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**Unformatted text preview: **eraction
terms. The term suppressed by the smallest scale is the cubic
scalar term n ¼ 3, nA ¼ nh ¼ 0, which is suppressed by the
scale Ã5 ¼ ðMP m4 Þ1=5 , ^
The free graviton coupled to the source via ð1=MP Þh0 T
also survives the limit, as does the free decoupled vector.
We can now understand the origin of the Vainshtein radius
at which the linear expansion breaks down around heavy
point sources. The scalar couples to the source through the We also get a whole slew of interaction terms, third order
and higher in the ﬁelds, suppressed by various scales. We
always assume m < MP . always appears with two derivatives, A always appears with one derivative, and h always
appears with none, so a generic term, with nh powers of h ,
nA powers of A , and n powers of , reads
2
$ m2 MP hnh ð@AÞnA ð@2 Þn
4Ành À2nA À3n ^ nh $ Ã ^
^
h ð@AÞnA ð@2 Þn ; (7.4) where the scale suppressing the term is
Ã ¼ ðMP mÀ1 Þ1= ; ¼ 3n þ 2nA þ nh À 4
:
n þ nA þ nh À 2 Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 Kurt Hinterbichler: Theoretical aspects of massive gravity 694 ^
trace, ð1=MP ÞT . To linear order around a central source of
mass M, we have
M1
^
:
$
MP r (7.11) The nonlinear term is suppressed relative to the linear term by
the factor
^
@4
M1
$
:
M P Ã 5 r5
Ã5
5
5 (7.12) Nonlinearities become important when this factor becomes of
order 1, which happens at the radius
rV $ M
MP 1=5
1
GM 1=5
$
:
Ã5
m4 (7.13) When r & rV , linear perturbation theory breaks down and
nonlinear effects become important. This is exactly the
Vainshtein radius found in Sec. V.B by directly calculating
the second order correction to spherical solutions.
In the decoupling limit, the gauge symmetries (7.7) reduce
to their linear forms,
^
^
h ¼ @ þ @ ; ^
^
A ¼ @ Ã; ¼ 0:
(7.14) Even though is gauge invariant in the decoupling limit, the
fact that it always comes with two derivatives means that the
global Galileon symmetry (7.8) is still present, as is the shift
symmetry on A .
B. Ghosts Note that the Lagrangian (7.10) is a higher derivative
action, and its equations of motion are fourth order. This
means that this Lagrangian actually propagates two
Lagrangian degrees of freedom rather than one, since we
need to specify twice as many initial conditions to uniquely
solve the fourth order equations of motion (de Urries and
Julve, 1998), and by Ostrogradski’s theorem (Ostrogradski,
1850; Woodard, 2007), one of these degrees of freedom is a
ghost. The decoupling limit contains 6 degrees of freedom:
two in the massless tensor, two in the free vector, and two in
the scalar. This matches the number of degrees of freedom in
the full theory as determined in Sec. V.C, so the decoupling
limit we have taken is smooth. The extra ghostly scalar
degree of freedom is the Boulware-Deser ghost. Note that
at linear order, the higher derivative scalar terms for the scalar
are not visible, so the linear theory has...

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