Unformatted text preview: ackground metric, which is a
solution to the full nonlinear equations, about which we may
expand the action. Often the solution metric we are expanding
around will be the same as the absolute metric, but if we were
expanding around a different solution, say a black hole, there
would be two distinct structures, the black hole solution
metric and the absolute metric.
If we add matter to the theory and agree to use only
minimal coupling to the metric g , then the absolute metric
does not directly inﬂuence the matter. It is the geodesics and
lengths as measured by the full metric (i.e., the solution of the
ﬁeld equations) that we care about. In massive gravity, unlike
in GR, if we have a solution, we cannot perform a diffeomorphism on it to obtain a second solution to the same theory.
What we obtain instead is a solution to a different massive
gravity theory, one in which the absolute metric is related to
the original absolute metric by the same diffeomorphism.
Going to more general interactions beyond Eq. (5.1), our
main interest will be in adding interaction terms with no
derivatives, since these are most important at low energies.
The most general such potential which reduces to Fierz-Pauli
at quadratic order involves adding terms cubic and higher in
h in all possible ways
S ¼ 2 dD x ð ÀgRÞ À Àg0 m2 Uðgð0Þ ; hÞ ;
where the interaction potential U is the most general one that
reduces to Fierz-Pauli at linear order,
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 Uðgð0Þ ; hÞ ¼ U2 ðgð0Þ ; hÞ þ U3 ðgð0Þ ; hÞ þ U4 ðgð0Þ ; hÞ
þ U5 ðgð0Þ ; hÞ þ Á Á Á ; (5.4) U2 ðgð0Þ ; hÞ ¼ ½h2 À ½h2 ; (5.5) U3 ðgð0Þ ; hÞ ¼ þC1 ½h3 þ C2 ½h2 ½h þ C3 ½h3 ; (5.6) U4 ðgð0Þ ; hÞ ¼ þD1 ½h4 þ D2 ½h3 ½h þ D3 ½h2 2
þ D4 ½h2 ½h2 þ D5 ½h4 ; (5.7) U5 ðgð0Þ ; hÞ ¼ þF1 ½h5 þ F2 ½h4 ½h þ F3 ½h3 ½h2
þ F4 ½h3 ½h2 þ F5 ½h2 2 ½h
þ F6 ½h2 ½h3 þ F7 ½h5 ;
. (5.8) The square bracket indicates a trace, with indices raised with
gð0Þ; , i.e., ½h ¼ gð0Þ h , ½h2 ¼ gð0Þ h gð0Þ h , etc.
The coefﬁcients C1 , C2 , etc. are generic coefﬁcients. Note
that the coefﬁcients in Un ðgð0Þ ; hÞ for n > D are redundant by
1, because there is a combination of the various contractions,
the characteristic polynomial LTD ðhÞ (see the Appendix),
which vanishes identically. Thus one of the coefﬁcients in
Un ðgð0Þ ; hÞ for n > D (or any one linear combination) can be
set to zero.
If we want, we can reorganize the terms in the potential by
raising and lowering with the full metric g rather than the
absolute metric gð0Þ ,
S ¼ 2 dD x ð ÀgRÞ À Àg m2 V ðg; hÞ ; (5.9)
V ðg; hÞ ¼ V2 ðg; hÞ þ V3 ðg; hÞ þ V4 ðg; hÞ þ V5 ðg; hÞ
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