Journal ArticleDOI

# Vacuum polarization on topological black holes

AbstractWe investigate quantum effects on topological black hole space-times within the framework of quantum field theory on curved space-times. Considering a quantum scalar field, we extend a recent mode-sum regularization prescription for the computation of the renormalized vacuum polarization 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 results with those for a spherically-symmetric black hole.

Topics: Black hole (68%), Event horizon (66%), Vacuum polarization (59%), Scalar field (58%), Quantum field theory (55%)

### 1. Introduction

• In quantum field theory on curved space-time , the renormalized stress-energy tensor (RSET) 〈T̂µν〉 is an object of central importance.
• Their method makes heavy use of WKB approximations and the RSET is given as a sum of two parts, the first of which is analytic and the second of which requires numerical computation.
• Nonetheless, the VP shares many features with the RSET, for example, if the VP diverges on a horizon, then it is likely that the RSET will also diverge there.
• The corresponding calculations on the three-dimensional, 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-coupled 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 expressions have been used to study the back-reaction of the quantum field on the space-time geometry via the semi-classical Einstein equations (1.1) [50–52].

### 2. Topological black holes

• The authors consider static black hole solutions of the vacuum Einstein equations with a negative cosmological constant Λ: Gµν + Λgµν = 0. (2.1).
• The metric in this case is the usual Schwarzschild anti-de Sitter spacetime.
• Here θ again takes on all positive real values.
• The authors will parameterize their black holes by (k, α,M) rather than (k, L,M).

### 3. Quantum scalar fields on topological black hole space-times

• The authors wish to consider the VP for a quantum Klein-Gordon field in the Hartle-Hawking state propagating on the classical topological black hole space-times discussed in the previous section.
• In the quantum theory, the field gets promoted to an operator-valued distribution, φ(x) → φ̂(x), upon which is imposed a set of canonical commutation relations (see, for example, [65]).
• Extracting a meaningful finite quantity from this is the essence of the regularization problem which the authors discuss in the following section.
• For the remainder of this section, the authors will focus on constructing the Green’s function in the Hartle-Hawking state |HH〉.
• An important point in the construction of the mode-sum representation of the Green’s function is that the pathological behaviour in the coincidence limit manifests as the nonconvergence of the mode-sums in (3.6).

• The näıve expression for the VP (3.5) is ill-defined and requires a prescription that (i) removes the singular terms (a process called regularization) and (ii) absorbs the terms introduced in order to cure the divergences into some other parameters in the theory (a process called renormalization).
• The conceptual framework for achieving this in a curved space-time, known as the point-splitting scheme, dates back to seminal work by DeWitt and Christensen [1–3].
• A more general axiomatic variant of their original point-splitting prescription, known as Hadamard regularization [4–6], is the approach adopted here.

### 4.4. k = −1 regularization parameters

• Factoring out the time dependence using an appropriate addition theorem is a little more subtle than in the other two cases.
• This is unlike the other cases where the appropriate addition theorems involved convergent sums, albeit sums that are only slowly converging.
• The result is however the same as that obtained by using (4.37) with ψ = ψ′ and ignoring the fact that the sum does not converge.

### 5. Renormalized vacuum polarization

• To compute the renormalized VP numerically, the authors therefore need to find the mode functions pnλ(ζ), qnλ(ζ), the normalization constants Nnλ, and the regularization parameters Ψnλ(i, j|r), before combining all these quantities into the sums in (5.1).
• The authors describe their numerical methodology for each of these parts before discussing their results.

• Since the ODE (3.9) is singular at the horizon, the authors start their integration close to the horizon and use the Frobenius series (5.2) to give the initial values of pnλ and its first derivative.
• Unlike the asymptotically flat case, infinity is a regular singular point of the ODE (3.9).
• On pure adS space-time, either transparent or reflective boundary conditions can be imposed on a massless, conformally-coupled scalar field at time-like infinity [66].
• The effect on the renormalized VP of choosing alternative boundary conditions will be explored elsewhere [67] (see also [49] for the effect of boundary conditions on the renormalized VP on a BTZ black hole).

### 5.2. Mode sums

• Having found the radial functions pnλ and qnλ, the regularization parameters Ψnλ(i, j|r) (4.40) are readily computed in Mathematica since the authors have analytic expressions for these quantities.
• The relative error between these two mode sums is typically O(10−9) or smaller.
• Second, approximating the integrand by an interpolating function between the grid points in λ also introduces an error into the evaluation of the integral.
• Here δλ and ǫλ are constants which are estimated by fitting this power law (using the Mathematica function FindFit) to the values of the integrand computed numerically for λ ∈ (40, 50).
• Once the sum/integral over λ has been performed, the final, trivial, part of the computation of the renormalized VP is to subtract the term −f ′(r)/48π2r in (5.1).

### 5.3. Results for the vacuum polarization

• The authors now present their results for the renormalized VP (5.1) on a selection of topological black holes.
• In all their plots below, dots denote values computed numerically, while the curves interpolate between these values.
• Alternatively, for large α and small L, the black hole is large compared with the adS curvature length-scale.
• For all values of α, the authors find that the renormalized VP is monotonically decreasing from its value on the horizon to the asymptotic value (5.6).
• 8), both of which leave the dimensionless radial coordinate ζ unchanged.

### 6. Conclusions

• Conformally-coupled scalar field on an asymptotically adS topological black hole space-time, for which the event horizon is a two-surface of constant curvature (either positive, negative, or zero).the authors.
• This method extends in an elegant way to topological black holes having an event horizon with either zero or negative curvature, and enables the Hadamard parametrix to be written as a mode sum.
• This monotonically decreasing behaviour as the radial distance from the event horizon increases is the same as that observed for a massless conformallycoupled scalar field in the Hartle-Hawking state on an asymptotically flat Schwarzschild black hole [24].
• Either of these extensions to the work presented here would involve finding a representation of the tail part of the Hadamard parametrix, which can be ignored when calculating the renormalized VP for a massless, conformally-coupled scalar field.

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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 [
13], 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 [13]. 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 a t Schwarzsch i l d black hole backgrounds, for both quantum scalar
ﬁelds [
913] and ﬁel d s of higher spin [1417]. 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, [
2227]) 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 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
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
4143] 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 [4449]). 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) [5052].
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, [
5362]).
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 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 [
5362]:
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)

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