REVIEW
Ghosts in Massive Gravity
High Energy Physics - Theory · Submitted 16 May 2005 · 19 references
Pith's verdict: The ghost-around-a-source computation is solid and important; the abstract's 'independent of the choice of non-linear terms' is proven explicitly only to quartic order, with the all-orders step asserted by counting.
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Referee report
The paper
The paper claims
Lorentz-invariant massive gravity is undone by a ghost: around any heavy source the graviton's scalar mode goes negative-energy, at distances far outside the Vainshtein radius where the theory was meant to recover general relativity.
The reading
What carries the argument
The Stückelberg/Goldstone description of the mass term. Local coordinate invariance is restored by four Goldstone fields $π_μ = A_μ + ∂_μ φ$ entering through the symmetric tensor $H_{αβ} = h_{αβ} + ∂_α π_β + ∂_β π_α + ∂_α π^γ ∂_β π^γ$, so that the mass term is built from $H$ and the scalar $φ$ appears with two derivatives. Only the Fierz–Pauli choice cancels the four-derivative term for $φ$ at quadratic order, and the cancellation of $(∂^2 φ)^n$ at higher orders requires a specific infinite set of coefficients, with the residual freedom sitting in total-derivative combinations such as $(□φ)^3 - 3 □φ (∂_μ∂_ν φ)^2 + 2(∂_μ∂_ν φ)^3$. That residual freedom is exactly what the paper shows to be insufficient: expanding the equation of motion linear in the fluctuation and counting independent tensor structures against free coefficients gives the no-go.
Extended reading notes
Core claim
The claim is that ghost instabilities in Lorentz-invariant massive gravity are unavoidable, whatever nonlinear terms are added to the Fierz–Pauli Lagrangian. The argument runs in two languages. In the Goldstone language, the scalar $φ$ that restores broken diffeomorphism invariance has self-couplings of the form $(∂^2φ)^n$; cancelling these one order at a time raises the cutoff from $Λ_5 = (m_g^4 M_P)^{1/5}$ to $Λ_3 = (m_g^2 M_P)^{1/3}$, but around a nontrivial background any surviving term of this type contributes a four-derivative kinetic term for the fluctuation $φ - Φ$, hence a ghost of position-dependent mass $m_ghost}^2 ∼ Λ_5^5/∂^2Φ_c$ (or $Λ_3^6 Φ_c ∂^2 Φ_c$ in the improved theory). The authors then check explicitly, at cubic and quartic order in $H_{μν}$, that the freedom to add the total-derivative combinations $L^TD}_n$ with free coefficients $α_3, α_4, ...$ is not enough: the number of dangerous tensor structures grows faster than the number of tunable coefficients, so at least one four-derivative term survives in the fluctuation equation of motion. In unitary gauge the same object is the sixth mode, constrained to $h = 0$ at linear order by the trace of the Einstein equations but rendered dynamical by any curvature, with mass $m_{6th}^2 ∼ m_g^2 H_{μν}$ — matching the Goldstone result once the background is written in unitary gauge. An ADM Hamiltonian analysis closes the loop: keeping the lapse linear (so that its equation remains a constraint rather than a determining equation) is possible at quadratic and cubic order but fails at quartic order, where the relevant matrix has rank 4 and its image does not contain the inhomogeneous term.
Load-bearing premise
That the obstruction found explicitly at cubic and quartic order really persists at every order: the all-orders claim rests on a counting argument — more dangerous tensor structures than tunable coefficients, and a rank-4 matrix whose image misses the inhomogeneous term — rather than on an explicit computation at each order.
Editorial extensions
If this is right
- General relativity is never a good approximation in massive gravity: the Vainshtein mechanism can only screen the scalar inside $R_V$, but the effective theory has already broken down before that radius is reached.
- Any use of massive gravity below $R_ghost}$ requires postulating new physics at the local ghost mass, which means lowering the cutoff below $Λ_3$ and abandoning the predictivity the higher-order tuning was meant to buy.
- The sixth mode is not an artifact of the linearised theory: it propagates on any curved background, with a mass set by the background curvature, $m_{6th}^2 ∼ m_g^2 H_{μν}$.
- Raising the cutoff from $Λ_5$ to $Λ_3$ by cancelling the $(∂^2φ)^n$ self-couplings does not cure the problem; it shrinks the Vainshtein radius but leaves the ghost radius above it.
- No nonlinear extension of the Fierz–Pauli action keeps the ADM lapse a Lagrange multiplier beyond quartic order, so the sixth degree of freedom cannot be tuned away in the Hamiltonian formulation.
Reading between the lines
- The diagnostic used here — expand the scalar sector to quadratic order around the source background and look for more than two time derivatives per fluctuation — is a cheap screen that could be applied to any proposed infrared modification of gravity before asking about its cosmology.
- Because the ghost mass is largest far from the source and shrinks inward, the theory is least trustworthy exactly where it was supposed to become general relativity; this inverts the usual intuition that infrared modifications are safest on short scales.
- The no-go is tied to Lorentz invariance. A Lorentz-violating mass term changes the dispersion of the helicity-zero mode and can evade the argument, so the interesting question becomes which of the two — Lorentz invariance or a healthy scalar sector — a viable theory has to give up.
- The numbers are stark enough to be a filter on phenomenology: for $m_g ∼ H_0$ the ghost radius around the Sun is of order the horizon, so essentially every astrophysical or solar-system test of a graviton mass lies inside the region where the effective description has already failed.
Original abstract
In the context of Lorentz-invariant massive gravity we show that classical solutions around heavy sources are plagued by ghost instabilities. The ghost shows up in the effective field theory at huge distances from the source, much bigger than the Vainshtein radius. Its presence is independent of the choice of the non-linear terms added to the Fierz-Pauli Lagrangian. At the Vainshtein radius the mass of the ghost is of order of the inverse radius, so that the theory cannot be trusted inside this region, not even at the classical level.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
In the context of Lorentz-invariant massive gravity we show that classical solutions around heavy sources are plagued by ghost instabilities. The ghost shows up in the effective field theory at huge distances from the source, much bigger than the Vainshtein radius. Its presence is independent of the choice of the non-linear terms added to the Fierz-Pauli Lagrangian. At the Vainshtein radius the mass of the ghost is of order of the inverse radius, so that the theory cannot be trusted inside this region, not even at the classical level.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In the context of Lorentz-invariant massive gravity we show that classical solutions around heavy sources are plagued by ghost instabilities. | result 0.90 | 0.92 | 0.88 | asserted 1.00 |
| s1 | The ghost shows up in the effective field theory at huge distances from the source, much bigger than the Vainshtein radius. | result 0.99 | 0.82 | 0.12 | asserted 1.00 |
| s2 | Its presence is independent of the choice of the non-linear terms added to the Fierz-Pauli Lagrangian. | result 0.99 | 0.79 | 0.11 | asserted 1.00 |
| s3 | At the Vainshtein radius the mass of the ghost is of order of the inverse radius, so that the theory cannot be trusted inside this region, not even at the classical level. | result 0.52 | 0.79 | 0.24 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
93 arXiv papers cite this in 156 sentences
Papers whose main result relates to this one's
States a conflicting result 1 paper
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This implies the absence of the pathological Boulware-Deser ghost to all orders.
Resolving the Ghost Problem in non-Linear Massive Gravity · 0.85
Papers citing it · 119 sentences without a confident label
Contests it 3 papers
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Therefore the cure for the ghost instability proposed in [ref] does not work for the case $a=-1$ of the ghost condensate/ekpyrotic scenario.
The New Ekpyrotic Ghost · Appendix. Exorcising ghosts? · contest 0.93
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In fact, as pointed out in [ref] , the properties ([ref]) of the tensors $X_μν$ guarantee that there are no ghosts in the lagrangian ([ref]) of the decoupling limit theory.[ This is contrary to [ref] , which claims that a ghost is still present at quartic order.
Theoretical Aspects of Massive Gravity · The appearance of galileons and the absence of ghosts · contest 0.93
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The attempt by [ref] for constructing non-linear interactions of the helicity-0 mode yielded a negative conclusion due to an erroneous decomposition of the massive graviton into its helicity modes.
A systematic approach to generalisations of General Relativity and their cosmological implications · Introduction · contest 0.93
Uses its method or tool 2 papers
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To identify the cutoff scale it is most useful to perform a decoupling limit analysis, as was done in detail in [ref] for the spin-2 case, based on earlier analyses [ref] .
Positivity Constraints on Interacting Spin-2 Fields · Cycle and Line EFTs of Interacting Spin-2 Fields · 0.92
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Following [ref] , let's consider the stability of the classical solutions to ([ref]) around a massive point source.
Theoretical Aspects of Massive Gravity · Ghosts · 0.84
Uses its data or values 1 paper
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In the original model, the ghost starts propagating at energy scales higher than $(1000km)^-1$ [ref] , leading to strong conflict with observations.
Dynamical diffeomorphisms · Introduction · 0.80
Compares with it 3 papers
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This is in contrast to the flat space decoupling limit, where choosing the ghost-free structure raises the strong coupling scale from $Λ_5=\(m^4\)^1/5$ to $Λ_3=\(m^2\)^1/3$ [ref] .
On the (A)dS Decoupling Limits of Massive Gravity · Interacting Proca in the decoupling limit · 0.96
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We also note that the term of concern of Ref. [ref] , found in that work in the Lagrangian formalism, is nothing but the term already found in [ref] in the Hamiltonian formalism, that was addressed above.].
Nonlinear Dynamics of 3D Massive Gravity · Introduction and Summary · 0.89
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And it seems very natural that it is the model with the right number of physical degrees of freedom and without any ghosts that has the best high energy behavior (compare e.g. [ref] ).
On massive spin 2 electromagnetic interactions · Conclusion and discussion · 0.83
Cites it as support 3 papers
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In fact, the $Λ_5$ interactions also excite a ghost at the same scale $Λ_Q$ [ref] , so the scale where quantum corrections become important (which directly derives from $Λ_5$ around a given background) should indeed be seen as the effective cutoff scale of the theory in question [A detailed discussion of the relation between the cutoff of the theory—the scale where full unitarity breaks down—and the perturbative unitarity violation scales, as well as their relation to an `environmental' strong coupling scale, is beyond the scope of this paper.
On strong coupling scales in (minimal) massive gravity · Raising the strong coupling scale · 0.97
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This indicates that there is a regime where the theory considered, and hence also its solutions, is well approximated[Of course the discussion here is a bit loose, it will be made more precise when dealing with spherically symmetric solutions.] by retaining only the quadratic action and the cubic self interaction of $ϕ̃$, as noted in particular in Ref. [ref] .
The Vainshtein mechanism in the Decoupling Limit of massive gravity · Strong coupling and decoupling limit in the Goldstone picture · 0.98
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There is actually no choice of potential $𝒰$ which would keep it linear in the lapse beyond cubic order [ref] .
Massive Gravity · Primary constraint · 0.97
Cites it as background 21 papers
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In massive gravity, the standard problem is the Boulware–Deser (BD) ghost [ref] , see also [ref] .
New Kinetic Terms for Massive Gravity and Multi-gravity: A No-Go in Vielbein Form · Overview · 0.99
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Similarly, components of the massless particle associated with general coordinate reparametrization invariance, the graviton, could acquire mass via gravitational Higgs mechanism [ref] upon spontaneous breaking of diffeomorphism invariance by scalar vacuum expectation values[For earlier and subsequent related works, see, e.g., [ref] , and references therein.].
Massive Gravity in de Sitter Space via Gravitational Higgs Mechanism · Introduction and Summary · 0.92
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This is possible by taking a specific limit of the theory in such a way that the different degrees of freedom decouple, hence called the “decoupling limit". * In the decoupling limit: In this limit, AGS showed that the ghost could in principle be pushed beyond the scale $Λ_5$, however shortly after, it was argued that the ghost would inexorably reappear at the scale $Λ_3$, [ref] .
Helicity Decomposition of Ghost-free Massive Gravity · Introduction · 0.83
Discussion
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