Bigravity and Multimetric Interactions

Ghost-free bigravity promotes the reference metric of dRGT massive gravity to a second dynamical metric. Around a proportional Einstein background, its linear fluctuations diagonalize into one massless spin-2 field with two polarizations and one Fierz–Pauli massive spin-2 field with five. That clean $2+5$ split is background-dependent, while the nonlinear seven-mode count follows from the constraint structure.

Required background. Variational and Symplectic Geometry supplies constraint and phase-space counting, and Weak-Field Gravity supplies linear mode diagonalization.

Helpful background. Einstein Equations supplies Einstein backgrounds; Gravitational Actions and Boundary Terms fixes the two-metric action principle; Formulations of General Relativity supplies ADM counting; and Gravity Beyond General Relativity supplies the wider theory-space setting.

One interaction convention

We use

$$S=m_g^2\!\int\!d^4x\sqrt{-g}\,R(g) +m_f^2\!\int\!d^4x\sqrt{-f}\,R(f) -2m^4\!\int\!d^4x\sqrt{-g}\sum_{n=0}^{4}\beta_n e_n(S) +S_{\mathrm m},$$

where

$$S=\sqrt{g^{-1}f},\qquad \alpha=\frac{m_f}{m_g}, \qquad \mu^2=\frac{m^4}{m_g^2}.$$

A real, smooth square-root branch is assumed. The constants $m_g,m_f,m$ have mass dimension one, but $m$ is an interaction parameter rather than the physical Fierz–Pauli mass derived below. Since the Einstein terms have no $1/2$ prefactor, the reduced Planck masses are $M_g^2=2m_g^2$ and $M_f^2=2m_f^2$. With the site’s stress convention, the $g$ matter source enters its field equation as $T^{(g)}_{\mu\nu}/(2m_g^2)$; the review’s Eq. (5.24) uses half this stress tensor.

The potential preserves only the diagonal subgroup of the two diffeomorphism groups. Its special lapse dependence supplies one additional second-class constraint pair beyond the four diagonal gauge generators. The two spatial metrics and momenta give $24$ phase-space variables: four first-class constraints remove $8$, and the second-class pair removes $2$, leaving $(24-8-2)/2=7$ nonlinear degrees of freedom. The action, field equations, and constraint count are reviewed in Schmidt-May and von Strauss 2016, §§ 5.1–5.2, eqs. (5.1)–(5.8).

Proportional Einstein backgrounds

Set

$$\bar f_{\mu\nu}=c^2\bar g_{\mu\nu},\qquad c\gt0, \qquad \bar S=c\,\mathbf 1.$$

Take vacuum and the indicated positive square-root branch. The Bianchi constraint makes $c$ constant on the generic proportional branch. Both metric equations reduce to Einstein equations written with $\bar g_{\mu\nu}$, but they agree only if

$$\Lambda_g(c)=\Lambda_f(c),$$

with

$$\Lambda_g=\mu^2\left(\beta_0+3c\beta_1+3c^2\beta_2+c^3\beta_3\right),$$ $$\Lambda_f=\frac{\mu^2}{\alpha^2c^2} \left(c\beta_1+3c^2\beta_2+3c^3\beta_3+c^4\beta_4\right).$$

Here $\Lambda_f$ is the coefficient of $\bar g_{\mu\nu}$ in the $f$ equation; the cosmological constant multiplying $\bar f_{\mu\nu}$ is $\Lambda_f/c^2$. This algebraic background equation determines admissible values of $c$ for generic parameters. It must be solved before inserting a mass formula: choosing $c$ independently would generally not define a solution. Negative square-root eigenvalues require a separately specified branch, even though $f=c^2g$ alone cannot distinguish their sign. The reduction and background polynomial are derived in Schmidt-May and von Strauss 2016, § 5.3.1, eqs. (5.9)–(5.12).

Massless and massive combinations

Perturb both metrics,

$$g_{\mu\nu}=\bar g_{\mu\nu}+\delta g_{\mu\nu}, \qquad f_{\mu\nu}=c^2\bar g_{\mu\nu}+\delta f_{\mu\nu}.$$

Convenient unnormalized eigen-combinations are

$$\delta G_{\mu\nu}=\delta g_{\mu\nu}+\alpha^2\delta f_{\mu\nu}, \qquad \delta M_{\mu\nu}=\frac{\delta f_{\mu\nu}-c^2\delta g_{\mu\nu}}{2c}.$$

The sum-like field $\delta G_{\mu\nu}$ obeys the massless linearized Einstein equation. The difference-like field $\delta M_{\mu\nu}$ obeys a Fierz–Pauli equation with mass, measured relative to $\bar g_{\mu\nu}$,

$$m_{\mathrm{FP}}^2 =\mu^2\frac{1+\alpha^2c^2}{\alpha^2c^2} \left(c\beta_1+2c^2\beta_2+c^3\beta_3\right).$$

After imposing their respective constraints, $\delta G$ carries two and a nondegenerate $\delta M$ sector carries five polarizations. Exceptional massless or partially massless linear limits require separate counting. The exact diagonal equations and normalization conventions are given in Schmidt-May and von Strauss 2016, § 5.3.2, eqs. (5.13)–(5.22). The original construction displays the same one-massless/one-massive spectrum in Hassan and Rosen 2012, § 4, eqs. (4.1)–(4.7).

There is no unique nonlinear field redefinition that globally separates “the massive metric” from “the massless metric.” The eigenstates above are linear notions tied to a proportional background. On a general solution the perturbations can mix and need not diagonalize into two independent spin-2 equations.

Matter couplings select the observed metric

A standard safe choice is

$$S_{\mathrm m}=S_{\mathrm m,g}[g,\Psi_g]+S_{\mathrm m,f}[f,\Psi_f],$$

with independent matter sectors. If ordinary matter couples only to $g_{\mu\nu}$, then $g$ is the physical metric, but a $g$-source generally excites both linear eigenstates because $\delta g$ is a mixture of $\delta G$ and $\delta M$. Coupling the same dynamical matter field minimally to both metrics generically destroys the constraint that removes the Boulware–Deser ghost. These distinctions and the canonical reason for them are set out in Schmidt-May and von Strauss 2016, § 5.4.1, eqs. (5.23)–(5.25).

Special composite effective-metric couplings can be meaningful as EFT interactions below a stated scale, but “ghost-free in a decoupling limit” is weaker than an all-scale nonlinear constraint proof. Matter coupling, background stability, and observational interpretation must therefore be stated together.

From two metrics to many

For many metrics, pairwise Hassan–Rosen interactions can be represented by a graph: vertices are metrics and edges are bimetric potentials. Chain and centre constructions, and more generally acyclic combinations of them, permit compatible shift redefinitions and the needed constraints. Closing the graph into a cycle generically obstructs that construction in the metric formulation and can restore a Boulware–Deser mode. The explicit tree-like actions and the three-metric loop diagnostic appear in Schmidt-May and von Strauss 2016, § 8.2.1, eqs. (8.11)–(8.14).

This graph criterion concerns the interaction and constraint structure. It does not guarantee positive masses, stable backgrounds, acceptable matter couplings, or a sufficiently high cutoff. A viable multimetric model must pass all of those tests separately.

Exercises

1. Invert the eigenstate map. Express $\delta g_{\mu\nu}$ and $\delta f_{\mu\nu}$ in terms of $\delta G_{\mu\nu}$ and $\delta M_{\mu\nu}$. Which modes does a source coupled to $g$ excite?

Solution

From $2c\delta M=\delta f-c^2\delta g$, substitute $\delta f=2c\delta M+c^2\delta g$ into $\delta G=\delta g+\alpha^2\delta f$. This gives

$$\delta g=\frac{\delta G-2\alpha^2c\delta M}{1+\alpha^2c^2}, \qquad \delta f=\frac{c^2\delta G+2c\delta M}{1+\alpha^2c^2}.$$

Therefore a linear coupling $\delta g_{\mu\nu}T^{\mu\nu}$ contains both the massless field and, unless its mixing coefficient vanishes in a limit, the massive field. Canonical normalization changes the coupling coefficients but not this conclusion.

2. Symmetric background check. Set $c=\alpha=1$. Derive the background condition and $m_{\mathrm{FP}}^2$.

Solution

Equality of the cosmological terms gives

$$\beta_0+3\beta_1+3\beta_2+\beta_3 =\beta_1+3\beta_2+3\beta_3+\beta_4,$$

or $\beta_0+2\beta_1-2\beta_3-\beta_4=0$. The mass is

$$m_{\mathrm{FP}}^2=2\mu^2(\beta_1+2\beta_2+\beta_3).$$

The background equation and positive $m_{\mathrm{FP}}^2$ are separate conditions; neither establishes the absence of all curved-background instabilities.

References

• Hassan, S. F., and Rachel A. Rosen. “Bimetric Gravity from Ghost-Free Massive Gravity.” Journal of High Energy Physics 02 (2012): 126. doi:10.1007/JHEP02(2012)126; 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.