partial^\mu h_{\mu\nu}=A\,\partial_\nu h.

The trace of the field equation then becomes

(1−A)□h+m22(1−4A)h=0.(1-A)\Box h+\frac{m^2}{2}(1-4A)h=0.

For A≠1A\ne1, this is a differential equation: the trace is an additional propagating scalar. For A=1A=1 and m≠0m\ne0, it instead imposes the algebraic constraint h=0h=0, after which the divergence relation becomes ∂μhμν=0\partial^\mu h_{\mu\nu}=0. Thus the ten components of a symmetric tensor obey four transverse constraints and one trace constraint,

10−4−1=5.10-4-1=5.

Those five components form the irreducible massive spin-2 representation. To see their physical meaning, work in the rest frame of a positive-mass Fourier mode. Transversality sets h0μ=0h_{0\mu}=0, and tracelessness leaves a symmetric trace-free 3×33\times3 matrix: 6−1=56-1=5 amplitudes. The remaining equations reduce to (□−m2)hμν=0(\Box-m^2)h_{\mu\nu}=0, giving ω2=k2+m2\omega^2=\mathbf k^2+m^2 with the site’s signature. Thus m2>0m^2\gt0 is the ordinary massive branch used in this count. This equation-of-motion count agrees with the Hamiltonian analysis in Hinterbichler 2012, § 2.1, pp. 14–16. Away from the Fierz–Pauli point the trace equation restores a sixth mode.

Stückelberg diagnosis of the ghost

Section titled “Stückelberg diagnosis of the ghost”

The Stückelberg formulation makes the health of that sixth mode visible. Restore the broken linearized diffeomorphism by replacing

hμν⟼hμν+2∂(μχν),χμ=Bμm+∂μπm2.h_{\mu\nu}\longmapsto h_{\mu\nu}+2\partial_{(\mu}\chi_{\nu)}, \qquad \chi_\mu=\frac{B_\mu}{m}+\frac{\partial_\mu\pi}{m^2}.

Under hμν↦hμν+∂(μξν)h_{\mu\nu}\mapsto h_{\mu\nu}+\partial_{(\mu}\xi_{\nu)} one chooses χμ↦χμ−ξμ/2\chi_\mu\mapsto\chi_\mu-\xi_\mu/2. The pure-π\pi part of the generic mass term is

Lπ,hd=−12m2[(∂μ∂νπ)2−A(□π)2].\mathcal L_{\pi,\mathrm{hd}} =-\frac{1}{2m^2}\left[ (\partial_\mu\partial_\nu\pi)^2-A(\Box\pi)^2 \right].

At A=1A=1, (∂μ∂νπ)2−(□π)2(\partial_\mu\partial_\nu\pi)^2-(\Box\pi)^2 is a total derivative. At any other AA, a term proportional to (1−A)(□π)2(1-A)(\Box\pi)^2 remains, producing fourth-order equations and an Ostrogradsky pair with one wrong-sign mode. This is the short Stückelberg proof of uniqueness; see de Rham 2014, § 2.2.2, eqs. (2.39)–(2.46).

At the tuned point, the scalar has not disappeared. It obtains a healthy two-derivative kinetic term through mixing with hμνh_{\mu\nu}. After the field shift hμν=h~μν+πημνh_{\mu\nu}=\widetilde h_{\mu\nu}+\pi\eta_{\mu\nu}, the kinetic sector contains

−14h~Eh~−34(∂π)2−18FμνFμν,Fμν=∂μBν−∂νBμ.-\frac14\widetilde h\mathcal E\widetilde h -\frac34(\partial\pi)^2 -\frac18F_{\mu\nu}F^{\mu\nu}, \qquad F_{\mu\nu}=\partial_\mu B_\nu-\partial_\nu B_\mu.

It therefore displays 22 helicity-2, 22 helicity-1, and 11 helicity-0 degrees of freedom. The diagonalization, including the mass mixings suppressed in the displayed kinetic sector, is given in de Rham 2014, § 2.2.2, eqs. (2.47)–(2.48).

What the tuning does and does not establish

Section titled “What the tuning does and does not establish”

Fierz–Pauli tuning removes the linear sixth-mode ghost about the background on which the quadratic expansion was performed. It does not prove that an arbitrary nonlinear potential retains a constraint, nor that every background is stable, causal, or within an effective field theory’s cutoff. The nonlinear Boulware–Deser question is answered only after the Hamiltonian constraint structure is checked.

The limit m→0m\to0 is also subtle. The helicity-1 variables decouple from a conserved source at linear order, but the helicity-0 field couples to the trace TT. That coupling leads directly to the vDVZ discontinuity and motivates the nonlinear Vainshtein analysis on the next page.

Exercises

Section titled “Exercises”

1. Constraint count at the tuned point. Couple the field to a conserved source and consider the vacuum region outside it. Starting from

Eμνρσhρσ+m22(hμν−ημνh)=0,\mathcal E_{\mu\nu}{}^{\rho\sigma}h_{\rho\sigma} +\frac{m^2}{2}(h_{\mu\nu}-\eta_{\mu\nu}h)=0,

derive the constraints and count the polarizations.

Solution

Taking a divergence annihilates the kinetic operator and gives ∂μhμν−∂νh=0\partial^\mu h_{\mu\nu}-\partial_\nu h=0. The trace of the kinetic operator is □h−∂μ∂νhμν\Box h-\partial^\mu\partial^\nu h_{\mu\nu}, which vanishes after using that relation. The trace of the mass term is −3m2h/2-3m^2h/2, so h=0h=0. The divergence relation then gives ∂μhμν=0\partial^\mu h_{\mu\nu}=0. Four transverse conditions plus one trace condition reduce ten components to five. None is a gauge choice because the mass term has broken the linearized diffeomorphism symmetry.

2. Test the detuning. Repeat the trace step for generic AA and explain why merely taking AA close to one is not enough.

Solution

The divergence gives ∂μhμν=A∂νh\partial^\mu h_{\mu\nu}=A\partial_\nu h. Hence the kinetic trace is (1−A)□h(1-A)\Box h, and the full trace is (1−A)□h+m2(1−4A)h/2=0(1-A)\Box h+m^2(1-4A)h/2=0. For every nonzero 1−A1-A, the trace has its own second-order propagation equation. The Stückelberg form shows that this comes from a fourth-derivative scalar sector with an extra ghostlike canonical pair. A small coefficient can move the associated scale, but it does not restore the missing constraint; only A=1A=1 does.

References

Section titled “References”