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**Unformatted text preview: **degrees of freedom, the
same counting as in the linear theory. The nonlinear theory
contains the same number of degrees of freedom as the
linearized theory.
Now looking at the mass term, in ADM variables we have
ðh h À h h Þ
¼ ik jl ðhij hkl À hik hjl Þ þ 2ij hij
À 2N 2 ij hij þ 2Ni ðgij À ij ÞNi ;
where hij gij À ij . The action becomes
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 (5.47) (5.48) C
;
m2 ij hij Ni ¼ 1 ij
ðg À ij ÞÀ1 Cj :
m2 (5.49) When these values are plugged back into Eq. (5.48), we
have an action with no constraints or gauge symmetries at all,
so all the phase space degrees of freedom are active. The
resulting Hamiltonian is
H¼ Æt 1 Z D ij
_
d xp gij À N C À Ni Ci
2 2
m2 ik jl
½ ðhij hkl À hik hjl Þ þ 2ij hij
À
4
À 2N 2 ij hij þ 2Ni ðgij À ij ÞNi : In the m Þ 0 case, the Fierz-Pauli term brings in contributions to the action that are quadratic in the lapse and shift
(but still free of time derivatives). Thus the lapse and shift no
longer serve as Lagrange multipliers, but rather as auxiliary
ﬁelds, because their equations of motion can be algebraically
solved to determine their values, (5.42) Æt Æt S¼ 689 1Zd 1
C2
1 i ij
C ðg À ij ÞÀ1 Cj
d x 2 ij þ
2
2
2m hij 2m2
þ m2 ik jl
½ ðhij hkl À hik hjl Þ þ 2ij hij ;
4 (5.50) which is nonvanishing, unlike in GR. In four dimensions, we
thus have 12 phase space degrees of freedom, or 6 real
degrees of freedom. The linearized theory had only 5 degrees
of freedom, and we have here a case where the nonlinear
theory contains more degrees of freedom than the linear
theory. It should not necessarily be surprising that this can
happen, because there is no reason nonlinearities cannot
change the constraint structure of a theory or that kinetic
terms cannot appear at higher order.
As argued by Boulware and Deser (1972), the
Hamiltonian (5.50) is not bounded, and since the system is
nonlinear, it is not surprising that it has instabilities
(Gabadadze and Gruzinov, 2005). The nature of the instability, i.e., whether it is a ghost of a tachyon, what backgrounds
it appears around, and its severity, is hard to see in the
Hamiltonian formalism. But in Sec. VII.B we see that this
instability is a ghost, a scalar with a negative kinetic term, and
that its mass around a given background can be determined. It
turns out that around ﬂat space, the ghost degree of freedom is
not excited because its mass is inﬁnite, but around nontrivial
backgrounds its mass becomes ﬁnite. This ghostly extra
degree of freedom is referred to as the Boulware-Deser ghost
(Boulware and Deser, 1972).
There is still the possibility that adding higher order interaction terms such as h3 terms and higher can remove the
ghostly sixth degree of freedom. Boulware and Deser analyzed a large class of various mass terms, showing that the
sixth degree of freedom remained (Boulware and Deser,
1972), but they did not consider the most general poss...

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