![FIG. 1. Carter-Penrose diagrams of the spacelike stretched black hole spacetime for the case r0 < r− < r+ on the left and of the manifold M with mirrors described in the text on the right (adapted from Ref. [16]).](/figures/fig-1-carter-penrose-diagrams-of-the-spacelike-stretched-307om29m.webp)
Renormalized vacuum polarization on rotating
warped AdS
3
black holes
Hugo R. C. Ferreira
∗
and Jorma Louko
†
School of Mathematical Sciences, University of Nottingham,
Nottingham NG7 2RD, United Kingdom
(Dated: Revised December 2014)
Abstract
We compute the renormalized vacuum polarization of a massive scalar field in the Hartle-Hawking
state on (2+1)-dimensional rotating, spacelike stretched black hole solutions to Topologically Mas-
sive Gravity, surrounded by a Dirichlet mirror that makes the state well defined. The Feynman
propagator is written as a mode sum on the complex Riemannian section of the spacetime, and a
Hadamard renormalization procedure is implemented by matching to a mode sum on the complex
Riemannian section of a rotating Minkowski spacetime. No analytic continuation in the angular
momentum parameter is invoked. Selected numerical results are given, demonstrating the numer-
ical efficacy of the method. We anticipate that this method can be extended to wider classes of
rotating black hole spacetimes, in particular to the Kerr spacetime in four dimensions.
∗
pmxhrf@nottingham.ac.uk
†
jorma.louko@nottingham.ac.uk
1
arXiv:1410.5983v2 [gr-qc] 28 Jan 2015
I. INTRODUCTION
The study of quantum field theory on black hole spacetimes has mostly been restricted
to static, spherically symmetric spacetimes. Nevertheless, there have been attempts at
considering stationary black hole spacetimes, with the main focus on the Kerr spacetime
[1–6]. One important task is the computation of expectation values of the renormalized
stress-energy tensor for a matter field in a given quantum state [7, 8]. This has proven to
be very challenging and, so far, almost all calculations have only addressed the differences
between expectation values for different quantum states [6] and the large field mass limit [9].
In [10], the stress-energy tensor for the rotating BTZ black hole [11, 12] was renormalized
with respect to AdS
3
, by using the fact that the black hole corresponds to AdS
3
with
discrete identifications, but this method cannot be used for more general classes of rotating
black hole solutions. We could summarize the main difficulties in three points: (i) the
technical complexity of the computations required for the Kerr spacetime, due to the lack of
spherical symmetry, (ii) the nonexistence of generalizations of the (globally defined, regular
and isometry-invariant) Hartle-Hawking state defined in static spacetimes, and (iii) the
unavailability of Euclidean methods which simplify the task in static spacetimes.
To tackle point (i), we focus on a rotating black hole spacetime in 2+1 dimensions,
the spacelike stretched black hole [13]. This is a vacuum solution of topologically massive
gravity (TMG) [14, 15], a deformation of (2+1)-dimensional Einstein gravity, and it can
be thought of as a “warped” version of the BTZ black hole. In contrast to the BTZ so-
lution, the causal structure of the spacelike stretched black hole is similar to that of the
Kerr spacetime [16]. In this setting, the matter field equations can be solved in terms of
hypergeometric functions, which considerably simplify the technical issues in comparison
with the Kerr spacetime. These black hole solutions are known to be classically stable to
massive scalar field perturbations and, in particular, classical superradiance does not give
rise to superradiant instabilities [17]. In this paper, we study a quantum scalar field on this
black hole spacetime.
Concerning point (ii), the Hartle-Hawking vacuum state in the Schwarzschild spacetime
is well known not to generalize to the Kerr spacetime [18]. As reviewed in [5], this is linked
to the existence of a speed-of-light surface, outside of which no observer can corotate with
the Kerr horizon. However, if we surround the Kerr hole by a mirror that is inside the
speed-of-light surface, and we introduce appropriate boundary conditions at the mirror,
then a Hartle-Hawing state (regular at the horizon and invariant under the isometries of the
spacetime) exists inside the mirror. Further, this Hartle-Hawking state is known to be free
from superradiant instabilities for a massless field [5, 6, 19] and the same conclusion may
well extend to a massive field. In this paper we introduce a similar mirror on the (2 + 1)-
dimensional spacelike stretched black hole, and we consider the similar Hartle-Hawking state
inside this mirror. This (2 + 1)-dimensional Hartle-Hawking state is known to be free of
superradiant instabilities for massless as well as massive fields [17].
Finally, regarding point (iii), while Kerr does not admit a real section with a positive
definite metric [20], it does admit a real section with a complex Riemannian metric to which
the Feynman propagator in the Hartle-Hawking state inside a mirror can be analytically
continued [21–23]. This complex Riemannian, or “quasi-Euclidean”, section on Kerr, hence,
serves as the counterpart of the more familiar Euclidean (or Riemannian) section of static
black hole spacetimes. In this paper we introduce the similar complex Riemanian section
of the spacelike stretched black hole, and we exploit this section to renormalize the vacuum
2
expectation value of a massive scalar field. The crucial point is that the complex Riemannian
section of the spacelike stretched black hole has a unique Green’s function, and this Green’s
function is expressible as a discrete mode sum whose divergence at the coincidence limit
can be matched to that of a corresponding mode sum on a complex Riemannian section of
a rotating flat spacetime. The renormalization procedure in the Hartle-Hawking state can,
hence, be carried out using this discrete mode sum.
In summary, in this paper we shall compute the renormalized vacuum polarization hΦ
2
(x)i
of a massive scalar field Φ in the Hartle-Hawking state on a spacelike stretched black hole
surrounded by a mirror with Dirichlet boundary conditions, implementing the Hadamard
renormalization prescription on the complex Riemannian section of the spacetime. In the
first instance, this calculation can be taken as a warm-up for the computation of the renor-
malized stress-energy tensor on the spacelike stretched black hole. In the longer perspective,
we believe that all the conceptual aspects of our method are applicable to wide classes of
rotating black hole spacetimes, and in particular to Kerr in four dimensions. An implemen-
tation of our method in more than three dimensions will of course face new technical issues
due to the more complicated functions that arise in the separation of the wave equation.
The contents of the paper are as follows. We begin in Sec. II with the quantization
of a massive scalar field on the spacelike stretched black hole bounded by a mirror, in-
cluding a short description of the Hadamard renormalization. In Sec. III, we outline the
quasi-Euclidean method we use to obtain the complex Riemannian section of the black hole
spacetime and renormalize the vacuum polarization. This is followed in Sec. IV with the
numerical evaluation of the renormalized vacuum polarization. Finally, our conclusions are
presented in Sec. V. Technical steps in the analysis are deferred to five appendices. Through-
out this paper we use the (−, +, +) signature and units in which ~ = c = G = k
B
= 1.
II. SPACELIKE STRETCHED BLACK HOLES AND SCALAR FIELDS
In this section, we first give a short description of topologically massive gravity and
review the basic features of the spacelike stretched black hole solutions, including their
causal structure. We then proceed to quantize the massive scalar field and outline the
Hadamard renormalization procedure.
A. Spacelike stretched black holes
The (2+1)-dimensional rotating black holes we focus in this paper are vacuum solutions
of topologically massive gravity, whose action is obtained by adding a gravitational Chern-
Simons term to the Einstein-Hilbert action with a negative cosmological constant [14, 15]
S = S
E-H
+ S
C-S
, (2.1)
with
S
E-H
=
1
16πG
Z
d
3
x
√
−g
R +
2
`
2
, (2.2)
S
C-S
=
`
96πGν
Z
d
3
x
√
−g
λµν
Γ
ρ
λσ
∂
µ
Γ
σ
ρν
+
2
3
Γ
σ
µτ
Γ
τ
νρ
. (2.3)
3
FIG. 1. Carter-Penrose diagrams of the spacelike stretched black hole spacetime for the case
r
0
< r
−
< r
+
on the left and of the manifold M with mirrors described in the text on the right
(adapted from Ref. [16]).
G is Newton’s gravitational constant, ν is a dimensionless coupling, g is the determinant of
the metric, R is the Ricci scalar, ` > 0 is the cosmological length (which will be set to ` ≡ 1
from now on), Γ
ρ
λσ
are the Christoffel symbols, and
λµν
is the Levi-Civita tensor in three
dimensions.
The spacelike stretched black hole is one of the several types of warped AdS
3
black hole
solutions [13]. Its metric, in coordinates (t, r, θ), is given by
ds
2
= −N
2
(r)dt
2
+
dr
2
4R
2
(r)N
2
(r)
+ R
2
(r)
dθ + N
θ
(r)dt
2
, (2.4)
with t ∈ (−∞, ∞), r ∈ (0, ∞), (t, r, θ) ∼ (t, r, θ + 2π) and
R
2
(r) =
r
4
h
3(ν
2
− 1)r + (ν
2
+ 3)(r
+
+ r
−
) − 4ν
p
r
+
r
−
(ν
2
+ 3)
i
, (2.5a)
N
2
(r) =
(ν
2
+ 3)(r − r
+
)(r − r
−
)
4R
2
(r)
, (2.5b)
N
θ
(r) =
2νr −
p
r
+
r
−
(ν
2
+ 3)
2R
2
(r)
. (2.5c)
There are outer and inner horizons at r = r
+
and r = r
−
, respectively, where the
coordinates (t, r, θ) become singular, and a singularity at r = r
0
. The dimensionless coupling
ν ∈ (1, ∞) is the warp factor, and in the limit ν → 1 the above metric reduces to the metric
of the BTZ black hole in a rotating frame. More details about this black hole solution can
be found in [13, 17, 24–29]. Here, we just describe a few relevant features.
4
The Carter-Penrose diagram for this spacetime when r
0
< r
−
< r
+
is shown in Fig. 1,
which is essentially of the same form of those of asymptotically flat spacetimes in 3+1
dimensions.
Consider the exterior region r > r
+
. ∂
t
and ∂
θ
are Killing vector fields. However, ∂
t
is
spacelike everywhere, even though surfaces of constant t are still spacelike. Consequently,
there is no stationary limit surface and no observers following orbits of ∂
t
(the usual “static
observers” in other spacetimes) anywhere. In fact, it is easy to show that there is not any
timelike Killing vector field in the exterior region of the spacetime.
Nonetheless, there are observers at a given radius r following orbits of the vector field
ξ(r) = ∂
t
+ Ω(r) ∂
θ
, which are timelike as long as
Ω
−
(r) < Ω(r) < Ω
+
(r) , (2.6)
with
Ω
±
(r) = −
2
2νr −
p
r
+
r
−
(ν
2
+ 3) ±
p
(r − r
+
)(r − r
−
)(ν
2
+ 3)
. (2.7)
Ω(r) is negative for all r > r
+
, approaches zero as r → +∞, and tends to
Ω
H
= −
2
2νr
+
−
p
r
+
r
−
(ν
2
+ 3)
(2.8)
as r → r
+
. In view of these observations, we can regard Ω
H
as the angular velocity of the
outer horizon with respect to stationary observers close to infinity.
One particular important class of observers are the “locally non-rotating observers”
(LNRO), whose worldlines are everywhere normal to constant-t surfaces. Because of this,
they are sometimes also known as “zero angular momentum observers” (ZAMO). In this
case, Ω(r) = −N
θ
(r), which satisfies (2.6). They are the closest to the concept of “static
observers” in this spacetime.
Even though there is no stationary limit surface, there is still a speed-of-light surface,
beyond which an observer cannot corotate with the outer horizon. Given the information
above it is easy to check that the vector field χ = ∂
t
+ Ω
H
∂
θ
is the Killing vector field which
generates the horizon. χ is null at the horizon and at
r = r
C
=
4ν
2
r
+
− (ν
2
+ 3)r
−
3(ν
2
− 1)
, (2.9)
which is the location of the speed-of-light surface.
In the context of quantum field theory, as it is detailed below, the nonexistence of an
everywhere timelike Killing vector field in the exterior region of the spacetime is directly
related to the nonexistence of a well defined quantum vacuum state which is regular at the
horizon and is invariant under the isometries of the spacetime. For the Kerr spacetime, this
has been proven in [18]. A vacuum state with these properties can however be defined if
we restrict the spacetime by inserting an appropriate mirrorlike boundary which respects
the Killing isometries of the spacetime. The simplest example is a boundary M at constant
radius r = r
M
, in which the scalar field satisfies Dirichlet boundary conditions, Φ(t, r
M
, θ) =
0. If we choose the radius such that r
M
∈ (r
+
, r
C
), then χ is a timelike Killing vector
field up to the boundary, and a vacuum state with the above properties is well defined.
Moreover, the introduction of a mirror with reflective boundary conditions also serves to
5