Ghost-Free Nonlinear Massive Gravity

The de Rham–Gabadadze–Tolley (dRGT) potential is distinguished by a nonlinear constraint pair that removes the Boulware–Deser scalar. The key Hamiltonian fact is not that the potential looks linear in the original lapse—it does not—but that an invertible shift redefinition makes lapse linearity manifest while leaving the new shift equations lapse-independent. Preserving the resulting primary constraint produces a secondary constraint, so a generic regular branch propagates five rather than six modes.

Required background. Variational and Symplectic Geometry supplies Poisson brackets and first-/second-class constraint counting, and Weak-Field Gravity supplies the Fierz–Pauli limit that the nonlinear theory must reproduce.

Helpful background. Einstein Equations fixes the dynamical metric convention; Gravitational Actions and Boundary Terms supplies the canonical action principle; Formulations of General Relativity supplies ADM variables; and Gravity Beyond General Relativity supplies the wider consistency questions.

Square-root potential and its domain

Use the chapter-wide bimetric normalization

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

with

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

On this page $f_{\mu\nu}$ is prescribed, so its Einstein–Hilbert term is omitted; $m_f$ and $\alpha$ become relevant when $f$ is dynamical. The $e_n$ are elementary symmetric polynomials, for example $e_0=1$, $e_1=[S]$, and $e_2=([S]^2-[S^2])/2$. This $\beta_n$ convention and its relation to other common parameterizations are given in de Rham 2014, § 6, eqs. (6.3), (6.7), and (6.22)–(6.24).

The proof has a domain hypothesis: $g^{-1}f$ must admit a real, smooth square root on a chosen branch. A real square root is not guaranteed for every pair of Lorentzian metrics, and switching branches through a degeneracy is not licensed by the local argument. We also assume that the ADM shift change below exists and is locally invertible.

The coefficient $m_g^2$ equals $M_g^2/2$ for the reduced Planck mass $M_g$; the symbol $m$ in the potential is not itself the physical graviton mass. The defining identity $\det(\mathbf1+tS)=\sum_{n=0}^4e_n(S)t^n$ is a useful way to compute all five polynomials. For example, on the positive proportional branch $S=c\mathbf1$, $e_n(S)=\binom4n c^n$. Moreover $\sqrt{-g}\,e_4(S)=\sqrt{-f}$ on that branch, so the $\beta_4$ term has no dynamical $g$ variation when $f$ is fixed. The $\beta_0$ term acts as a $g$ cosmological term.

ADM variables and the decisive shift change

Write the dynamical and reference metrics in ADM form with lapses $(N,L)$, shifts $(N^i,L^i)$, and spatial metrics $(\gamma_{ij},{}^3f_{ij})$. Only $\gamma_{ij}$ has canonical momentum $\pi^{ij}$ in massive gravity. Hassan and Rosen introduce three variables $n^i$ through

$$N^i-L^i=\left(L\delta^i{}_j+ND^i{}_j\right)n^j,$$

where $D^i{}_j$ is a square-root matrix determined algebraically by $\gamma_{ij}$, ${}^3f_{ij}$, and $n^i$. On a patch where the Jacobian of this map is invertible, the Lagrangian takes the form

$$\mathcal L=\pi^{ij}\dot\gamma_{ij}-\mathcal H_0(\gamma,\pi,n) +N\,\mathcal C(\gamma,\pi,n).$$

The important conjunction is:

1. $N$ occurs linearly; and
2. the $n^i$ equations are independent of $N$.

The second property means that solving the three $n^i$ equations and substituting back cannot regenerate nonlinear powers of $N$. The explicit redefinition, its Jacobian condition, and the lapse-independent shift equations are established in Hassan and Rosen 2012, § 2.2–2.3, eqs. (2.7)–(2.21). The equivalent general-reference-metric construction appears in de Rham 2014, § 7.1, eqs. (7.18)–(7.30).

Primary and secondary constraints

After the new shifts are eliminated, variation with respect to the lapse gives

$$\mathcal C(\gamma,\pi)\approx0.$$

This is a primary constraint on the spatial metric phase space, not an equation that determines $N$. Consistency requires it to be preserved:

$$\dot{\mathcal C}(x)=\{\mathcal C(x),H\}\approx0.$$

For the dRGT potential, the self-bracket obeys $\{\mathcal C(x),\mathcal C(y)\}\approx0$, so this condition does not determine the lapse. It instead yields the genuine secondary constraint

$$\mathcal C_{(2)}(x)=\{\mathcal C(x),H_0\}\approx0.$$

The next preservation condition fixes $N$ on a generic branch rather than producing a tertiary constraint. Equivalently, the matrix of brackets between $\mathcal C$ and $\mathcal C_{(2)}$ is nondegenerate there: they form a second-class pair. The primary-to-secondary step, including the crucial self-bracket, is proved in Hassan and Rosen 2012, § 3.1, eqs. (3.1)–(3.5), with the explicit secondary constraint and no-tertiary analysis in Hassan and Rosen 2012, §§ 3.2–3.3, pp. 10–11.

Five-mode count

The spatial metric and its momentum provide $6+6=12$ phase-space dimensions per point. A second-class constraint removes one phase-space dimension, so the pair removes two:

$$12-2=10=2\times5.$$

Thus the nonlinear theory has five configuration-space degrees of freedom on a generic regular branch, matching Fierz–Pauli theory. The lapse and shifts are auxiliary variables and are not added to this reduced phase-space count. On exceptional backgrounds a constraint bracket can degenerate or a kinetic coefficient can vanish; then linear perturbation theory may display fewer modes because some are strongly coupled. Such a background does not invalidate the generic count, but it does require its own analysis.

Precisely what has been proved

The Hamiltonian constraint and secondary constraint remove the Boulware–Deser ghost. This only establishes the generic five-mode count; it does not imply stability or causality. In particular, the proof does not establish positivity of every remaining kinetic term, absence of tachyonic or gradient instabilities, causal characteristic cones, existence of a global real square-root branch, or control above the EFT cutoff. Nor does it guarantee that arbitrary matter couplings preserve the constraint. These are logically independent tests.

The decoupling-limit Galileon structure supplies a complementary diagnostic: its equations remain second order and expose the helicity-0 interactions at $\Lambda_3$. But a decoupling-limit calculation alone is not the full nonlinear Hamiltonian proof. Conversely, the Hamiltonian proof does not establish that any selected background lies within the decoupling-limit or EFT regime.

Exercises

1. Do the canonical count. Suppose a deformation of the fixed-reference-metric ADM system starts with the twelve variables $(\gamma_{ij},\pi^{ij})$. Compare the physical count when it has (a) no extra scalar constraints, (b) only one scalar constraint, and (c) the dRGT second-class pair.

Solution

(a) Twelve phase-space dimensions describe six configuration modes, the sixth being the Boulware–Deser scalar. (b) One second-class scalar constraint would leave eleven phase-space dimensions, an odd-dimensional reduced space that cannot describe ordinary canonical pairs; consistency therefore demands another condition or a degeneracy. (c) Two second-class constraints remove two dimensions, leaving ten phase-space dimensions or five propagating modes. They are second class, so there is no accompanying gauge quotient that would remove two dimensions per constraint.

2. Why is lapse linearity alone insufficient? Assume $\mathcal L=\pi\dot\gamma-H_0(n)+N\mathcal C(n)$, but the $n$ equation depends on $N$. Explain the danger.

Solution

If the solution is $n=n(\gamma,\pi,N)$, substituting it back produces $H_0[n(N)]-N\mathcal C[n(N)]$, which is generically nonlinear in $N$. Then the lapse equation can determine $N$ rather than constrain $(\gamma,\pi)$. dRGT avoids this because the $n^i$ equations factor so that they are independent of $N$; eliminating $n^i$ preserves linearity and leaves $N$ as a multiplier of $\mathcal C$.

References

• de Rham, Claudia. “Massive Gravity.” Living Reviews in Relativity 17 (2014): 7. doi:10.12942/lrr-2014-7; Open PDF.
• Hassan, S. F., and Rachel A. Rosen. “Confirmation of the Secondary Constraint and Absence of Ghost in Massive Gravity and Bimetric Gravity.” Journal of High Energy Physics 04 (2012): 123. doi:10.1007/JHEP04(2012)123; Open PDF.