REVIEW
Lorentz-violating graviton masses: getting around ghosts, low strong coupling scale and VDVZ discontinuity
High Energy Physics - Theory · Submitted 13 Jul 2004 · 22 references
Pith's verdict: A clean linearized result — the m0=0 trick kills the ghost and the VDVZ discontinuity — with the strong-coupling scale argued by normalization rather than by vertices.
T0 · AI review
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
Setting one graviton mass term to zero may give a ghost-free massive gravity with no VDVZ discontinuity and a strong-coupling scale of $({mM_{Pl}})^{1/2}$.
The reading
What carries the argument
The mechanism is the choice $m_0=0$, which makes $h_{00}=psi$ enter the quadratic action linearly and therefore act as a Lagrange multiplier; its constraint expresses $sigma$ in terms of $tau$, and the field $v$ is eliminated through its own equation of motion, leaving a single dynamical scalar $tau$ with Lagrangian (17). That Lagrangian carries terms enhanced by $1/m^2$, so the canonically normalised field satisfies $taupropto (m/M_{Pl}),tau_c$ instead of the Fierz–Pauli $taupropto tau_c/M_{Pl}$. The same enhancement appears for the vector mode, where integrating out $u_i$ gives $w_ipropto mM_{Pl},s_{ci}$, and for the second scalar invariant $Phi$. These normalisation factors, not the dispersion relations, are what control both the VDVZ discontinuity and the strong-coupling scale; a Stückelberg-style analysis of the pure-gauge part $h_{muu}=partial_mu pi_u+partial_u pi_mu$ reproduces the same conclusion by showing that the mass terms themselves supply healthy kinetic terms for all components of $pi$, including the longitudinal one that is degenerate in the Fierz–Pauli case.
Extended reading notes
Core claim
The paper shows that a graviton mass term which breaks Lorentz invariance but preserves the Euclidean symmetry of three-space, with the coefficient $m_0$ of $h_{00}h_{00}$ set to zero, yields a linearized theory about Minkowski with four advertised properties. First, the spectrum has a gap of order $m$ in every sector, so all modes are massive. Second, at momenta $p^2gg m^2$ the dispersion relations are of the form $omega^2propto p^2$: tensor modes are relativistic with mass $m_2$, while vector and scalar modes propagate with unconventional, generally subluminal or superluminal, maximum velocities set by ratios of the masses. Third, the VDVZ discontinuity is absent — in the massless limit taken with fixed mass ratios, the gauge-invariant fields $w_i$, $tau$ and $Phi$ generated by a conserved source reduce exactly to the general-relativity expressions of eqs. (21)–(23), not the Fierz–Pauli ones of eqs. (24)–(25). Fourth, because the normalisation factors relating the original fields to canonically normalised ones are at most of order $(mM_{Pl})^{-1}$ in every sector, the strong-coupling scale is $(mM_{Pl})^{1/2}$, matching the naive estimate rather than falling below it. Under the positivity conditions $m_1^2>m_4^2>0$, $m_2^2>m_3^2$ and $4m_2^2>m_4^2$, all kinetic and gradient terms in the reduced actions have the correct sign, so there are no ghosts or tachyons at this order.
Load-bearing premise
The entire case rests on the quadratic action about flat space: if the ghost-freedom and the $(mM_{Pl})^{1/2}$ strong-coupling scale do not survive once cubic and higher-order terms are included, the claim that this is a healthy theory of massive gravity does not follow.
Editorial extensions
If this is right
- If the construction holds, a Higgs phase of gravity with a Minkowskian background and broken Lorentz invariance can describe massive gravitons with an intrinsic strong-coupling scale of $(mM_{Pl})^{1/2}$, rather than the much lower scales forced by Fierz–Pauli or by brane-world infrared modifications.
- Gravitational interactions between conserved sources in the massless limit reduce to the Einstein-Hilbert form, so no fifth-force-like scalar remnant survives and no Vainshtein-type nonlinear screening is needed to hide it.
- The tensor sector remains relativistic with mass $m_2$, while vector and scalar modes acquire direction-dependent maximum velocities, giving a distinctive, in principle distinguishable, propagation signature.
- The special case $m_0\neq 0$ with $m_1=0$ is noted as having no physical scalar or vector excitations at all, only tensor modes of mass $m_2$, and is flagged as deserving separate study.
Reading between the lines
- The paper's own framing invites the next question it does not answer: whether the ghost-freedom and the $(mM_{Pl})^{1/2}$ scale survive when cubic and higher terms are included, since a Boulware–Deser-type ghost or a new strong-coupling problem could appear only beyond quadratic order.
- Because Lorentz violation is imposed in the mass term by hand rather than arising from a Higgs mechanism, the construction is a bottom-up existence proof; a natural follow-up is to ask which scalar-field condensates could generate exactly this pattern of masses with $m_0=0$.
- The direction-dependent maximum velocities for vector and scalar modes suggest a cosmological signature: if such a graviton mass existed, the stochastic gravitational-wave background would show mode-dependent propagation speeds, which is a sharper target than a generic mass bound.
- The observation that $m_0\neq 0$, $m_1=0$ leaves only tensor excitations hints at a broader classification problem: enumerating which sign patterns of the five masses give ghost-free, tachyon-free spectra at quadratic order, which the paper only partially surveys.
Original abstract
A theory with the action combining the Einstein--Hilbert term and graviton mass terms violating Lorentz invariance is considered at linearized level about Minkowskian background. It is shown that with one of the masses set equal to zero, the theory has the following properties: (i) there is a gap of order $m$ in the spectrum, where $m$ is the graviton mass scale; (ii) the dispersion relations at ${\bf p}^2 \gg m^2$ are $\omega^2 \propto {\bf p}^2$, the spectrum of tensor modes being relativistic, while other modes having unconventional maximum velocity; (iii) the VDVZ discontinuity is absent; (iv) the strong coupling scale is $(mM_{Pl})^{1/2}$. The latter two properties are in sharp contrast to the Lorentz-invariant gravity with the Pauli--Fierz mass term.
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
A theory with the action combining the Einstein--Hilbert term and graviton mass terms violating Lorentz invariance is considered at linearized level about Minkowskian background. It is shown that with one of the masses set equal to zero, the theory has the following properties: (i) there is a gap of order $m$ in the spectrum, where $m$ is the graviton mass scale; (ii) the dispersion relations at ${\bf p}^2 \gg m^2$ are $\omega^2 \propto {\bf p}^2$, the spectrum of tensor modes being relativistic, while other modes having unconventional maximum velocity; (iii) the VDVZ discontinuity is absent; (iv) the strong coupling scale is $(mM_{Pl})^{1/2}$. The latter two properties are in sharp contrast to the Lorentz-invariant gravity with the Pauli--Fierz mass term.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A theory with the action combining the Einstein--Hilbert term and graviton mass terms violating Lorentz invariance is considered at linearized level about Minkowskian background. | contribution 0.65 | 0.46 | 0.04 | asserted 0.68 |
| s1 | It is shown that with one of the masses set equal to zero, the theory has the following properties: (i) there is a gap of order $m$ in the spectrum, where $m$ is the graviton mass scale; (ii) the dispersion relations at ${\bf p}^2 \gg m^2$ are $\omega^2 \propto {\bf p}^2$, the spectrum of tensor modes being relativistic, while other modes having unconventional maximum velocity; (iii) the VDVZ discontinuity is absent; (iv) the strong coupling scale is $(mM_{Pl})^{1/2}$. | result 1.00 | 0.91 | 0.82 | asserted 1.00 |
| s2 | The latter two properties are in sharp contrast to the Lorentz-invariant gravity with the Pauli--Fierz mass term. | interpretation 0.89 | 0.51 | 0.16 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
How the literature uses it
70 arXiv papers cite this in 144 sentences
Papers citing it · 94 sentences without a confident label
Builds on it 1 paper
-
Foundational work has already been done on this previously [ref] for broad sectors of mass-parameter space in D = 3, but we would like to extend the analysis to arbitrary spatial dimensions and analyze both the free field dynamics and linear response to a conserved energy-momentum tensor.
Medium Generated Gap in Gravity and a 3d Gauge Theory · Lorentz Violating Massive Gravity · 0.98
Uses its method or tool 2 papers
-
Then, we have to consider the noncovariant perturbation around the Minkowski space as [ref] g_ij= δ_ij+ h_ij, N= 1+ n, N_i= n_i with
Critical gravity as van Dam-Veltman-Zakharov discontinuity in anti de Sitter space · vDVZ discontinuity in Minkowski space · 0.92
-
Instead, to understand the dynamical degrees of freedom, we will express $φ_μν$ as irreducible $SO(3)$ representations [ref] : [equation] Here, $φ^TT_ij$ are the transverse-traceless tensor (under $SO(3)$) modes of the field, $u_i$ and $A_i$ are transverse vectors, $w, σ, π$ are three dimensional scalars, and [equation] Plugging this decomposition of the field into the FP Lagrangian in Eq. [ref], and after integrating out the non-propagating fields (cf. App. [ref]), we arrive at separate Lagrangians for the propagating degrees of freedom in each sector [ref] .
Massive graviton dark matter searches with long-baseline atom interferometers · Lorentz invariant massive gravity · 0.81
Compares with it 3 papers
-
In both cases, namely, in the model analyzed in [ref] , as well as the one discussed here, the action can be expressed as
Vacuum degeneracy in Massive gravity=Multiplicity of fundamental scales · Lorentz symmetry broken spontaneously · 1.00
-
The interesting ranges propagating only one scalar or no scalars at all are the same as those in [ref] , and for generic mass terms with two scalars, there is always a gradient instability at high energies as happens in the Lorentz preserving case [ref] .
Bigravity and Lorentz-violating Massive Gravity · Scalar Modes · 0.91
-
We briefly compare such massive theories with the known approaches to massive gravitons [ref] .
Fakeons, unitarity, massive gravitons and the cosmological constant · Introduction · 0.99
Cites it as background 33 papers
-
Or again, a similar effect can be induced by Lorentz-violating terms, like in [ref] .
Lorentz-Violating Massive Gravity in Curved Space · Introduction · 0.91
-
This case corresponds to the situation where the action respects the Lorentz symmetry and at the same time, the symmetry is not satisfied at the background level due to the spacetime dependence of additional scalar fields corresponding to the Goldstone modes [ref] .
Vacuum degeneracy in Massive gravity=Multiplicity of fundamental scales · Lorentz symmetry broken spontaneously · 0.91
-
The models of this class are naturally called "massive gravity" since the graviton generically acquires a non-zero mass due to interactions with the scalar fields [ref] .
Black Hole Thermodynamics and Massive Gravity · Introduction · 0.82
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.