Massive Spin-2 and Multimetric Gravity

Giving a relativistic spin-2 field a mass adds helicity-1 and helicity-0 polarizations, changes its response to matter, and makes nonlinear consistency highly restrictive. This chapter develops the chain from the unique healthy quadratic mass term to dRGT constraints, bigravity mass eigenstates, screening, and branch-dependent black-hole and cosmological solutions. Its central lesson is methodological: removing one ghost is necessary, but background existence, perturbative stability, regularity, causal evolution, and EFT control remain separate tests.

Helpful background. Einstein Equations supplies the GR comparison; Gravitational Actions and Boundary Terms supplies the action principle; Formulations of General Relativity supplies ADM constraints; Gravity Beyond General Relativity locates this theory family; and Variational and Symplectic Geometry supplies the phase-space language used in the nonlinear count.

Enter this chapter

At the linear level, a symmetric tensor in four dimensions begins with ten components. The Fierz–Pauli mass tuning supplies five constraints and leaves the five polarizations of a massive spin-2 representation. A detuned trace becomes dynamical and ghostlike. Yet even the tuned theory has a discontinuous linear $m\to0$ source amplitude because its helicity-0 polarization still couples to the trace of stress energy.

Nonlinear interactions do two jobs that must not be confused. Near a source, helicity-0 derivative interactions can screen the extra force inside a Vainshtein radius. At the level of the full theory, the special dRGT potential preserves a primary and secondary second-class constraint, eliminating the Boulware–Deser mode. The first job explains a regime of phenomenological recovery; the second fixes the canonical degree-of-freedom count.

Shared square-root convention

The bimetric pages use

$$\begin{aligned} S_{\mathrm{bi}}={}&m_g^2\int d^4x\sqrt{-g}\,R(g) +m_f^2\int d^4x\sqrt{-f}\,R(f)\\ &-2m^4\int d^4x\sqrt{-g} \sum_{n=0}^{4}\beta_ne_n(S)+S_{\mathrm m}, \end{aligned}$$

with

$$S=\sqrt{g^{-1}f},\qquad S^\mu{}_\rho S^\rho{}_\nu=g^{\mu\rho}f_{\rho\nu}, \qquad \alpha=\frac{m_f}{m_g}.$$

Massive gravity treats $f_{\mu\nu}$ as prescribed and omits its Einstein–Hilbert term; bigravity makes it dynamical. Every use of the metric potential assumes a specified real, smooth square-root branch. The elementary-symmetric form and its parameter conventions are reviewed in de Rham 2014, § 6, eqs. (6.3)–(6.24), while the two-dynamical-metric action appears in Schmidt-May and von Strauss 2016, § 5.1, eqs. (5.1)–(5.7).

The action coefficients are not reduced Planck masses squared: $M_g^2=2m_g^2$ and $M_f^2=2m_f^2$. In the uncoupled $g$ theory, $m_g^2=(16\pi G_g)^{-1}$. With the site’s matter stress convention, the Einstein source is $T_{\mu\nu}/(2m_g^2)$. Also distinguish the interaction parameter $m$ from the background-dependent physical mass $m_{\mathrm{FP}}$. These normalizations matter in cosmology even when they are invisible in vacuum constraint counting.

Route through the six pages

1. Fierz–Pauli Massive Gravity derives the generic trace equation and uses Stückelberg fields to show why only the exact Fierz–Pauli tuning has five healthy linear modes.
2. The vDVZ Discontinuity and Vainshtein Limit compares the $-1/3$ and $-1/2$ trace weights in source exchange, then derives the cubic-Galileon Vainshtein scaling.
3. Ghost-Free Nonlinear Massive Gravity gives the proof architecture: an invertible shift change, lapse linearity, lapse-independent shift equations, and a primary–secondary constraint pair.
4. Bigravity and Multimetric Interactions promotes $f$ to a dynamical metric and diagonalizes proportional-background fluctuations into a massless two-mode and massive five-mode field.
5. Black-Hole Branches in Massive and Bigravity separates proportional, nonbidiagonal, and hairy solutions, then applies square-root, horizon, asymptotic, and spectral tests.
6. Cosmology, Strong Coupling, and Viability derives the FLRW Bianchi branches and distinguishes Higuchi, kinetic, gradient, strong-coupling, and cutoff conditions.

The order matters. Screening should be understood after the linear discontinuity, the $2+5$ spectrum after the nonlinear constraint, and solution viability after both.

A layered consistency test

Question	Required calculation	What a positive result does not establish
Is the intended number of modes present?	Constraint algebra or a controlled linear count	Positive kinetic energy or a regular background
Does the background solve both metric equations?	Branch equation, Bianchi condition, and boundary data	Perturbative stability
Are perturbations healthy?	Reduced kinetic, gradient, mass, and characteristic matrices	Nonlinear endpoint or ultraviolet completion
Is a black hole or cosmology globally admissible?	Real square root, regular horizon or lapse, and specified asymptotics	Validity above the EFT cutoff
Does the theory approach GR in the claimed regime?	Controlled decoupling, heavy-mass, or screened limit	Agreement in a singular limit or on another branch

This hierarchy prevents two common overstatements. “Ghost-free” in dRGT means that the Boulware–Deser canonical pair is absent; it does not mean every solution is stable. Conversely, a linearly stable background does not repair a missing nonlinear constraint in a different theory.

Connections to other gravity theories and observations

The chapter addresses classical theory construction, analytic limits, and the theoretical status of compact and homogeneous branches. Empirical likelihoods and numerical parameter bounds belong to Tests of Gravity, where assumptions about matter coupling and branch selection can be confronted with data. A stable low-energy quantum-gravity EFT or ultraviolet completion belongs to Quantum Gravity. Generic quantum-field-theory prerequisites receive only the bounded bridges required for the massive-spin-2 calculations here.

The two pages on open problems use results available through 2026-08-19. Their conclusions could change when nonlinear evolutions settle disputed endpoints, complete perturbation analyses change a branch’s stability, or a cutoff calculation changes the regime in which a classical solution is predictive.

Exercise

Transfer a branch polynomial carefully. Let $\beta_1=1$, $\beta_2=-1$, and $\beta_3=0$. The same polynomial $P(z)=\beta_1+2\beta_2z+\beta_3z^2$ occurs in the spherical black-hole and FLRW Bianchi factorizations. Find its root and explain why this does not identify the two physical branches.

Solution

$P(z)=1-2z$, so $z=1/2$. In the nonbidiagonal black-hole branch, $z=D/r$ is the angular-radius ratio and the root is used with a nonzero off-diagonal metric function $C$; horizon and asymptotic equations remain to be solved. In the algebraic FLRW branch, $z=y=b/a$ is a scale-factor ratio; the lapse and Friedmann equations remain to be solved. Equal algebraic roots reflect the same interaction polynomial, not equal geometries, boundary conditions, or stability operators.

References

• de Rham, Claudia. “Massive Gravity.” Living Reviews in Relativity 17 (2014): 7. doi:10.12942/lrr-2014-7; Open PDF.
• Schmidt-May, Angnis, and Mikael von Strauss. “Recent Developments in Bimetric Theory.” Journal of Physics A: Mathematical and Theoretical 49 (2016): 183001. doi:10.1088/1751-8113/49/18/183001; Open PDF.