JHEP09(2018)131
u v
Future
Interior
Past
Interior
Left
Exterior
Right
Exterior
Figure 1. Penrose diagram for the two-sided black branes we consider. This geometry is dual to
a thermofield double state |T F Di made by entangling two copies of the boundary theory.
null trajectory very close to the past horizon, as we will now see. Let (u, v) = (u
0
, v
0
) be
the initial position of the perturbation in Kruskal coordinates. Under time evolution these
coordinates change as (u
0
, v
0
) → (e
−
2π
β
t
u
0
, e
2π
β
t
v
0
), and this means that, as time passes, the
perturbation gets more and more localized at u = 0, and stretched along the v-direction.
Besides that, from the point of view of the t = 0 frame, the energy of the perturbation
increases exponentially as t
0
,
7
moves farther into the past, i.e. E = E
0
e
2π
β
t
0
. As a result,
for t
0
far enough into the past, the energy-momentum tensor of the perturbation can be
very well approximated by
T
shock
uu
= E
0
e
2π
β
t
0
δ(u) a(x
i
) , (3.8)
where E
0
is the asymptotic energy of the perturbation and a(x
i
) is some function rep-
resenting the localization of the operator O
2
(t
0
). Note that the shock wave divides the
geometry into two halves: the causal future of the shock wave (the region u > 0), and its
causal past (the region u < 0).
The backreaction to this perturbation on the geometry is actually very simple and can
be described by a shift v → v + α(t, x
i
) in the causal future of the shock wave, while the
causal past is unaffected.
8
This is illustrated in figure 2.
The shock wave metric is simply given by [24, 51, 52]
ds
2
= 2A(u, v)dudv + G
ij
dx
i
dx
j
− 2A(u, v)α(t, x
i
)δ(u)du
2
, (3.9)
where the shock wave profile α(t, x
i
) has to obey
"
G
ii
∂
i
∂
i
−
2π
β
2
G
ii
(r
H
)G
0
ii
(r
H
)
G
0
tt
(r
H
)
#
α(t, x
i
) =
8πG
N
E
A(r
H
)
e
2πt/β
a(x
i
) , (3.10)
7
In our convention the Killing time coordinate t runs forward on the right boundary and backwards on
the left. Hence, a perturbation on the left boundary at the time t
0
> 0 is in the past of the t = 0 slice of
the geometry.
8
This was first done in [51] for Minkowski spacetime, and then generalized for generic curved spacetimes
in [52]. More details about the case of anisotropic metrics can be found, for instance, in [24].
– 7 –
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