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**Unformatted text preview: **n. The linear theory violates the principle of continuity of the physics in the parameters of the theory. This is known
as the vDVZ discontinuity. The discontinuity was soon traced
to the fact that not all of the degrees of freedom introduced by
the graviton mass decouple as the mass goes to zero. The
massive graviton has ﬁve spin states, which in the massless
limit become the two helicity states of a massless graviton, two
helicity states of a massless vector, and a single massless
scalar. The scalar is essentially the longitudinal graviton, and
it maintains a ﬁnite coupling to the trace of the source stress
tensor even in the massless limit. In other words, the massless
limit of a massive graviton is not a massless graviton, but rather
a massless graviton plus a coupled scalar, and the scalar is
responsible for the vDVZ discontinuity.
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 If the linear theory is accurate, then the vDVZ discontinuity represents a true physical discontinuity in predictions,
violating our intuition that physics should be continuous in
the parameters. Measuring the light bending in this theory
would be a way to show that the graviton mass is mathematically zero rather than just very small. However, the linear
theory is only the start of a complete nonlinear theory,
coupled to all the particles of the standard model. The
possible nonlinearities of a real theory were studied several
years later by Vainshtein (1972), who found that the nonlinearities of the theory become stronger and stronger as the mass
of the graviton shrinks. What he found was that around any
massive source of mass M, such as the Sun, there is a new
length scale known as the Vainshtein radius rV $
2
ðM=m4 MP Þ1=5 . At distances r & rV , nonlinearities begin to
dominate and the predictions of the linear theory cannot be
trusted. The Vainshtein radius goes to inﬁnity as m ! 0, so
there is no radius at which the linear approximation tells us
something trustworthy about the massless limit. This opens
the possibility that the nonlinear effects cure the discontinuity. To have some values in mind, if we take M the mass of
the Sun and m a very small value, say the Hubble constant
m $ 10À33 eV, the scale at which we might want to modify
gravity to explain the cosmological constant, we have
rV $ 1018 km, about the size of the Milky Way.
Later the same year, Boulware and Deser (1972) studied
some speciﬁc fully nonlinear massive gravity theories and
showed that they possess a ghostlike instability. Whereas the
linear theory has 5 degrees of freedom, the nonlinear theories
they studied turned out to have 6, and the extra degree of
freedom manifests itself around nontrivial backgrounds as a
scalar ﬁeld with a wrong sign kinetic term, known as the
Boulware-Deser ghost.
Meanwhile, the ideas of effective ﬁeld theory were being
developed, and it was realized that a nonrenormalizable
theory, even one with apparent instabilities such as massive
gravity, can be made s...

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