List of Footnotes
1 | In the case of ‘discretization’ or ‘deconstruction’ the higher dimensional approach has been useful only ‘after the fact’. Unlike for DGP, in massive gravity the extra dimension is purely used as a mathematical tool and the formulation of the theory was first performed in four dimensions. In this case massive gravity is not derived per se from the higher-dimensional picture but rather one can see how the structure of general relativity in higher dimensions is tied to that of the mass term. | |
2 | The equation of motion with respect to ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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3 | This is already a problem at the classical level, well before the notion of particle needs to be defined, since classical
configurations with arbitrarily large ![]() ![]() |
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4 | In this review, the notion of fully non-linear coordinate transformation invariance is equivalent to that of full diffeomorphism invariance or covariance. | |
5 | Up to other Lovelock invariants. Note however that ![]() ![]() |
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6 | Strictly speaking, the notion of spin is only meaningful as a representation of the Lorentz group, thus the theory of massive
spin-2 field is only meaningful when Lorentz invariance is preserved, i.e., when the reference metric is Minkowski. While the
notion of spin can be extended to other maximally symmetric spacetimes such as AdS and dS, it loses its meaning for
non-maximally symmetric reference metrics ![]() |
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7 | This procedure can of course be used for any reference metric, but it fails in identifying the proper physical degrees of freedom when dealing with a general reference metric. See Refs. [145*, 154*] as well as Section 8.3.5 for further discussions on that point. | |
8 | In the normal branch of DGP, this brane-bending mode turns out not to be normalizable. The normalizable brane-bending mode which is instead present in the normal branch fully decouples and plays no role. | |
9 | Note that in DGP, one could also consider a smooth brane first and the results would remain unchanged. | |
10 | The local gauge invariance associated with covariance leads to ![]() ![]() ![]() |
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11 | Discretizing at the level of the metric leads to a mass term similar to (2.83*) which as we have seen contains a BD ghost. | |
12 | This special fully non-linear and Lorentz invariant theory of massive gravity, which has been proven in all generality to be free of the BD ghost in [295*, 296*] has since then be dubbed ‘dRGT’ theory. To avoid any confusion, we thus also call this ghost-free theory of massive gravity, the dRGT theory. | |
13 | The analysis performed in Ref. [95*] was unfortunately erroneous, and the conclusions of that paper are thus incorrect. | |
14 | In the previous section we obtained directly a theory of massive gravity, this should be seen as a trick to obtain a consistent theory of massive gravity. However, we shall see that we can take a decoupling limit of bi- (or even multi-)gravity so as to recover massive gravity and a decoupled massless spin-2 field. In this sense massive gravity is a perfectly consistent limit of bi-gravity. | |
15 | The field redefinition is local so no new degrees of freedom or other surprises hide in that field redefinition. | |
16 | See Refs. [169, 434, 242, 340] for additional work on deconstruction in five-dimensional AdS, and how this tackles the strong coupling issue. | |
17 | Some dofs may ‘accidentally’ disappear about some special backgrounds, but dofs cannot disappear non-linearly if they were present at the linearized level. | |
18 | More recently, Alexandrov impressively performed the full analysis for bi-gravity and massive gravity in the vielbein language [15*] determining the full set of primary and secondary constraints, confirming again the absence of BD ghost. This resolves the potential sources of subtleties raised in Refs. [96, 351, 349, 348]. | |
19 | We stress that multiplying with the matrix ![]() ![]() ![]() |
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20 | If only ![]() ![]() ![]() ![]() ![]() ![]() |
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21 | Technically, only one of them generates a first class constraint, while the ![]() ![]() |
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22 | This is actually precisely the way ghost-free massive gravity was originally constructed in [137*, 144*]. | |
23 | The non-renormalization theorem protects the parameters ![]() |
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24 | Note that the Vainshtein mechanism does not occur for all parameters of the theory. In that case the massless limit does not reproduce GR. | |
25 | This result has been checked explicitly in Ref. [146*] using dimensional regularization or following the log divergences.
Taking power law divergences seriously would also allow for a scalings of the form ![]() ![]() ![]() |
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26 | If ![]() |
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27 | We thank the authors of [177*, 178*] for pointing this out. | |
28 | This is not to say that perturbations and/or perturbativity do not break down earlier in the quartic Galileon, see for instance Section 11.4 below as well as [88, 58], which is another sign that the Vainshtein mechanism works better in that case. | |
29 | The minimal model does not have a Vainshtein mechanism [435] in the static and spherically symmetric configuration so
in the limit ![]() ![]() |
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30 | In taking this limit, it is crucial that the second metric ![]() |
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31 | In the context of DGP, the Friedmann equation was derived in Section 4.3.1 from the full five-dimensional picture, but one would have obtained the correct result if derived instead from the decoupling limit. The reason is the main modification of the Friedmann equation arises from the presence of the helicity-0 mode which is already captured in the decoupling limit. | |
32 | See [478*] for a recent review and more details. The convention on the parameters ![]() ![]() ![]() |
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33 | Notice that this is not an issue in massive gravity with a flat reference metric since the analogue Friedmann equation does not even exist. | |
34 | Notice that even if massive gravity is formulated without the need of a reference metric, this does not change the fact that one copy of diffeomorphism invariance in broken leading to additional degrees of freedom as is the case in new massive gravity. | |
35 | See also [335, 195, 49, 50, 51] for other ghost-free non-local modifications of gravity, but where the graviton is massless. |