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**Unformatted text preview: **ense of as an effective ﬁeld theory,
valid only at energies below some ultraviolet cutoff scale Ã.
In 2003, Arkani-Hamed, Georgi, and Schwartz (2003)
brought to attention a method of restoring gauge invariance
to massive gravity in a way that makes it very simple to see
what the effective ﬁeld theory properties are. They showed
that massive gravity generically has a maximum UV cutoff of
Ã5 ¼ ðMP m4 Þ1=5 . For Hubble scale graviton mass, this is a
length scale ÃÀ1 $ 1011 km. This is a very small cutoff,
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parametrically smaller than the Planck mass, and goes to
zero as m ! 0. Around a massive source, the quantum effects
become important at the radius rQ ¼ ðM=MPl Þ1=3 ð1=Ã5 Þ,
which is parametrically larger than the Vainshtein radius at
which nonlinearities enter. For the Sun, rQ $ 1024 km.
Without ﬁnding a UV completion or some other resummation, there is no sense in which we can trust the solution inside
this radius, and the usefulness of massive gravity is limited. In
particular, since the whole nonlinear regime is below this
radius, there is no hope to examine the continuity of physical
quantities in m and explore the Vainshtein mechanism in a
controlled way. On the other hand, it can be seen that the mass
of the Boulware-Deser ghost drops below the cutoff only
when r & rQ , so the ghost is not really in the effective theory
at all and can be consistently excluded. Kurt Hinterbichler: Theoretical aspects of massive gravity Putting aside the issue of quantum corrections, there has
been continued study of the Vainshtein mechanism in a purely
classical context. It has been shown that classical nonlinearities do indeed restore continuity with GR in certain circumstances. In fact, the ghost degree of freedom can play an
essential role in this, by providing a repulsive force in the
nonlinear region to counteract the attractive force of the
longitudinal scalar mode.
By adding higher order graviton self-interactions with
appropriately tuned coefﬁcients, it is in fact possible to raise
the UV cutoff of the theory to Ã3 ¼ ðMP m2 Þ1=3 , corresponding to roughly ÃÀ1 $ 103 km. In 2010, the complete action
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of this theory in a certain decoupling limit was worked out by
de Rham and Gabadadze (2010a), and they show that, remarkably, it is free of the Boulware-Deser ghost. Recently, it
was shown that the complete theory is free of the BoulwareDeser ghost. This Ã3 theory is the best hope of realizing a
useful and interesting massive gravity theory.
The subject of massive gravity also naturally arises in
extra-dimensional setups. In a Kaluza-Klein scenario such
as GR in 5d compactiﬁed on a circle, the higher Kaluza-Klein
modes are massive gravitons. Brane world setups such as the
Dvali-Gabadadze-Porrati (DGP) model (Dvali, Gabadadze,
and Porrati, 2000a) give more intricate gravitons with resonance masses. The study of such models has complemented
the study of pure 4d massive gravity and has pointed toward
new research directions....

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