JHEP03(2016)194
where, as the notation indicates, the exponent ∆ is the conformal dimension of the dual
operator and is given by
∆ =
d
2
+
r
d
2
4
+ m
2
L
2
. (2.6)
Now, the usual holographic dictionary relates the first coefficient φ
0
to the coupling λ, and
φ
1
to the expectation value of the relevant operat or O
∆
— the explicit relations will b e
given below in section
2.1.1. Implicitly with eq. (2.6) and throughout this subsection, we
are assuming the rel e vant operators have d/2 < ∆ < d. As commented before, the cases
of (d − 2)/2 < ∆ < d/2 and ∆ = d/2 will be treated separately in subsequent subsections.
In the above discussion, the ansatz has been fairly general so that it could describe
expectation values which vary across the boundary spacetime , e.g., hO
∆
i(x). However, in
the problem of interest, we want to probe the boundary theory by examini ng the entan-
glement entropy of spheres that are much smaller than the scale of any such variations.
Hence for simplicity, we will assume that our holographic background is invariant under
translations in both space and ti me in the following. Hence t he metric perturbations g
(k)
ab
in eq. (
2.4) and the coefficients φ
0,1
in the scalar (2.5) are constants, and our met r i c ansatz
in eq. (
2.3) is simplified with g
ab
(z, x) reduced to g
ab
(z). We also impose the boundary
condition that g
ab
(z) → η
ab
asymptotically to recover the flat boundary metric i n eq. (
2.4).
Note, however, that we are not otherwise r e st ri c t i ng the metric perturbations, which will
allow us to c onsi de r states which are anisotropic and stationary (but not static).
With these choices, the Einstein equations be come , e.g., [
28, 29]:
0 = g
′′
ab
−
d − 1
z
g
′
ab
− g
cd
g
′
ca
g
′
db
+
1
2
g
cd
g
′
cd
g
′
ab
−
1
z
g
cd
g
′
cd
g
ab
+
g
ab
(d − 1)z
2
m
2
L
2
Φ
2
+
κ
3
Φ
3
0 = g
ab
g
′′
ab
−
1
z
g
ab
g
′
ab
−
1
2
g
ab
g
′
bc
g
cd
g
′
da
+ Φ
′ 2
+
1
(d − 1)z
2
m
2
L
2
Φ
2
+
κ
3
Φ
3
(2.7)
Similarly, the scalar fiel d equation following from eqs. (
2.1) and (2.2) becomes
Φ
′′
−
(d − 1)Φ
′
z
+
g
ab
g
′
ab
2
Φ
′
−
(mL)
2
Φ
z
2
−
κΦ
2
2z
2
= 0. (2.8)
Moreover, we note that these equations are redund ant. For example, the second equation
in eq. (2.7) will autom ati c al l y be solved if the first set of equations, as well as the scalar
field equation (
2.8), are solved. However, this equation may still be used as a consistency
check for the soluti ons.
As commented above, we will construct the solutions perturbatively in the amplitude
in the scalar field. Hence we introduce a small expansion parameter ε, so th at the scalar
is written as
Φ(z) = ε φ
0
z
d−∆
+ ε φ
1
z
∆
+ O(ε
2
) (2.9)
as z approaches zero — we re-iterate that φ
0
and φ
1
are now simply constants. Solving
eq. (
2.8) to first order in ε yields the usual solution for ∆ given in eq. (2.6). Next, we solve
eq. (
2.7) to second order in ε. Her e , we formulate the most general ansatz for the metric
functions g
ab
(z) which approach η
ab
as z → 0,
g
ab
= η
ab
+ ε
2
m
ab
z
d
+ a
ab
φ
2
0
z
2∆
+ b
ab
φ
2
1
z
2(d−∆)
+ c
ab
φ
0
φ
1
z
d
+ O(ε
3
) , (2.10)
– 6 –
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