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2 679 (2.51) Quantum mechanically, this results from the Fadeev-Popov
gauge ﬁxing procedure. We have
L þ LGF ¼ 1h hh À 1hhh;
2
4 (2.52) whose equations of motion are (2.50). Note, however, that the
classical gauge condition we have been using is not obtained
as an equation of motion and must be imposed separately if
solutions are to be compared.
We can write the gauge ﬁxed Lagrangian as L þ LGF ¼
1
~
h O; h , where
2
~
O; ¼ h½1ð þ Þ À 1 : (2.53)
2
2
Going to momentum space and inverting, we obtain the
propagator
Ài 1
1
;
D ;¼ 2 ð þ ÞÀ
D À 2
p2
(2.54)
~
which satisﬁes the Eq. (2.43) with O in place of O.
This propagator grows similar to $1=p2 at high energy.
Comparing the massive and massless propagators,
Eqs. (2.44) and (2.54), and ignoring for a second the terms
in Eq. (2.44) which are singular as m ! 0, there is a difference in coefﬁcient for the last term, even as m ! 0. For
D ¼ 4, it is 1=2 vs 1=3. This is the ﬁrst sign of a discontinuity
in the m ! 0 limit.
III. LINEAR RESPONSE TO SOURCES We now add a ﬁxed external symmetric source T ðxÞ to
the action (2.1),
Z
1
S ¼ dD x À @ h @ h þ @ h @ h À @ h @ h
2
1
1
þ @ h@ h À m2 ðh h À h2 Þ þ h T :
2
2
(3.1)
Àð
Here ¼ MP DÀ2Þ=2 is the coupling strength to the source.5
The equations of motion are now sourced by T , hh À @ @ h À @ @ h þ @ @ h
þ @ @ h À hh À m2 ðh À hÞ ¼ ÀT :
(3.2)
In the case m ¼ 0, acting on the left with @ gives identically
zero, so we must have the conservation condition @ T ¼ 0
if there is to be a solution. For m Þ 0, there is no such
condition.
5
The normalizations chosen here are in accord with the general
pﬃﬃﬃﬃﬃﬃﬃﬃ
relativity deﬁnition T ¼ ð2= ÀgÞL=g , as well as the normalization g ¼ 2h . Kurt Hinterbichler: Theoretical aspects of massive gravity 680 A. General solution to the sourced equations We now ﬁnd the retarded solution of Eq. (3.2), to which the
homogeneous solutions of Eq. (2.2) can be added to obtain the
general solution. Acting on the equation of motion (3.2) with
@ , we ﬁnd
@ h À @ h ¼ 2 @ T :
(3.3)
m
Plugging this back into Eq. (3.2), we ﬁnd
hh À @ @ h À m2 ðh À hÞ
¼ ÀT þ 2 ½@ @ T þ @ @ T À @@T ;
m
where @@T is short for the double divergence @ @ T .
Taking the trace of this we ﬁnd
h¼À
DÀ2
@@T:
TÀ 4
m 2 ð D À 1Þ
m DÀ1 Applying this to Eq. (3.3), we ﬁnd
@ h ¼ À 2
@ T þ 2 @ T
m ðD À 1Þ
m
DÀ2
@ @@T;
À4
m DÀ1 (3.4) The general solution for a conserved source is then
Z dD p
1
eipx 2
ð2ÞD
p þ m2
p p
1
þ 2 T ðpÞ ; (3.9)
Â T ðpÞ À
DÀ1
m h ðxÞ ¼ (3.5) which when applied along with Eq. (3.4) to the equations of
motion gives
@ @
1
À 2 T
ð@2 À m2 Þh ¼ À T À
DÀ1
m
þ 2 @ @ T þ @ @ T
m
...

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