General-Relativistic Limits and Decoupling Limits

A parameter value that makes a modified action resemble the Einstein–Hilbert action does not by itself establish a general-relativistic limit. The physical modes, source couplings, selected solution branch, observables, and domain of validity must approach their GR counterparts together. Decoupling limits are a complementary tool: they keep selected interactions finite while other scales are sent away, exposing which mode obstructs or enables recovery. Neither kind of limit is meaningful without canonical normalization and a stated cutoff.

Required background. Einstein Equations provides the GR solutions and sourced response used as the limiting baseline; Gravitational Actions and Boundary Terms provides the action and boundary data that must converge with the bulk terms; Variational and Symplectic Geometry provides the reduced phase space on which a limit of physical modes is defined.

Helpful background. Formulations of General Relativity supplies the gauge, constraint, and initial-value variables needed to compare solution branches on a common domain.

Four limits that must agree

Let a family of theories be labeled by $\varepsilon$, with GR expected at $\varepsilon=0$. At least four logically separate statements are involved.

Limit	Question	Typical failure
Action or equation	Do the coefficients and differential operators approach their GR form?	A canonically normalized extra field keeps a finite coupling even when its written coefficient vanishes.
Phase-space	Do constraints, gauge orbits, and physical degrees of freedom approach those of GR?	A constraint changes rank or an extra mode becomes infinitely strongly coupled.
Solution	With mapped initial and boundary data, does the chosen branch approach a GR solution?	The GR branch is disconnected, singular, or surrounded by a shrinking boundary layer.
Observable	Do dimensionless predictions for the same sources and measuring metric converge?	Fields converge in one frame while source couplings, charges, or detector units do not.

A useful observable criterion is uniform convergence on a common physical domain $\mathcal D$:

$$\lim_{\varepsilon\to0} \sup_{x\in\mathcal D} \frac{\left|\mathcal O_\varepsilon(x)-\mathcal O_{\mathrm{GR}}(x)\right|} {\mathcal O_{\mathrm{scale}}(x)}=0.$$

$\mathcal D$ must remain inside the EFT cutoff and include the region where the prediction is used. Pointwise convergence outside a shrinking transition layer is not enough. Nor is convergence of a gauge-dependent field component: the observable, source, branch, and frame map must be shared.

Three phrases should therefore be kept distinct.

• A parameter limit sends couplings or masses to specified values.
• A GR limit makes the physical GR degrees of freedom, solutions, and observables emerge on a controlled common domain.
• A decoupling limit scales several parameters together so that selected canonically normalized interactions remain finite while other interactions disappear. It diagnoses a sector; it is not automatically the full theory’s GR limit.

Linear massive gravity and the vDVZ discontinuity

The cleanest counterexample begins with a massive spin-2 field coupled to a conserved source. Ignoring a common normalization and writing only the tensor structure, single-particle exchange in four dimensions gives

$$\mathcal A_m(p) \propto \frac{1}{p^2+m^2} \left(T_{ab}T'^{ab}-\frac13TT'\right),$$

whereas the massless GR exchange is

$$\mathcal A_{\mathrm{GR}}(p) \propto \frac{1}{p^2} \left(T_{ab}T'^{ab}-\frac12TT'\right).$$

The coefficient multiplying the trace does not approach the massless value as $m\to0$. The helicity-0 polarization retains a finite coupling to $T$. For two slow nonrelativistic sources, the two tensor structures are respectively $2TT'/3$ and $TT'/2$ up to the common sign convention. Matching the Newtonian force therefore rescales the massive coupling by $3/4$. A traceless source, such as electromagnetic radiation, does not couple to the helicity-0 trace; after that Newtonian matching its response is only $3/4$ of the GR value. This is the van Dam–Veltman–Zakharov discontinuity, reviewed with the explicit source exchange in de Rham 2014, §2.2.3, pp. 16–18, equations (2.49)–(2.66).

The calculation proves a failure of the linearized observable limit. It does not prove that every nonlinear massive-gravity solution fails to approach GR. The linear expansion itself can become nonuniform near a source.

The Λ₃ decoupling limit

In ghost-free massive gravity, introduce Stückelberg fields and canonically normalized helicity-2, helicity-1, and helicity-0 variables. The most useful degree-of-freedom-preserving scaling is

$$M_{\mathrm{Pl}}\to\infty, \qquad m\to0, \qquad \Lambda_3=(M_{\mathrm{Pl}}m^2)^{1/3} \quad\text{fixed},$$

with $T_{ab}/M_{\mathrm{Pl}}$ held fixed. Free graviton self-interactions then vanish while the helicity-0 Galileon-type interactions at $\Lambda_3$ and its coupling to the source trace remain. Schematically,

$$\mathcal L_{\mathrm{dec}} \supset -\frac12h^{ab}\mathcal E_{ab}^{\phantom{ab}cd}h_{cd} -\frac32(\partial\pi)^2 +\frac{(\partial\pi)^2\Box\pi}{\Lambda_3^3} +\frac{h_{ab}T^{ab}+\pi T}{M_{\mathrm{Pl}}}.$$

The precise coefficients depend on the dRGT potential parameters and field basis; the scaling and retained sectors are the invariant point. It preserves the interactions responsible for the Vainshtein mechanism and makes the helicity-0 obstruction visible. Scaling limits and the $\Lambda_3$ construction are developed in de Rham 2014, §8.1 and §§8.3.1–8.3.3, pp. 69–77.

For a source of mass $M$, balancing the linear and cubic scalar terms gives, up to model-dependent numerical factors,

$$r_V\sim\left(\frac{GM}{m^2}\right)^{1/3}.$$

Outside $r_V$ the linear helicity-0 response is visible. Inside $r_V$ its nonlinear derivative interactions dominate and can suppress the scalar force relative to the tensor force, allowing observables to approach GR even though the linear $m\to0$ limit is discontinuous. The orders $m\to0$, $r/r_V\to0$, and linearization therefore do not commute. Vainshtein’s original resolution of the linear discontinuity is in Vainshtein 1972, pp. 393–394.

A checklist for controlled recovery

For any proposed GR regime, record the following data explicitly.

1. Scaling path: which dimensionful and dimensionless parameters tend to which values, and what combinations are held fixed?
2. Canonical variables: how do kinetic eigenvalues, constraint rank, and physical mode count behave? A vanishing kinetic term can mean strong coupling rather than decoupling.
3. Sources and frame: which metric couples to matter, and which normalized currents or stress tensors are fixed?
4. Branch and data: do regular center or horizon data, asymptotic charges, and initial data map continuously to a named GR branch?
5. Common observables: which dimensionless, gauge-invariant quantities are compared, and is convergence uniform over their physical domain?
6. Validity: do all local energies, curvatures, gradients, and nonlinear scales remain below the cutoff? Is the perturbative ordering uniform?
7. Independent discriminator: does a traceful versus traceless source, a different background, or a different order of limits expose a hidden mode?

A Maxwell–Proca comparison is a useful control. As a Proca mass tends to zero, the longitudinal polarization decouples from a conserved current, so the source exchange approaches Maxwell theory. Massive spin-2 fails for a generic traceful source because its helicity-0 polarization retains the trace coupling. The mere persistence of a formal polarization is therefore not the criterion; its canonical coupling and effect on observables are.

Common pitfalls

Delete the modification in the action and declare GR recovered. A canonically normalized extra mode or a disconnected solution branch can retain a finite effect after the written coefficient tends to zero.

Call every scaling limit a decoupling limit. A useful decoupling limit states which modes and interactions survive. Sending a mass to zero while an unwanted mode becomes strongly coupled is not evidence that it decouples.

Use screening outside EFT control. Nonlinear suppression can restore a GR observable while simultaneously exciting higher operators. The screened solution is predictive only where its gradients and fluctuation scales remain inside the declared EFT domain.

Exercises

Use the two exchange amplitudes above for (i) two nonrelativistic sources and (ii) a nonrelativistic source interacting with a traceless source. Normalize the massive theory to reproduce the Newtonian interaction in case (i). What does it then predict relative to GR in case (ii), and why does this not settle the nonlinear theory?

Solution

For a slow source, $T_{00}$ dominates and $T\simeq-T_{00}$ in the site signature. Apart from a shared positive source product, the massive tensor structure is

$$1-\frac13=\frac23,$$

while the massless GR structure is

$$1-\frac12=\frac12.$$

To make the Newtonian amplitudes equal, the overall massive coupling squared must be multiplied by $(1/2)/(2/3)=3/4$. If the second source is traceless, then $T'=0$ and both unnormalized tensor structures reduce to $T_{ab}T'^{ab}$. The Newtonian normalization remains, so the massive prediction is $3/4$ of the GR response even as $m\to0$.

This is a genuine discontinuity of linear Fierz–Pauli exchange. It does not settle the nonlinear theory because the perturbation series about the linear solution ceases to be uniform inside the Vainshtein radius. There the helicity-0 self-interactions retained in the $\Lambda_3$ decoupling limit are as important as its linear term. One must solve on the screened branch, verify regular boundary data and EFT control, and compare the same observable before claiming either recovery or failure of GR.

References

• de Rham, Claudia (2014), “Massive Gravity,” Living Reviews in Relativity 17, article 7. DOI; Open PDF.
• Vainshtein, Arkady I. (1972), “To the Problem of Nonvanishing Gravitation Mass,” Physics Letters B 39, 393–394. DOI.
• van Dam, Hendrik, and Martinus Veltman (1970), “Massive and Mass-less Yang–Mills and Gravitational Fields,” Nuclear Physics B 22, 397–411. DOI.