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**Unformatted text preview: **e-Deser ghost, around any
¨
background. This can also been seen in the Stuckelberg
language (de Rham, Gabadadze, and Tolley, 2011a).
C. The Ã3 Vainshtein radius We now derive the scale at which the linear expansion
breaks down around heavy point sources in the Ã3 theory. To
linear order around a central source of mass M, the ﬁelds still
have their usual Coulomb form
M1
^^
:
; h $
MP r (8.20) The nonlinear terms in Eqs. (8.10) or (8.18) are suppressed
relative to the linear term by a different factor than in the Ã5
theory,
^
@2
M1
$
:
(8.21)
3
M P Ã 3 r3
Ã3
3
Nonlinearities become important when this factor becomes of
the order of 1, which happens at the radius
M1
GM 1=3
$
:
(8.22)
r ð3 Þ $
V
MP Ã 3
m2
This is parametrically larger than the Vainshtein radius found
in the Ã5 theory.
It is important that the decoupling limit Lagrangian is
ghost free. To see what could go wrong if there were a ghost,
^
expand around some spherical background ¼ ÈðrÞ þ ’
14 This is contrary to Creminelli et al. (2005) who claims that a
ghost is still present at quartic order. As remarked, however, by de
Rham and Gabadadze (2010a), they arrive at the incorrect decoupling limit Lagrangian, which can be traced to a minus sign mistake
in their Eq. 5, which should be as in Eq. (8.4). Kurt Hinterbichler: Theoretical aspects of massive gravity 700 and similarly for h . The cubic coupling and quartic couplings could possibly give fourth order kinetic contributions
of the schematic form, respectively, These would correspond to ghosts with r-dependent masses, We can see the Vainshtein mechanism at work already by
calculating the ratio of the ﬁfth force due to the scalar to the
force from ordinary Newtonian gravity,
8 r 3=2
r ( r ð3 Þ ;
^
V
F
0 ðrÞ=MP < ðrðV3Þ Þ
¼
$
(8.29)
2
:
FNewton
M=MP r2
1
r ) r ð3 Þ :
V Ã3
Ã6
3
3
;
;
(8.24)
È
[email protected] È
or, given that the background ﬁelds are similar to È $
ðM=MP Þð1=rÞ,
2
M
MP
Ã 6 r4 :
(8.25)
m2 ðrÞ $ P Ã3 r;
ghost
3
3
M
M There is a gravitational strength ﬁfth force at distances much
farther than the Vainshtein radius, but the force is suppressed
at distances smaller than the Vainshtein radius.
This suppression extends to all scalar interactions in the
presence of the source. To see how this comes about, we study
perturbations around a given background solution ÈðxÞ.
Expanding 1
È ð @2 ’ Þ 2 ;
Ã3
3 1
È @ 2 È ð @2 ’ Þ 2 :
Ã6
3 (8.23) m2 ðrÞ $
ghost Thus the ghost mass sinks below the cutoff Ã3 at the radius
1=2
M1
M
1
rð3Þ $
;
:
(8.26)
ghost
MP Ã 3
MP
Ã3
As happened in the Ã5 theory, these radii are parametrically
larger than the Vainshtein radius. This is a fatal instability
which renders the whole nonlinear region inaccessible, unless
we lower the cutoff of the effective theory so that the ghost
stays above it, in which case unknown quantum corrections
would also kick in at $rð3Þ , swamping the entire nonlinear
ghost
Vainshtein region.
D. Th...

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