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Morley, T., Taylor, P. and Winstanley, E. orcid.org/0000-0001-8964-8142 (2018) Vacuum
polarization on topological black holes. Classical and Quantum Gravity, 35 (23). 235010.
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https://doi.org/10.1088/1361-6382/aae45b
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Vacuum polarization on topological black holes
Thomas Morley
Consortium for Fundamental Physics, School of Mathematics and Statistics,
Hicks Building, Hounsfield Road, Sheffield. S3 7RH United Kingdom
E-mail: TMMorley1@sheffield.ac.uk
Peter Taylor
Centre for Astrophysics and Relativity, School of Mathematical Sciences,
Dublin City University, Glasnevin, Dublin 9, Ireland
E-mail: Peter.Taylor@dcu.ie
Elizabeth Winstanley
Consortium for Fundamental Physics, School of Mathematics and Statistics,
Hicks Building, Hounsfield Road, Sheffield. S3 7RH United Kingdom
E-mail: E.Winstanley@sheffield.ac.uk
Abstract. We investigate quant u m effects on topological black hole space-times
within the framework of quantum fiel d theory on curved space-times. Considering
a quantum scalar field, we extend a recent mo d e- su m regularization prescription for
the computation of the renormalized vacuum polarizat i on to asymptotically anti-de
Sitter black holes with nonspherical event horizon topology. In particular , we calculate
the vacuum polarization for a massless, conformally-coupled scalar field on a four-
dimensional topological Schwarzschild-anti-de Sitter black hole background, comparing
our resul t s with those for a spherically-symmetric black hole.
PACS numbers: 04.62.+v, 04.70.D y
Keywords: va cu u m polarization, topological black holes
1. Introduction
In quantum field theory on curved space-ti m e (QFTCS), the renormalized stress-energy
tensor (RSET) h
ˆ
T
µν
i is an object of central importance. Via the semi-classical Einstein
equations
G
µν
+ Λg
µν
= 8πh
ˆ
T
µν
i, (1.1)
(where G
µν
is the Einstein tensor, Λ the cosmological constant and g
µν
the metric
tensor) the RSET governs the back-reaction of the quantum field on the space-time
Vacuum polarization on topological black holes 2
geometry. The stress-energy tensor oper at o r
ˆ
T
µν
involves products of field operators
evaluated at the sam e space-time point and is therefore divergent. In the point-splitting
approach p i o n eer ed by DeWitt and Christensen [
1–3], this divergence is regularized by
considering the operators acting on two closely-sepa r at ed space- ti m e points. The RSET
is computed by subtracting off th e divergences, wh i ch ar e purely geometrical in nature
and independent of the quantum state under consi d er at i o n . The parametrix encoding
these geometric d i vergent terms was orig i n al l y constructed using a DeWitt-Schwinger
expansion [1–3]. This prescripti o n was made mor e p r eci s e by Wald [4,5] who g ave a set of
axioms that (almost) uniquely det er m i n e t h e R S ET. Wald further showed that encoding
the divergences using Hadamard elementary solutions, of which the DeWitt-Schwinger
expansion can be thought of as a special case, produced an RS ET satisfying these
axioms. Th e Hadamard prescription provides a more elegant and general prescri p t i o n
than the DeWitt-Schwinger approach in that it can be applied for arbitrary field m a ss
and arbitrary dimensions (see, for example, [
6]).
In practice, the computation of the RSET on space-times other than those with
maximal symmetry (see, for example, [
7,8]) is a challenging task. Of particular interest
are black hole space-times, and there is a long history of RSET computations on
asymptotically fl a t Schwarzsch i l d black hole backgrounds, for both quantum scalar
fields [
9–13] and fiel d s of higher spin [14–17]. In four space-time dim en si o n s, Anderson,
Hiscock and Samuel (AHS) [18, 19] have developed a general methodology for finding
the RSET on a stati c, spherically-symmetric black hole. Their method makes heavy
use of WKB approximati o n s and the RSET is given as a sum of two parts, the first
of which is analytic and the second of which requi re s numerical computation. More
recently, Levi, Ori and collaborators [20, 21] have developed a new method (dubbed
“pragmatic mode-sum regul a r i za ti o n ” [22]) for finding the RSET which does not rely
on WKB approximations and has th e advantage that it can be applied to stationary as
well as static black holes [21].
Given th e challenges of computing the RSET, it is instr u ct i ve to consider instead
the vacuum polarization (VP) h
ˆ
φ
2
i of a quantum scalar field
ˆ
φ. Unlike the tensor RSET,
the VP is a scalar object and hence cannot distinguish between the future and past event
horizons of a black hole. Nonetheless, the VP shares many f eat u r es with the RSET,
for example, if the VP diverges on a horizon, then it is likely that the RSET will also
diverge there. The VP has been compu t ed on asymptotically flat, spherically-symmetric,
four-dimensional black hole space-times (see, for examp l e, [
22–27]) using bot h the AHS
and Levi-Ori methods. The AHS method uses a Euclideanized black hole space-time,
and is therefore most amenable for the computation of the VP in the Hartle-Hawking
state [
28], while the Levi-Ori method employs a Lorentzian metri c and h a s been applied
to the Boulware [29] and Unruh [30] states. The AHS method has the disad vantage
that, unless th e field is massless and conformally coupled, the analytic part of the
expression for the VP diverges at the black hole event horizon. The numeric part also
diverges at the horizon , resul t i n g in a q u a ntity which is fi n i t e overal l [
31]. Using Green-
Liouville asymptotics, Breen and Ottewill [32] im p r oved the AHS method by writ i n g
Vacuum polarization on topological black holes 3
the VP as a sum of terms, each of which is m a n i fest l y finite. Their method also extends
to the RSET [33, 34]. Another approach is to match the Hadamard parametrix to a
mode-sum in Minkowski spa ce- ti m e in a suitab l e choice of coo rd i n a t es [
35,36], but this
approach is not very general and requires a bespoke mapp i n g between Minkowski space-
time and the black hole being con si d er ed . However, this matching method has been
successfully applied to nonsphericall y - sy m m et ri c space-times where the AHS method is
not applicable [
35, 36].
With the exception of [
36], the above discussion focu ses on asympt o t i cal l y flat
black hole geometries. Asymptotically anti-de Sitter (adS) black holes have received
rather less attention in the QFTCS literature, despite their inherent interest du e to
the adS/CFT (conformal field theory) correspondence (see, for example, [37] for a
review). The VP h a s been computed for a massless, conformally-co u p l ed scalar field
on a four-dimensional spherically-symmetric Schwarzschild-adS black hole using the
AHS method [
38] and also on a static, spherically-symmetric adS black hole with
asymptotically Lifschitz geometry [
39]. An AHS-like meth od has also been used to
find the VP on an asymptotically adS black hole with cylindrical rather than spher i ca l
symmetry [
40]. The cor r esponding calculations on th e three-dimen si on a l , asymptotically
adS BTZ black hole [
41–43] are rather simpler than those for four-dimensional black
holes and both the VP and RSET for a conformally -c ou pl ed scalar field can be found in
closed form (both when there is a black hole event horizon and in the naked singularity
case [44–49]). These closed-form ex p r essi o n s have been used to study the back-reaction
of the quantum field on th e space-time geometry via the semi-classical Einstein equations
(1.1) [50–52].
In asymptotically adS space-time, unli ke asympt o t i cal l y flat space-time, black holes
do not necessarily have spherical event horizon topology (see, for example, [
53–62]).
This context has received little attention in the literature on semi-classical effects,
but p rovides an interesting dichotomy between th e classical and semi-classical Einstein
equations. For example, given an a sym p to t i ca l l y adS black hole with an event horizon
whose geometry is flat, it is straightforward to make identificat i o n s that pr oduce a black
cylinder or a black torus. However, as these identifications correspond to a choice of
boundary cond i ti o n s, the classical equations themsel ves are not sensitive to this choice
since the PDEs are quasi-local in nat u r e (though obviously a particular solution is
picked out by the choice of boundary conditions). On the other hand, the semi -cl a ssi c al
Einstein equatio n s (
1.1) are sourced by the expectation value of a fiel d operato r in a given
quantum state, a state which requires global information in order to be defined and which
is sensitive t o any such id entifications that di st i n gu i sh between, say, a cylinder and a
torus. Since the quantum stress-energy tensors on backgrounds with different topologies
(but with the same geometry) are different, the back-reaction effects will be very much
sensitive to the g l ob a l topology. Thou gh we do not consider different identifications for
the same black hole geometry in this paper, this context does provide an a d d i ti o n a l
motivation and as a first step in these directions, it is necessa r y to consider quantum
fields on the backgrounds of asymptotically adS black holes with horizons of nonspherical
Vacuum polarization on topological black holes 4
topology. More speci fi ca l l y, we calculate the VP for a massless, conformally-coupled
quantum scalar field on the backgrounds of Schwarzschild-adS black holes wit h event
horizons of spherical, planar and hyberboloidal topology.
We work in the Hartle-Hawking state [28] and follow the recent “extended
coordinates” meth odology of [
63, 64], which gives an effi ci ent numerical method for
finding the VP on the Eucli d e an i ze d space-time. In [
63, 64], the extended coordinates
method is developed for static, spherically-symmetric black holes in four or more space-
time dimensions. Here we apply the extended coordin a t es app r oa ch to black holes wit h
nonspherical event horizon topology. For our purposes, a distinct advantage of th e
extended coordinates approach is t h at the asymptotically adS nature of the space-times
we consider does not present any particular difficulties.
The outlin e of the paper is as follows. We rev i ew the geometry of topological b l a ck
holes in Section
2, before deriving an expression for the Euc l i d ean Gre en ’ s fun ct i o n fo r
a massless, conformal l y -c ou p l ed , scalar field on these ba ckgrounds in S e ct i on
3. The
renormalized VP will be calcul at ed using Hadamard renormali za ti o n , and in Section
4
we introduce the Hadamard parametrix, emp l oying the extended coordinates method
of [
63,64] to write this as a mode sum, which is amenable to numerical evaluatio n . Our
numerical methodology is outlined in Section
5, together with our numerical results.
Section
6 contains our conclusions and further discussion. Throughout this paper, we
use mostly plus space-time conventio n s, and units in which G = ~ = c = k
B
= 1.
2. Topological black holes
We consider static black hole solutions of the vacu u m Ei n st ei n equat i o n s with a negative
cosmological constant Λ:
G
µν
+ Λg
µν
= 0. (2.1)
The event horizon is a two-surface of constant curvature, and the metric takes the
following form, in Schwarzschild-like coordinates [
53–62]:
ds
2
= −f (r) dt
2
+
dr
2
f(r)
+ r
2
dΩ
2
k
, (2.2)
where k ∈ {−1, 0, 1}, corresponding to negative, zero and positive horizon curvature
respectively. The function f(r) is given by
f(r) = k −
2M
r
+
r
2
L
2
, (2.3)
where M is the black hole mass and L =
p
−3/Λ is the adS curvature l en g th - sca l e. The
two-metric dΩ
2
k
is
dΩ
2
k
= dθ
2
+ F
2
k
(θ) d ϕ
2
, (2.4)
with
F
k
(θ) =
sin θ, k = 1,
θ, k = 0,
sinh θ, k = −1.
(2.5)
