JHEP11(2019)098
where (r
∗
, r
s
) are functions of the boundary time t and of the length l. We will denote
(x
+
(r), v
+
(r)) and (x
−
(r), v
−
(r)) as branches 1 and 2 of the geodesic, respectively. At
initial time t = 0, the geodesic is entirely in AdS and
r
∗
(t = 0) =
2
l
. (2.10)
2.2 BTZ geodesics
For v > 0, the Vaidya spacetime is a BTZ black hole:
ds
2
= −r
2
1 −
r
2
h
r
2
dv
2
+ 2 dv dr + r
2
dx
2
. (2.11)
The event horizon of the black hole is located at r = r
h
and the Hawking temperature
is T =
r
h
2π
.
The part of the HRT surface in the Vaidya spacetime for v > 0 is given by the space-like
geodesic in the BTZ geometry [30]:
x
±
(r) =
1
4r
h
2 ln
r
2
− J r
2
h
±
q
r
4
+ (E
2
−J
2
−1) r
2
h
r
2
+ J
2
r
4
h
r
2
+ J r
2
h
±
q
r
4
+ (E
2
−J
2
−1) r
2
h
r
2
+ J
2
r
4
h
+ ln
(J + 1)
2
−E
2
(J − 1)
2
−E
2
,
(2.12)
v
±
(r) = t +
1
2r
h
ln
r − r
h
r + r
h
r
2
− (E + 1) r
2
h
±
q
r
4
+ (E
2
−J
2
−1) r
2
h
r
2
+ J
2
r
4
h
r
2
+ (E − 1) r
2
h
±
q
r
4
+ (E
2
−J
2
−1) r
2
h
r
2
+ J
2
r
4
h
, (2.13)
with E and J being two integration constants arising from the equations of motion (see
appendix A). Depending on the values of E, J in (2.12), (2.13), the structure of the
geodesic changes; it is useful to distinguish four regions [30], see figure 1. In our nota-
tion, we have translated the solutions in x in such a way that they are symmetric under
the exchange x → −x.
Let us start for simplicity with E = 0, which corresponds to geodesics lying on t-
constant slices. By symmetry, it is not restrictive to choose J > 0 and then there are only
two kinds of such geodesics (see figure 1): the ones with J > 1 (region I) and the ones
with J < 1 (region III).
2
In figure 2 we show the plot of the geodesic (2.12) for both the
cases J > 1 and J < 1. By direct calculation, we find that the minimal value of r along
the geodesic is:
r
0
=
(
J r
h
, J > 1
r
h
, J < 1 .
(2.14)
The geodesics relevant as HRT surfaces for the static BTZ black hole are the ones in region
I, because they have minimal length compared to the ones in region III. Note that a
space-like geodesic with E = 0 in a static BTZ spacetime never penetrates inside the black
2
In the special case E = 0 and J = 1, the geodesic is singular. We shall see that this value will be never
attained in our context.
– 5 –
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