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Quantum-improved Schwarzschild-(A)dS and Kerr-(A)dS spacetimes

Jan M. Pawlowski1,2 and Dennis Stock3,1

1Institut für Theoretische Physik, Universität Heidelberg,

Philosophenweg 16, 69120 Heidelberg, Germany

2ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung mbH,

Planckstr. 1, 64291 Darmstadt, Germany

3University of Bremen, Center of Applied Space Technology and Microgravity (ZARM),

28359 Bremen, Germany

(Received 21 August 2018; published 12 November 2018)

We discuss quantum black holes in asymptotically safe quantum gravity with a scale identification based

on the Kretschmann scalar. After comparing this scenario with other scale identifications, we investigate in

detail the Kerr–(A)dS and Schwarzschild–(A)dS spacetimes. The global structure of these geometries is

studied as well as the central curvature singularity and test particle trajectories. The existence of a Planck-

sized, extremal, zero-temperature black hole remnant guarantees a stable end point of the evaporation

process via Hawking radiation.

DOI: 10.1103/PhysRevD.98.106008

I. INTRODUCTION

The consistent quantization of gravity is an open

challenge to date. One of the candidates is the asymptotic

safety (AS) scenario for quantum gravity [1], its attraction

being the possible quantum-field theoretical ultraviolet

completion of the standard model with gravity. If realized,

it is the minimal UV closure of high-energy physics

including gravity within a purely field-theoretical setup.

One of the prominent and characteristic properties of

asymptotically safe gravity is its ultraviolet scaling regime

for momentum scales klarger than the Planck scale Mpl.In

the AS setup, the latter is defined as the scale beyond which

quantum gravity corrections dominate the physics and agrees

well with the classical Planck scale. In this regime, the

Newton’s coupling Gand cosmological constant Λ,aswell

as all further couplings of terms, e.g., of the higher curvature

invariants Rn, run according to their canonical scaling. For

the Newton’s coupling and cosmological constant in par-

ticular, this entails GðkÞ∝1=k2and ΛðkÞ∝k2, respectively,

instead of the classical constant behavior. Consequently, the

physics at these scales looks rather different to that of general

relativity.

Black holes offer one of the few possibilities where

such deviations from classical general relativity may be

observed as they feature large curvatures. Asymptotically

safe quantum black holes have been amongst the first

applications of asymptotically safe gravity after its first

explicit realization within the functional renormalization

group [2]. Within such a renormalization group setting, the

Newton’s coupling and cosmological constant are naturally

elevated to couplings running with the momentum (RG)

scale k. Then, classical solutions of the Einstein field

equations are quantum improved by replacing Newton’s

and the cosmological constant by functions depending on a

respective length scale. The k-dependent RG-runnings,

equipped with an identification between momentum and

length scales, serve as an ansatz for these functions. The

earliest works investigated the Schwarzschild spacetime

[3,4] followed by studies of the Kerr spacetime [5] and

Schwarzschild–(A)dS geometries [6]. Black holes in higher

dimensions have been studied in [7]. All works, summa-

rized in [8], match the classical results of general relativity

in the low energy limit, but show significant changes for the

number of horizons, test particle trajectories, the Hawking

temperature, and the entropy around the Planckian regime.

There is evidence for a cold, extremal Planck-sized

remnant, which is a smallest black hole with zero temper-

ature, a possibly promising answer to the end point of black

hole evaporation. By studying dynamical, nonvacuum

solutions such as the Vaidya spacetime, the processes of

black hole formation [9] and evaporation [10] can be

addressed directly, leading to the same conclusions as

above. The quantum effects render the central curvature

singularity at r¼0less divergent, some scenarios lead to a

complete resolution. A detailed study on the implications

for the laws of black hole thermodynamics was performed

in [11]. Most of the above results for a quantum-improved

spacetime were obtained by using a cutoff identification

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PHYSICAL REVIEW D 98, 106008 (2018)

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based on a classical spacetime. This was addressed in

[12,13], where a consistent framework with an underlying

quantum spacetime was introduced.

In this work, we present a new scale identification based

on the quantum-improved classical Kretschmann scalar.

This approach takes the running of the couplings into

account which removes unphysical features in the resulting

geometries. For the first time in this quantum gravity setup,

the Kerr–(A)dS geometry, as the most general vacuum

black hole solution including a cosmological constant, is

studied in great detail. As a special case (a¼0), the results

for Schwarzschild–(A)dS are presented separately. The

ordinary Schwarzschild and Kerr solutions are also con-

tained by setting the cosmological constant to zero.

This work is structured as follows: we start with a brief

review of the AS scenario of quantum gravity in Sec. II, and

discuss the studied geometries in Sec. III. The novel scale

identification is discussed in Sec. IV. Results on horizons

and the GR-limit are presented in Sec. V, the global structure

in Sec. VI, test particle trajectories in Sec. VI C, the

curvature singularity in Sec. VII, and Hawking temperatures

and the black hole evaporation process in Sec. VIII. Some

technical details are deferred to the appendices which

contain in particular a discussion of proper distance match-

ings; see Appendix C.

II. ASYMPTOTIC SAFE QUANTUM GRAVITY

By now, asymptotically safe quantum gravity has been

studied in an impressive wealth and depth of approximations

including higher derivative terms, the full fðRÞpotential as

well as the inclusion of matter, see e.g., [14–19] and

references therein. The specific shape of the running of

GðkÞand ΛðkÞdepends on the regularization scheme or

regulator which also defines part of the scale identification.

Moreover, despite the advances in the approximation

schemes used in recent computations, the systematic error

estimates are still relatively large. However, while these

details do not affect the results of this work qualitatively, all

runnings have to meet the following general constraints:

(1) The existence of a UV fixed point, that is, the

dimensionless couplings gand λbecome constant in

the UV-limit:

ðg; λÞ⟶

k→∞ðg;λÞ:ð1Þ

(2) The effective theory should recover the classical

theory of general relativity in the IR-limit, i.e., Gand

Λapproach Newton’s constant G0and a cosmo-

logical constant Λ0respectively, reducing the effec-

tive action to the Einstein-Hilbert action:

G;Λ⟶

k→0G0;Λ0⇔g∼k2and λ∼k−2:ð2Þ

The running of gðkÞand λðkÞis typically obtained numeri-

cally. In the following, we approximate them by analytical

expressions, which show the same features and are com-

patible with the above constraints in the UV and IR. For

instance, a comparison with the results of the systematic

vertex expansion up to the fourth order in [20] is provided

in Fig. 18 in the Appendix. The following scale runnings

are used:

gðkÞ¼ G0gk2

gþG0k2⇔GðkÞ¼ G0g

gþG0k2;

λðkÞ¼Λ0

k2þλ⇔ΛðkÞ¼Λ0þλk2:ð3Þ

The functional dependence of gðkÞwas already used in [4]

and λðkÞagrees with the expression used in [8] without

the logarithmic term. G0and Λ0are the IR values of the

gravitational and cosmological coupling, whereas gand λ

are the fixed point values of the dimensionless couplings.

In the following analysis, we choose the numerical values at

the fixed point to be the ones for the background couplings

obtained in Appendix B of [20], together with their

identification scheme in (34):

g¼1.4;λ¼0.1:ð4Þ

The dependence of Newton’s coupling GðkÞand cosmo-

logical constant ΛðkÞon the running scale kreflects the

nontrivial dependence of the full effective action at vanish-

ing cutoff scale on the Laplacian Δ, as well as the existence

of higher-order terms. As in earlier works, we use the

following strategy to take into account these terms: we use

solutions to the Einstein field equations and assume that

quantum gravity effects can be modeled by momentum-

dependent Gand Λ, equipped with a relation to convert the

momentum into a length scale. The now r-dependent Gand

Λare inserted back into the classical solution, yielding a

quantum-improved spacetime. This procedure is the ana-

logue of the Uehling’s correction in QED, see [21,4] for

more details. In the context of asymptotically save gravity, it

has been shown in [6], that a quantum-improved metric in

the above sense can be a solution to the field equations

derived from the quantum-improved Einstein-Hilbert action

in the UV-limit, at least in the spherically symmetric case.

Furthermore, the quantum-improved metric, together with

its observables, approach the results obtained from general

relativity in the IR, and thus show the correct low

energy limit.

In the following we need the couplings GðrÞand ΛðrÞas

functions of radius rrather than momentum scale k. Thus,

we have to establish a relation kðrÞin order to arrive at

GðkðrÞÞ;ΛðkðrÞÞ. A commonly used ansatz for kðrÞis

kðrÞ¼ ξ

DðrÞ;ð5Þ

with constant ξand a r-dependent function Dwith

momentum dimension minus one (length), encoding the

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physical scales. Our choice ξ¼1=ﬃﬃﬃﬃﬃ

λ

pis further motivated

in Appendix A.

III. INVESTIGATED GEOMETRIES

In this work, we study geometries based on solutions of

the Einstein equations with cosmological constant, but

vanishing stress-energy tensor. Depending on the sign of

the cosmological constant, the spacetime is called asymp-

totically de Sitter (dS), flat, or anti–de Sitter (AdS). As the

stress-energy tensor is zero, the black hole is allowed to

have a mass and angular momentum, but no charge. Thus,

we study the Schwarzschild–(A)dS spacetime of a non-

rotating black hole and the Kerr–(A)dS spacetime for a

rotating black hole.

The Kerr–(A)dS geometry is the most general vacuum

black hole solution, which includes a cosmological con-

stant. Hence the Schwarzschild–(A)dS as well as the

Schwarzschild and Kerr solutions in flat space can be

obtained from Kerr–(A)dS by either setting the rotations

parameter aor the cosmological coupling Λto zero. In our

analysis, we discuss the quantum-improved Schwarzschild–

(A)dS and Kerr–(A)dS solution, but the results can be

easily extended to asymptotically flat spacetimes. Below

we briefly summarize some basic properties of these

geometries.

A. Schwarzschild–(A)dS

The Schwarzschild–(A)dS solution is a two-parameter

family of solutions of the nonvacuum Einstein equations,

labeled by ðM; ΛÞ. It is explicitly given by

ds2¼−fðrÞdt2þf−1ðrÞdr2þr2dΩ2;

fðrÞ≔1−2MG

r−

Λ

3r2;ð6Þ

with t∈ð−∞;∞Þ,r∈ð0;∞Þ, Newton’s constant G, the

cosmological constant Λ, and dΩ2the metric on S2.

This solution is spherically symmetric and displays a

curvature singularity at r¼0if M≠0.ForΛ¼0,it

reduces to the Schwarzschild solution in flat space and for

M¼0but Λ≠0, one obtains the metric describing AdS or

dS, depending on the sign of Λ. Therefore, this metric

interpolates between a Schwarzschild solution on small

scales and an (A)dS solution on large scales. Horizons are

solutions to fðrÞ¼0.

B. Kerr–(A)dS

The Kerr–(A)dS solutions form a three parameter family,

labeled by (M,J,Λ). Unlike in the flat case, Mand J

cannot be interpreted as mass and angular momentum of

the black hole anymore; however, for convenience, we still

refer to them as mass and angular momentum in the text

below. The metric is given by [22],

ds2¼−

Δr

ρ2Ξ2ðdt−asin2θdϕÞ2þρ2

Δr

dr2þρ2

Δθ

dθ2

þΔθsin2θ

Ξ2ρ2ðadt−ðr2þa2ÞdϕÞ2;ð7Þ

with

a≔J

M;

ρ2≔r2þa2cos2θ;

Δr≔ðr2þa2Þ1−

Λ

3r2−2GMr;

Δθ≔1þΛ

3a2cos2θ;

Ξ≔1þΛ

3a2:ð8Þ

The parameter ais referred to as rotation parameter and is

restricted by

1

3Λa2>−1;ð9Þ

in order to preserve the Lorentzian signature of the metric.

The coordinate ranges are t∈ð−∞;∞Þ,r∈ð0;∞Þ,θ∈

½0;πand ϕ∈½0;2πÞ. It can be shown that this solution

reduces to a Kerr black hole in the limit of small r, whereas

for large rit gives back the metric of (A)dS. In the case of

a¼0, one recovers the Schwarzschild–(A)dS metric of a

nonrotating black hole (6).ForΛ¼0, the metric reduces to

the one of a Kerr black hole in flat space. For M¼0and

a¼0, we recover (A)dS. For M≠0, there is a curvature

singularity at r¼0in the equatorial plane θ¼π

2. Horizons

correspond to solutions of Δr¼0.

IV. SCALE IDENTIFICATION

In pure gravity systems, i.e., systems with vanishing

stress-energy tensor, there is no unique way to fix the scale

identification. In fact, it turns out that physical features of

the spacetime such as the number of horizons, Hawking

temperatures and the strength of the curvature singularity

actually do depend on the particular choice of kðrÞ.

Motivated by dimensional analysis, one simple way to

identify the momentum scale of the FRG setup with a

length scale is an inverse proportionality. However, this

ansatz is completely insensitive to typical scales of the

underlying spacetime. Therefore, different scale setting

procedures have been brought forward, for instance on

the level of the field equations, e.g., [6]. A more feasible

approach to account for spacetime features is to use proper

distance integrals. As such, they give rise to diffeomor-

phism invariant quantities. Proper distance integrals based

on classical spacetimes were suggested in [4]. Later, it was

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pointed out in [12,13], that this procedure can be upgraded

to a consistent setting by computing the proper distance

already in the quantum-improved geometry.

Here, we investigate this approach for Schwarzschild–

(A)dS and Kerr–(A)dS spacetimes. However, using two

different integration contours for the computation of the

proper distance in the upgraded scheme yields ill-defined

quantities. In the case of a radial integration path, we find

diverging surface gravities for all horizons. This results in

divergent Hawking temperatures, independent of the black

hole parameters. In the case of a path prescribed by the

timelike geodesic of an infalling observer, we find an

identically vanishing eigentime. The analysis and results

for the proper distances are given in Appendix C.

In light of these results, a different identification scheme

is required. Such a scheme has to be based on other

diffeomorphism invariant quantities, for example on cur-

vature scalars. In cosmological contexts, the Ricci scalar R

has been used [23,24]. However, the classical Ricci scalar

cannot be used, since it vanishes identically for vacuum

solutions of the Einstein field equations. Thus, in the

following analysis, we will base our scale identification

on the Kretschmann scalar K¼Rαβγδ Rαβγδ, a diffeomor-

phism invariant quantity of momentum dimension four.

This motivates the scale identification

DKðrÞ¼ 1

χðK−K∞Þ1=4;ð10Þ

with a constant χ, chosen to be χ¼ð

1

8Þ1=4in the following

calculations, and K∞¼Kðr¼∞Þ¼8=3Λ2

0, using (11).

We subtract the Kretschmann scalar at r→∞; otherwise,

DðrÞwould approach a constant in the IR and therefore G

and Λwould fail to display the correct IR-limit G0and Λ0,

respectively, cf. (3). For simplicity, we base the matching

on the classical Kretschmann scalar in the equatorial plane

(θ¼π=2). For both Kerr–(A)dS and Schwarzschild–(A)

dS, we arrive at

K¼8

3Λ2þ48M2

r6G2:ð11Þ

The quantum-improved version of the classical

Kretschmann scalar (11), referred to as Kqu, provides a

consistent framework accounting for typical scales of the

underlying (quantum) geometry. Of course, it would be

desirable to use the true Kretschmann scalar, computed

directly from the quantum-improved metric. This is left for

future work. On a technical level, the RG-improved version

turns (10) into a functional equation for DKðrÞ. In order for

this equation to have a positive, real solution, χmust be

constrained to χ<ð3=8Þ1=4, such that the expression under

the root in the UV-expression in Table II remains positive.

In Appendix A, we discuss the impact of χon the

results. Also, the quantum-improved version of classical

Kretschmann scalar (11) approaches the classical version

for r→∞, but this does not hold for DK, given by (10),

because Kqu →K∞is faster than Kcl →K∞. The curvature

near the singularity, the construction of the Penrose dia-

grams, and the UV-limits for each proper distance are

discussed in Appendix E.

V. LAPSE FUNCTION AND NUMBER

OF HORIZONS

With the running couplings Gand Λfrom the previous

section, physical properties of the quantum-improved

spacetimes can be deduced. Central tools are the lapse

functions fðrÞand ΔðrÞ, whose roots determine the

location of horizons in the spacetime. These zeros are

shown to be Killing horizons in Appendix B, implying that

they can be assigned a constant surface gravity, which turns

out to be proportional to the first derivative of the lapse

function evaluated at the horizon. This can be used to

address thermodynamical processes such as the end point

of black hole evaporation via Hawking radiation. Another

interesting question is that of the similarity of the quantum-

improved geometry to the classical geometry in general

relativity, serving as a metric ansatz for the quantum

improvement.

In this section, we will discuss the lapse functions fðrÞ

and ΔðrÞfor the Kretschmann matching by determining the

number of horizons and comparing them with the lapse

functions of general relativity. We first start with asymp-

totically AdS spacetimes, i.e., Λ0<0, and comment on the

results for Λ0>0subsequently. The results for all other

matchings can be found in Appendix C.

A. Schwarzschild–AdS

Classically, i.e., for constant G&Λ0<0, the lapse

function fðrÞshows just one zero corresponding to the

event horizon of the black hole, whereas the quantum-

improved Schwarzschild geometry shows up to two hori-

zons, if a consistent matching is adopted; see Fig. 1.

Starting at very large masses, well above the Planck mass,

we find two horizons, generated by a minimum of the lapse

function. Comparing with the classical lapse function in

Fig. 2shows that the outer horizon of the quantum-

improved spacetime coincides with the event horizon of

the classical black hole. The larger the mass, the better the

agreement and the more the inner horizon moves towards

zero. Hence, increasing the mass makes the black holes

more classical. Decreasing the mass causes the minimum to

shrink and the horizons to move towards each other. There

exists a critical mass Mcaround two Planck masses,

Mc≈2MPl, when the minimum is also a zero of the lapse

function. Then, both horizons merge and fðrÞhas a double

root. We will see later, that this geometry is similar to a

classical, extreme Reissner-Nordström black hole in AdS.

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For masses below the critical mass, the minimum is above

zero and no horizons are present.

The results for matchings computed in spacetimes with

running couplings agree with the matchings based on

spacetimes with constant couplings on the position of

the outer horizon, but differ significantly for smaller radii.

These differences emerge because in the latter case, the

matching is based on a classical geometry, whereas we

actually study a quantum geometry with running couplings.

Varying the amplitude for negative Λ0does not affect the

qualitative results, but changes the scale.

B. Kerr–AdS

A classical, nonextremal Kerr–AdS spacetime has two

horizons: a Cauchy horizon inside the black hole event

horizon. In contrast to the Schwarzschild case discussed

above, the quantum improvement of this spacetime does

not allow for more horizons than in the classical geometry.

Since the proper distances vanish identically in the con-

sistent scenarios, we show only the results for the

Kretschmann matching in Fig. 3and the dependence on

the rotation parameter for fixed mass in Fig. 4. The results

for the linear matching can be found in Appendix C.In

general, the consistent quantum-improved version displays

the same behavior as the classical solution. However,

the inner horizon in the quantum-improved spacetime is

located at larger radii than the classical Cauchy horizon;

see Fig. 5.

C. Asymptotically de Sitter spaces

If we take the spacetime to be asymptotically de Sitter,

we find the possibility to get up to three horizons. The

additional horizon is generated by the positive cosmologi-

cal constant in the IR and appears in the classical regime at

large radii. The typical shapes of fðrÞand ΔrðrÞare

displayed in Figs. 6and 7for the Kretschmann matching,

the dependence on the amplitude of Λ0is shown in Figs. 8

FIG. 2. Comparison of fðrÞfor all matchings with the classical

result from general relativity for M¼10MPl and Λ0¼−0.1.

Matching based on the quantum geodesic in dark blue, classical

geodesic in dark green, quantum radial path in light blue, classical

radial path in purple, quantum Kretschmann scalar in light green,

classical Kretschmann scalar in dashed black, linear matching in

red and the result from general relativity in dashed dark blue. All

matchings, apart from the classical Kretschmann setting, agree

with the classical position of the outer black hole horizon.

FIG. 1. fðrÞfrom (6) based on the Kretschmann scalar

matching for increasing mass from top to bottom. Results

based on the quantum-improved Kretschmann scalar are given

by solid curves, whereas results based on the classical Kretsch-

mann scalar are dashed. The parameters are Λ0¼−0.1and

M¼0.1;1;2;5;9MPl. Curves of the same mass have the

same color.

FIG. 3. ΔrðrÞfrom (8) based on the Kretschmann scalar

matching for increasing mass from top to bottom. Results

based on the quantum-improved Kretschmann scalar are given

by solid curves, whereas results based on the classical Kretsch-

mann scalar are dashed. With parameters Λ0¼−0.1,a¼2and

M¼0.1;2;4;5;7;9MPl. Curves of the same mass have the

same color.

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and 9. Varying mcontrols the position of the two inner

horizons via the formation of a minimum, whereas Λ0

governs the location of the outer horizon. Thereby, the

interplay of the amplitudes of mand Λ0dictates the number

of horizons. Although we cannot provide an analytical

condition involving mand Λ0for the spacetime exhibiting

three horizons, it is suggestive to see it as the generalized

version of the condition for a classical Kerr-dS spacetime to

have three horizons. This also implies that both quantum-

improved spacetimes have two distinct extremal cases:

both inner horizons merge at a mass m¼Myielding an

extremal black hole inside the cosmological horizon. Or

both outer horizons merge at m¼M, forming the largest

Schwarzschild/Kerr-dS black hole possible, analogous to

the Nariai spacetime.

VI. GLOBAL STRUCTURE, PENROSE DIAGRAMS

AND PARTICLE TRAJECTORIES

In contrast to the classical Schwarzschild–(A)dS and

Kerr–(A)dS geometries of general relativity, the quantum-

improved counterparts can exhibit a different number of

horizons and hence may show a different global structure,

depicted in terms of Penrose diagrams. It turns out that both

geometries, i.e., one based on the Schwarzschild and the

other on the Kerr metric, have the same Penrose diagram.

The resulting diagram is equivalent to the classical

Reissner-Nordström or Kerr geometry. Hence, the quantum

FIG. 4. ΔrðrÞbased on the quantum Kretschmann scalar

matching for fixed mass M¼5MPl and Λ0¼−0.1, but increas-

ing a¼0, 1, 2, 3, 4, 5 from bottom to top.

FIG. 5. Comparison of ΔrðrÞfor the linear matching in dark

blue, the classical Kretschmann setting in red, the quantum

Kretschmann setting in green and the classical result from general

relativity in light blue, with M¼10MPl,a¼2and Λ0¼−0.1.

Apart from the classical Kretschmann setting, all other matchings

agree with the classical position of the outer horizon.

FIG. 6. fðrÞfor asymptotic dS with Λ0¼0.001 for increasing

mass M¼0.1;1;2;3;4;5;6;7;8;9;10MPl from top to bottom.

FIG. 7. ΔrðrÞfor asymptotic dS with Λ0¼0.001 and a¼30

for increasing mass M¼1;3;5;7;9;11;13;15;17;19MPl from

top to bottom.

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106008-6

improvements of the metric lead to a unified global

structure for quantum-improved black hole spacetimes

based on solutions of the Einstein field equations. Yet,

as it is shown in Sec. VI C below, particles move differently

in each geometry.

We start by determining whether the singularity is

timelike, spacelike or null. To that end we compute the

norm of the normal vector of a hypersurface of constant rin

the limit r→0. The norm turns out to be the rr-element of

the inverse metric grr, yielding

grr

Sch ¼

r→01&grr

Kerr ⟶

r→01

cos2θ:ð12Þ

Hence, the singularity is timelike in both cases, irrespective

of whether the spacetime is asymptotically AdS or dS. As is

shown in Appendix B, zeros of fand Δrcorrespond to

Killing horizons. The succession of sign changes of the

lapse function dictates how the hypersurfaces of constant r

change from timelike over null to spacelike.

A. Asymptotically anti–de Sitter spacetimes

The lapse function of Schwarzschild–AdS and the

Kerr–AdS spacetime share the same qualitative features,

resulting in the same Penrose diagram. The formal

construction of the maximally extended spacetime works

the same as for the classical Kerr spacetime, for instance

see [22,25], but now with an asymptotic AdS-patch. For

a mass larger than the critical mass Mc, the lapse

function has two distinct roots, so the spacetime exhibits

two horizons; see Fig. 10.Whenm¼Mc, both roots

coincide and we find an extremal black hole with just

one horizon. For even lower masses, that is m<M

c,no

horizon is present, but the singularity still exists,

cf. Sec. VII, leaving a spacetime with a naked singu-

larity. Later, via a heuristic argument, we will argue that

this unphysical spacetime cannot be formed by gravi-

tational collapse.

B. Asymptotically de Sitter spacetimes

The results for the Schwarzschild- and Kerr-dS geom-

etries agree with each other. The spacetime exhibits two

distinguished masses, M<M, at which two of the possible

three horizons merge. Starting with M<m<M

, the

spacetime has three distinct horizons, two of them are

associated with the black hole and one with the positive

cosmological constant on large scales; see Fig. 11. This case

is equivalent to the classical Kerr-dS geometry. For m¼M,

the outer black hole horizon and the cosmological horizon

merge. This leaves an extremal spacetime containing a

maximally sized black hole, Fig. 12, similar to the Nariai

spacetime. For even larger masses, there is just one horizon

left; see Fig. 13. On the other end, the de Sitter spacetime

contains an extremal black hole if m¼M.Form<M

,we

have a de Sitter geometry containing singularity, which is

naked for observers within the cosmological horizon. The

construction of the maximally extended spacetime is analo-

gous to the one for the classical Kerr-dS case, described for

instance in [22].

C. Particle trajectories

In order to investigate whether particles propagate

differently in the quantum spacetimes as compared to

general relativity, we study their trajectories. Although

most new effects in quantum-improved spacetimes happen

around the Planck scale, there are possibly deviations from

classical trajectories already on length scales well above.

Our setup in the following is a test mass with zero angular

FIG. 8. fðrÞfor asymptotic dS for increasing Λ0¼0.0001,

0.0005, 0.0008, 0.001, 0.0015, 0.002, 0.003, 0.004, 0.005 from

top to bottom and fixed mass M¼5MPl.

FIG. 9. ΔrðrÞfor asymptotic dS for increasing Λ0¼0.0001,

0.0005, 0.0006, 0.0007, 0.0009, 0.0015, 0.002, 0.003 from top to

bottom. Fixed mass M¼5MPl and a¼30.

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momentum Lalong its (timelike) geodesic in a nonextre-

mal geometry, neglecting all backreactions. Furthermore,

we are allowed to restrict the motion to the equatorial plane,

see [26] for more details. In order to classify orbits into

categories, for instance orbits terminating at the central

curvature singularity or bound ones, it suffices to study

only the change of the radial coordinate.

1. Schwarzschild

In the quantum-improved Schwarzschild geometry, the

equation for the radial motion of a test mass, starting with

zero angular momentum Lat some distance rwith energy

E, reads according to (F5)

˙

r2¼E2−fðrÞ;ð13Þ

where ˙

rdenotes the change of the radial coordinate along

the geodesic parametrized by the eigentime. This equation

is only dependent on rand can be thought of as an energy

equation per unit mass for the total energy Eof the test

particle in an effective, one-dimensional potential fðrÞ.As

was already found in [4] for the asymptotically flat case,

possible trajectories are the same as in the classical

Reissner-Nordström scenario, thereby differing signifi-

cantly from a classical Schwarzschild setup. The only

difference to the asymptotical flat case arises at large scales,

where the effective potential fðrÞ→∞, depending on

whether the spacetime is asymptotically de Sitter or anti–de

Sitter. Recalling the shape of fðrÞ, e.g., Fig. 1, we note that

the effective potential is repulsive close to the singularity. In

an asymptotically AdS geometry and for a test mass with

energy E, the following options are possible, all being

bound orbits in radial direction:

(1) If Eequals the minimum of the lapse function fmin ,

then the particle is on a circular, stable orbit in

the region between the horizons. The radius is

FIG. 10. Penrose diagrams for quantum-improved Schwarzschild–and Kerr–AdS spacetimes. Hypersurfaces r¼const are drawn in

grey, each diagram can be further extended in vertical direction. To the left the Penrose diagram for the nonextremal black hole with

outer horizon Hoand inner horizon Hi, the timelike singularity (r¼0) and conformal infinity (r¼∞). In the middle the diagram for the

extremal geometry with just one horizon H. The black dots are not part of the singularity. To the right, the diagram for AdS with a naked

singularity at r¼0.

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determined by the distance where the repulsive

singularity balances the repulsive negative asymp-

totical cosmological constant.

(2) For fmin <E<0, the particle is on a bound orbit,

remaining in the region between both horizons.

(3) If 0<E<1, the orbit will again be bound, but now

the particle periodically crosses horizons. For in-

stance, first starting in the region outside of the outer

horizon, the trajectory will first cross the outer

horizon, then the inner one. Because it cannot

overcome the repulsive barrier of the singularity,

it is bounced back and the radius is increasing again.

By crossing another horizon, it will end up in an

identical patch of the extended spacetime. This

motion continues indefinitely and the particle will

travel through infinitely many universes. We will

comment on the physicality of this scenario at the

end of this section.

(4) If E>1, the energy of the particle can overcome the

potential barrier and manages to approach the

singularity at r¼0with nonzero kinetic energy.

But in contrast to the classical Schwarzschild–AdS

scenario, the particle again follows a path through

infinitely many identical universes, reaching the

singularity in each of them.

For the case of a nonextremal black hole with asymptotic de

Sitter patch, we note that the maximum fmax is always

smaller than one. Therefore, we find scenarios one and two

from above, but also some differences:

(5) The case 0<E<f

max admits a bound orbit,

equivalent to scenario three with the outer turning

point of the particle being located between the

cosmological and the outer black hole horizon, as

well as an unbound one beyond the cosmological

horizon.

(6) For E¼fmax, the particle is at rest at the

distance, where the attracting force of the black

hole balances the attraction generated by the

positive cosmological constant on large scales.

This is an unstable equilibrium, since small

perturbations cause the particle either to move

inwards in a similar way to five, or to escape to

infinity.

(7) In contrast to all above cases, the orbit is unbound in

radial direction for E>f

max, and the particle can

escape to infinity. Depending on whether or not

E≷1, it can reach the singularity at r¼0.

2. Kerr

The equation for the change of the radial coordinate

along the geodesic of a test particle with energy Eand zero

angular momentum Lin the equatorial plane of the Kerr

geometry reads (cf. (G4)),

FIG. 11. Penrose diagram for quantum-improved Schwarzschild- and Kerr-dS geometry with the three horizons of a nonextremal

black hole configuration. Starting at the timelike singularity at r¼0, we first cross the inner horizon Hiand then the outer horizon Ho

before crossing the cosmological horizon Hcand reaching conformal infinity r¼∞. This diagram can be further extended into all

directions. Again, r¼const hypersurfaces are depicted by grey curves.

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˙

r2¼RðrÞ≔E2Ξ2½ðr2þa2Þ2−a2Δr−r2Δr

r4;ð14Þ

where we introduced the function RðrÞfor convenience.

For a fixed geometry ðG0;Λ0;M;aÞ, the energy Eof the

particle determines the allowed orbits. In the following, we

continue closely along the more detailed analysis of the

classical Kerr–(A)dS geometry carried out in [26]. Since

the above equation is quadratic in ˙

r, geodesics always have

to satisfy RðrÞ≥0. A simple root of RðrÞcorresponds to a

turning point, where the particle comes to rest. A circular

orbit of constant r¼r0requires both ˙

rand ̈rto vanish at r0,

translating via Eq. (14) into the condition of RðrÞhaving an

extremum as well as a zero at r0. Depending on whether

this extremum is a maximum or minimum, the circular orbit

will be stable or unstable. Hence, RðrÞhaving at least a

double zero at r0is a sufficient condition for a circular orbit.

The function RðrÞfor Kerr–AdS is displayed in Fig. 14.

At large radii, the repulsiveness of the effective AdS

spacetime prevents particles from escaping to infinity.

There exists a special energy E0, above which observers

inevitably fall into the singularity along a terminating orbit.

For E¼E0, three types of orbits are possible. RðrÞexhibits

a double zero at r0, allowing for an unstable, circular orbit.

For radii larger than r0, we find a bound orbit, crossing both

horizons. Particles starting at r<r

0are accelerated along

terminating trajectories and will end up in the singularity.

However, if E<E

0, the double root splits and we find the

possibility of having bound orbits as well as terminating

ones at radii below the inner horizon. For the smallest

energies, E→0, the particle moves from horizon to

horizon. The only difference for Kerr-dS compared to

the AdS case, is that particles can always escape to infinity;

see Fig. 15.

The trajectories have been calculated for an idealized,

pointlike observer, neglecting any backreaction on the

geometry. However, the location of the inner horizon is

typically at about the Planck scale, where backreaction

effects should be taken into account. The quantum-

improved Schwarzschild case turns out to be similar to

FIG. 12. Penrose diagram for quantum-improved Schwarzs-

child- and Kerr-dS geometry with the two horizons of an extremal

black hole configuration. Starting at the curvature singularity at

r¼0, we first cross the inner horizon Hiand then the outer one

Ho, before arriving at conformal infinity r¼∞. This diagram

can be further extended to the top and bottom as well. The black

dots are not part of the singularity. The displayed pattern of the

r¼const hypersurfaces is the one for m¼M. For m¼M, the

hypersurfaces between the horizons become spacelike.

FIG. 13. Penrose diagram for the quantum-improved Schwarzs-

child–and Kerr–AdS configuration showing only one horizon H,

always shielding the singularity at r¼0from an observer near

conformal infinity r¼∞.

FIG. 14. RðrÞfrom (14) for Kerr–AdS with G0¼1,

Λ0¼−0.1,M¼10MPl,a¼1and increasing particle energy

Efrom bottom to top.

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the classical Reissner-Nordström spacetime, for which it

was shown that there is a blueshift instability at the inner

(Cauchy) horizon. Additionally, it was shown in [27], that

perturbations of initial data cause the Cauchy horizon to be

replaced by a null singularity. Due to the similarities

between the quantum-improved Schwarzschild and the

classical Reissner-Nordström spacetime, it is tempting to

speculate that the classical findings might also hold for the

quantum case. Hence, one has to take the above results with

care, especially the many world trajectories. Summarizing,

there are differences between the classical and the quan-

tum-improved geometry, but they only become relevant at

very small length scales, where the results have to be taken

with a grain of salt.

VII. CURVATURE SINGULARITY & EFFECTIVE

ENERGY-MOMENTUM TENSOR

Since quantum gravity effects become important in high

curvature regimes, it is expected that they alter the nature of

the curvature singularity at r¼0. Previous results from

asymptotic safe quantum gravity [6–8] and other quantum

gravity scenarios, e.g., [28], predict a substantial weaken-

ing of the singularity or even its disappearance. A

weakening of the singularity manifests itself for instance

in changes of the Kretschmann scalar. We compute the

Ricci scalar Ras well as the Kretschmann scalar Kof the

quantum-improved geometries in the UV fixed point

regime, and compare the findings with the classical result

of general relativity. Table Ilists the highest degree of

divergence of the Ricci and Kretschmann scalar for both

investigated geometries for all discussed matchings. Upon

comparison with the classical result of general relativity, the

consistent quantum scenarios display a weakening of the

singularity but not a complete resolution.

In the quantum-improved spacetimes, the Ricci scalar is

diverging too, because we have changed the geometry

which is not a vacuum solution of the Einstein field

equations anymore. In fact, it is a geometry with an

effective energy-momentum tensor [29], induced by the

running couplings. Using the classical field equations, this

effective energy-momentum tensor Teff

μν can be computed

by calculating the Einstein tensor Gμν from the quantum-

improved metric,

Gμν þΛ0gμν ≕8πG0Teff

μν :ð15Þ

Note that Teff

μν is covariantly conserved, assuming a metric

connection, ∇μgμν ¼0, because the Einstein tensor sat-

isfies the Bianchi identity ∇μGμν ¼0by construction.

However, physical interpretations of this effective

energy-momentum tensor in terms of matter have to be

drawn with great care. For instance, it turns out that the Teff

rr

is diverging at horizons, fðrÞ¼0, because Grr ¼f−1þrf0

fr2

and grr ¼1=fðrÞ. Additionally, it has been shown in [5],

that Teff

μν in the quantum-improved flat Kerr geometry

violates the weak, the null, the strong and the dominant

energy condition. We expect similar results in the present

case, including the cosmological constant. These observa-

tions suggest that quantum gravity contributions to the

energy-momentum tensor are of a fundamentally different

nature than the ones of conventional matter and should not

be interpreted as matter. In fact, the running couplings

should be taken into account already on the action level,

resulting in different field equations. This is done, for

example, in Quantum Einstein Gravity (QEG) [16], based

on the quantum-improved Einstein Hilbert action

TABLE I. Ricci scalar Rand Kretschmann scalar Kfor Schwarzschild–and Kerr–AdS for different matchings compared to the

classical result.

Classical cl. Kretschmann qu. Kretschmann Linear cl. radial path qu. radial path cl. geodesic qu. geodesic

RSch 4Λ0∼const ∼r−3=2∼r−2∼const ∼r−2∼const ∼r−3=2

KSch ∼r−6∼r−6∼r−3∼r−4∼r−6∼r−4∼r−6∼r−3

RKerr 4Λ0∼r−3∼r−2∼r−4∼r−4∼r−4

KKerr ∼r−6∼r−6∼r−4∼r−8∼r−8∼r−8

FIG. 15. RðrÞfrom (14) for Kerr-dS with G0¼1,Λ0¼0.01,

M¼10MPl,a¼1and increasing particle energy Efrom bottom

to top.

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S¼Zd4xﬃﬃﬃﬃﬃﬃ

−g

pR−2ΛðrÞ

16πGðrÞ:ð16Þ

The resulting new field equations [30], based on the

runnings (3), read the same as (15) with

8πG0Teff

μν ¼−λk2ðrÞgμν þGðrÞð∇μ∇ν−gμν□Þ1

GðrÞ:ð17Þ

It has been shown in [31], that the covariant conservation

of the effective energy-momentum tensor in QEG is

equivalent to the following relation between the running

couplings,

R∇μ1

GðrÞ−2∇μΛðrÞ

GðrÞ¼0:ð18Þ

This relation is not satisfied by our quantum-improved

Schwarzschild–(A)dS and Kerr–(A)dS metrics, meaning

that they are not solutions to the new field equations (15)

with (18), derived in the Einstein-Hilbert truncation of a

potentially more complicated fundamental action.

VIII. HORIZON TEMPERATURES AND

BLACK HOLE EVAPORATION

In this section, we first establish the fact, that surface

gravities in spacetimes based on the quantum-improved

version of the radial path proper distance are divergent,

before discussing the Hawking temperatures in the

Kretschmann scenario. Finally, we will discuss implications

on the black hole evaporation process.

The Hawking temperature of a black hole in flat space

received by an observer at infinity is given by TH¼κ

2π[32],

with surface gravity κof the event horizon. For an observer

at finite distance rin the static region outside the black

hole, the above expression is modified by a redshift factor,

TH¼κ

2π

1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

gðK; KÞ

p;ð19Þ

where gðK; KÞis the norm of the static Killing vector K.In

more general terms, a surface gravity can be assigned to any

Killing horizon of a spacetime. Gibbons and Hawking

showed in [22], that cosmological horizons also emit

radiation which can be detected by an observer in the static

spacetime region. In general, emission is a consequence of

the observer not being able to access the spacetime hidden

behind the horizon(s), thereby being fundamentally unable

to measure the quantum state of the complete universe (see

[22] for a more detailed discussion). The notion of a horizon

temperature only appears to be meaningful for observers in

a static spacetime region, since only such observers detect

radiation of this temperature. Taking Reissner-Norström as

example, this is only the case for the region outside the

black hole. In between the horizons, the spacetime is not

static anymore and inside the inner horizon, the spacetime is

static again, but connected to the singularity. This would

require to impose boundary conditions at the singularity,

being far from obvious. Hence in the following, we only

refer to a horizon having a temperature, if the horizon is the

boundary of a static region, not connected to the singularity.

In Appendix B, horizons in the quantum-improved space-

time are shown to be Killing horizons, thus a surface gravity

can be assigned to each of them.

Technically, the surface gravity κof a Killing horizon can

be computed by taking the covariant derivative of the norm

of the Killing vector, or alternatively via a periodicity in

Euclidean time introduced in [33]. In any case, we find

κSch ¼1

2jf0ðr0Þj &κKerr ¼1

2jΔ0

rðr0Þj

ðr2

0þa2Þ;ð20Þ

r0being the radial coordinate of the horizon. Since

horizons are zeros of fðrÞand ΔrðrÞ, respectively, (C4)

implies that the derivative of the proper distance D0ðrÞ

diverges at the horizons for the quantum version of the radial

path. As addressed in Appendix Din detail, this does not

necessarily mean that the proper distance itself is diverging

at a horizon, unless the horizon is extremal. But computing

the surface gravity explicitly via (20) generates the following

terms, proportional to D0ðrÞ, and therefore diverging at the

horizons,

f0ðrÞ∼2

3r2−6G2

0gλMrDðrÞ

ðgλD2ðrÞþG0Þ2þr4

D3ðrÞD0ðrÞ;

Δ0ðrÞ∼2r2ða2þr2Þ

3D3ðrÞ−4G2

0gλMrDðrÞ

ðgλD2ðrÞþG0Þ2D0ðrÞ:ð21Þ

The terms in the brackets are in general nonvanishing at the

horizons. In particular, this holds also for arbitrary large

masses in the classical regime, where it is known that the

surface gravity and Hawking temperature stays finite. This is

the main reason why we consider the scale identification

based on the quantum radial path as unphysical. In contrast,

along with the proper distance based on a geodesic, the

construction based on the Kretschmann scalar shows no

divergent behavior at the horizons and therefore leads to

finite Hawking temperatures.

Next, we discuss the mass dependence of the surface

gravities, focusing on the quantum Kretschmann scenario

from now on. It suffices to look at the slope of the lapse

function at each horizon, since it is proportional to the

surface gravity. The results for Schwarzschild–AdS and

Schwarzschild-dS can be found in Figs. 16 and 17, the plots

for the Kerr cases are qualitatively the same. The whole

evolution, appearance and disappearance of horizons is

driven by the formation of a minimum of the lapse function.

The quantum-improved Schwarzschild–AdS spacetime

exhibits no horizon up to the critical mass Mc≈1.2MPl.

At M¼Mcrit, the minimum of the lapse function is at zero,

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hence the slope is zero and so are the surface gravities. For

growing mass, the slope becomes steeper because the

minimum expands, thus the surface gravities grow in

amplitude. In contrast, κcl in general relativity diverges

for M→0. However, the surface gravity of the outer horizon

matches the classical one for sufficiently large masses. The

Schwarzschild-dS scenario can have up to three horizons

and two special masses, M≈2MPl and M≈5.8MPl,at

which two of the three horizons merge. Starting in the M<

Mregime, there is no black hole, but only the cosmological

horizon. The case M¼Mcorresponds to the case M¼Mc

from above. For M<M<M

, there are three horizons

and the back hole gets bigger for increasing mass, until

M¼M, when the black hole has reached its maximal size

and its outer horizons merges with the cosmological horizon

to an extremal horizon with zero temperature.

In AdS spacetimes, an observer in the static region could

only measure a temperature coming from the black holes’

event horizon, whereas in dS spacetimes, the observer

would measure a mixture of two thermal spectra at different

temperatures, one coming from the back hole and one from

the cosmological horizon. In the static region outside the

black hole, one valid choice for the Killing vector in (19) is

K¼∂=∂t, yielding gðK; KÞ¼gtt. In the Schwarzschild

geometries, this implies that an observer located at a

horizon would measure an infinite temperature, in accor-

dance with general relativity. In Schwarzschild–AdS, the

temperature drops to zero for an infinitely distant observer,

as gtt diverges.

In the dS-scenario, there exists a distance between the

horizons, at which the observed temperature becomes

minimal, s because fhas a maximum. In the Kerr geom-

etries, ∂=∂tis timelike only outside the ergoregion. A static

Killing vector field for the entire region outside the black

hole can be obtained by linearly combining the two Killing

vectors of a Kerr spacetime; see Appendix B. Since all above

observations equally apply for classical as well as quantum

improved spacetimes, the is no qualitative difference for an

observer measuring horizon temperatures in a classical or a

quantum spacetime, except in the Planckian regime.

As final point, we would like to address the black hole

evaporation process. A standard mechanism to form black

holes is gravitational collapse. If the mass of a collapsing

object is larger than the Tolman-Oppenheimer-Volkoff

mass around 2M⊙, no other force can counterbalance

gravity and the object collapses to form a black hole.

Assuming that a macroscopic Schwarzschild or Kerr black

hole has formed via this process, well above the critical

mass, it will emit Hawking radiation and thereby lose

energy. This causes the black hole to shrink steadily, as its

mass is decreasing. This process continues, until the critical

mass Mcrit is reached. The temperature then becomes zero

and therefore the radiation stops. Hence, the naked singu-

larity case with M<M

crit can never be reached via this

process and we end up with a zero temperature, Planck-

sized, extremal black hole, often referred to as remnant.

This remnant serves as shield, guaranteeing that the cosmic

censorship conjecture remains satisfied. However, in [34] it

was shown that extremal black hole configurations with

zero temperature suffer from an instability at the extremal

horizon. Remnant end points were also found in other

studies within asymptotic safety [7,10] and beyond [35].

Based on a classical expression for the proper distance

it has been shown in [6,8] that the Schwarzschild–AdS

black hole evaporates completely. A more suitable setup to

FIG. 16. f0ðrÞas function of the mass Mfor the quantum -

improved Schwarzschild–AdS geometry for Λ0¼−0.1. Inner

horizon in blue, outer horizon in green. The outer horizon agrees

with the temperature of the event horizon in general relativity in

red for large masses. Taking absolute values yields the surface

gravities.

FIG. 17. f0ðrÞas function of the mass Mfor the quantum -

improved Schwarzschild-dS geometry for Λ0¼0.001. The

cosmological horizon in red, the inner black hole horizon in

blue and the outer black hole horizon in green. Taking absolute

values yields the surface gravities.

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discuss the evaporation process is given by the dynamical

Vaidya spacetime, used in [10]. There, a Planck-sized, cold

remnant as an end point has been found.

IX. SUMMARY

In this work, the quantum-improved Kerr–(A)dS black

hole was studied for the first time within a self-consistent

scale identification procedure. The latter is based on the

Kretschmann scalar. The Kerr–(A)dS geometry also

includes the Schwarzschild–(A)dS, as well as ordinary

Schwarzschild and Kerr spacetimes as special cases, by

setting either the rotation parameter aor the cosmological

constant Λ0to zero.

Both quantum-improved geometries show the same

global structure in terms of a timelike curvature singularity

at r¼0and the same number of horizons. Furthermore, it

has also been shown that the outer black hole horizon

corresponds to the classical black hole event horizon. The

timelike character of the singularity at r¼0in principle

allows particles to avoid the singularity. The quantum

corrections to the classical metric render the singularity

less divergent, but none of the studied scenarios was able to

resolve it completely. However, this singularity will always

be dressed by a horizon, such that there is no violation of

the cosmic censorship conjecture.

The horizons being Killing horizons admit a temper-

ature, causing the black hole to evaporate. In the Planckian

regime, however, the heat capacity of a tiny black hole stays

positive, ∂T

∂M>0, in contrast to the classical case. Thus, the

evaporation process comes to an end when the Hawking

temperature of the black hole is zero, leaving an extremal,

cold, Planck-sized remnant, serving as cosmic censor. This

is a thermodynamically stable end point because any

additional mass absorbed by the black hole will radiate

away until the temperature is again zero. It would be

interesting to see what implications for the black hole

information paradox can be drawn from the generic

existence of such remnants.

ACKNOWLEDGMENTS

We thank Alfio Bonanno, Kevin Falls, Domenico

Giulini, and Alessia Platania for discussions. This work

is supported by ExtreMe Matter Institute (EMMI) and is

part of and supported by the DFG Collaborative Research

Centre “SFB 1225 (ISOQUANT)”and also by the DFG

Research Training Group “Models of Gravity.”

APPENDIX A: CHOICE OF SCALE

IDENTIFICATION

Here we motivate our choice for kðrÞin (5). Inserting the

general parametrization kðrÞ¼ξ=DðrÞ, into (C5), we are

left with

fðrÞ¼1−2M

r

gðrÞD2ðrÞ

ξ2−r2

3

λðrÞξ2

D2ðrÞ

≈

UV

r→0

1−2M

r

gD2ðrÞ

ξ2−r2

3

λξ2

D2ðrÞ;

Δr≈

UV

r→0ðr2þa2Þ1−r2

3

λξ2

D2ðrÞ−2M

r

gD2ðrÞ

ξ2:ðA1Þ

The numerical values of gand λdepend on the particular

RG-trajectory and parametrization we have chosen and

therefore cannot be physical observables. However, the

product gλis an observable and hence independent of

the particular choice of the RG-trajectory. Its magnitude

turns out to be gλ≈0.1, e.g., in [16,20]. In this light, we

have two choices for ξin order to make (A1) solely

dependent on gλ,

ξ2¼gor ξ2¼1

λ

:ðA2Þ

Thus, in (5) we have chosen the second of the two

equivalent options. Varying ξfor a fixed geometry (G0,

Λ0,m,a), which is effectively done also in the quantum

Kretschmann scenario by introducing χ, turns out to have

only a weak impact on the position of the inner horizon.

Since it is typically located at small radii, we recall from

Table II, that varying ξmildly modifies the UV-limit.

Furthermore, we have an upper limit χ<ð3=8Þ1=4.

APPENDIX B: KILLING HORIZONS

In this section, we review the formal proof that every

zero of ΔrðrÞin (7) is a Killing horizon. This implies that a

constant surface gravity and thereby a temperature can be

associated to each horizon. The Schwarzschild–(A)dS case

is automatically contained by taking a→0.

TABLE II. UV-limits (r→0)ofDðrÞfor all investigated matchings.

Kretschmann Radial path Geodesic path

Classic Quantum Classic Quantum Classic Quantum

Schwarzschild

1

31=42χﬃﬃﬃﬃﬃﬃﬃ

MG0

pr3=2χ−4−8=3

48M2ðgλÞ21=8r3=4

2

3ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2G0M

pr3=22

ﬃﬃ3

prπ

2ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2G0M

pr3=2ð67

18MgλÞ1=4r3=4

Kerr r2

2a0π

4ar20

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Starting from the Kerr–(A)dS metric (7), assume that

ΔrðrÞhas jpositive roots, i.e., can be written as

ΔrðrÞ¼Y

j

i¼0ðr−riÞwith 0≤r0≤r1<…≤rj:ðB1Þ

The horizons are the hypersurfaces r¼ri¼const. Since

the spacetime is axisymmetric and stationary, we have two

commuting Killing vector fields: ð∂

∂tÞais stationary, at least

in some region of the spacetime, and ð∂

∂ϕÞamanifests the

symmetry axis. We now have to construct a Killing vector

field ξa, that is normal to, and null on these horizon

hypersurfaces. The most general form for ξawould be a

linear combination of both Killing vector fields,

ξa¼∂

∂ta

þα∂

∂ϕa

;ðB2Þ

with a constant α. We will fix this constant later by

requiring that ξashould vanish at the horizons. But first,

we must change from Boyer-Lindquist coordinates (7),to

coordinates that leave the metric regular at the horizons.

Such coordinates are induced by the principal null direc-

tions of the spacetime. The Kerr–(A)dS spacetime is of

algebraic type D, thus admits two distinct principal null

directions, referred to as ingoing and outgoing. They can be

represented in Boyer-Lindquist coordinates by the follow-

ing vectors,

nμ

¼r2þa2

Δr

Ξ;1;0;a

Δr

Ξ;ðB3Þ

where þ1is outgoing and −1ingoing. They now induce

outgoing and ingoing coordinates, being the Kerr–(A)dS

counterparts of Kerr-coordinates in flat space. We will

select the outgoing version, but in principle we could also

work with ingoing ones. The outgoing Kerr–(A)dS coor-

dinates ðv; χÞare defined as,

dv¼dtþΞr2þa2

Δr

dr

dχ¼dϕþΞa

Δr

dr: ðB4Þ

Inserting these back into (7), leaves us with the metric in

terms of Kerr–(A)dS coordinates ðv; r; θ;χÞ,

ds2¼−1

ρ2Ξ2ðΔr−Δθa2sin2θÞdv2þ2

Ξdvdr

−2asin2θ

ρ2Ξ2ððr2þa2ÞΔθ−ΔrÞdvdχ−2asin2θ

Ξdχdr

þsin2θ

ρ2Ξ2Δθðr2þa2Þ2−Δra2sin2θdχ2þρ2

Δθ

dθ2:

ðB5Þ

One can check that (B5) reduces to Kerr coordinates for

Λ¼0. The Killing vector field ξanow reads

ξa¼∂

∂va

þα∂

∂χa

:ðB6Þ

Requiring that ξais null on the horizons r¼riyields

ξ2jr¼ri¼½gvv þ2αgvχþα2gχχ r¼ri

¼Δθsin2θ

ρ2

iΞ2½a−αðr2

iþa2Þ2¼

!0ðB7Þ

and therefore

FIG. 18. Running of the dimensionless couplings gand λas a function of momentum scale kfor the analytical expressions from (3) in

green and from a fourth-order vertex expansion based on [20] in blue. Both approach their UV fixed point values, g¼1.4&λ¼0.1,

for k→∞.

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α¼a

r2

iþa2:ðB8Þ

Thus, we have found a family of vector fields ðξaÞi, being

null at one horizon at a time. In order to show that the

hypersurfaces r¼riare Killing horizons, it remains to be

checked if ξais hypersurface orthogonal, i.e., ξa¼ξμdxμ∼

drevaluated at the horizon,

ðξÞajr¼ri¼½gμν ξνdxμr¼ri¼1

Ξ1−a2sin2θ

r2

iþa2dr; ðB9Þ

with all other components vanishing. In summary, we are

able to construct a Killing vector field ξawhich is null on,

and normal to each horizon hypersurface r¼ri, and hence

have shown that the horizons corresponding to the roots of

Δrare indeed Killing horizons.

APPENDIX C: OTHER MATCHINGS

1. Linear matching

The simplest scaling is based on a dimensional analysis,

DLinðrÞ¼r; ðC1Þ

which has already been adopted for instance in [4]. In the

case of an identically vanishing cosmological coupling, is

the IR-limit of the classical proper distance along a radial

path [7]. But this matching does not take physical scales of

the underlying spacetime into account, for instance the

black hole scales given by M&a, or scales induced by the

gravitational or the cosmological coupling. Nevertheless,

this function already gives rise to many phenomena

observed for more complicated choices and hence can

serve as a toy model. The results for the lapse functions

based on the linear matching can be found in Figs. 19

and 20.

2. Proper Distances

We can also use the proper distance along a curve Cin

spacetime to specify DðrÞ,

DðrÞ¼Dprop ¼ZCﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jgμνdxμdxνj

q:ðC2Þ

This definition is diffeomorphism invariant and encodes the

spacetime structure, since the gravitational and cosmologi-

cal coupling typically appear in the metric. In most cases in

the literature, e.g., [6–8], the gravitational as well as

cosmological coupling have been fixed to be constants,

for instance the IR-values Λ0and G0. However, since the

FRG-flow generically gives rise to running couplings, it is

more natural and consequent to consider this running also

in the above integral, thus G→GðrÞand Λ→ΛðrÞ. In the

following, this quantum improvement procedure of proper

distances is extended to Schwarzschild–and Kerr–(A)dS

geometries. We will provide expressions for the proper

distance along a radial path and along the geodesic of a

radially infalling observer, both for constant, as well as

running Gand Λ. Additionally, the UV-limit of each proper

distance is obtained, cf. Table II.

a. Radial path

Inspired by the symmetry of the spacetime, we first take

the following radial path from 0 to ras integration contour

Cin (C2),

CSchw-ðAÞdS∶dt¼dΩ¼0;

CKerr-ðAÞds∶dt¼dϕ¼dθ¼0and θ¼π=2:ðC3Þ

The restriction to the equatorial plane in the Kerr case is

done for the sake of simplicity. Driven by the results of [5]

FIG. 19. fðrÞfrom (6) based on the linear matching for

increasing mass from top to bottom, with Λ0¼−0.1,

M¼0.1;1;2;3;4;5;6;7;8;9;10MPl.

FIG. 20. ΔrðrÞfrom (8) based on the linear matching for

increasing mass from top to bottom, with Λ0¼−0.1,a¼2and

M¼0.1;1;2;3;4;5;6;7;8;9;10MPl.

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106008-16

for the flat Kerr geometry, we assume that the varying θwill

not alter our results qualitatively. Applying the above

integration paths to (C2) yields,

DSchðrÞ¼Zr

0

d˜

rﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jg˜

r˜

rj

p¼Zr

0

d˜

r1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jfð˜

rÞj

p;

DKerrðrÞ¼Zr

0

d˜

rﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jg˜r˜rj

p¼Zr

0

d˜

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

˜

r2

jΔrð˜

rÞj

s;ðC4Þ

with the lapse functions

fðrÞ¼1−2GM

r−

Λ

3r2and

ΔrðrÞ¼ðr2þa2Þ1−

Λ

3r2−2MGr: ðC5Þ

In the following, this scenario with constant Gand Λwill

be referred to as the classical radial path because the

spacetime underlying the integral is a classical black hole

geometry with a cosmological constant.

Alternatively, we account for the running of the cou-

plings already in the proper distance, referred to as the

quantum radial path, with G¼GðrÞand Λ¼ΛðrÞin the

above integrals. This turns (C4) into integral equations for

DðrÞ, which can be transformed into a differential equation

by taking a derivative with respect to r. One can then easily

see that the derivative of DðrÞdiverges at every horizon,

where fðrÞand ΔðrÞvanish. Using the fixed point behavior

of Gand Λin the UV, these differential equations read for

small r,

D0

sch;quðrÞ¼ 1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

j1−2Mgλ

D2

sch;quðrÞ

r−r2

3D2

sch;quðrÞj

r;

D0

kerr;quðrÞ¼ 1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

j1þa2

r2−r2

3D2

kerr;quðrÞ−a2

3D2

kerr;quðrÞj

q:ðC6Þ

Both classical matchings as well as the one for the quantum

Schwarzschild scenario monotonously increase and satisfy

Dðr→0Þ¼0, as can be seen from the numerical results in

Fig. 23. In contrast, the proper distance is identically zero in

the quantum Kerr scenario, see (D2) Therefore, we only

show the results for the Schwarzschild-AdS geometry

in Fig. 21.

It turns out (cf. Sec. VIII) that the expression for the

Hawking temperature in a quantum-improved spacetime

contains terms proportional to the derivative of DðrÞ, hence

using the above construction for the proper distance leads

to diverging Hawking temperatures at all horizons.

Therefore, in the following we also discuss the proper

distance induced by the eigentime of a radially infalling

observer, where this feature is absent.

b. Radial timelike geodesic

The eigentime τof an observer, initially at rest at Rand

falling along a radial timelike geodesic into the singularity,

can also be used to identify the momentum cutoff scale

with a length scale by setting DðrÞ¼τðrÞ. Derived in

Appendix F, the eigentime for the Schwarzschild–(A)dS

scenario reads

FIG. 22. fðrÞbased on the radial geodesic matching for

increasing mass from top to bottom. Results, where DðrÞis

computed consistently in a quantum-improved spacetime, are

shown in solid, the dashed curves are the ones with a classically

computed DðrÞ. With parameters Λ0¼−0.1and M¼

0.1;1;2;3;4;5;6;7;8;9;10MPl. Curves of the same mass have

the same color.

FIG. 21. fðrÞbased on the radial path matching for increasing

mass from top to bottom. Results, where DðrÞis computed

consistently in a quantum-improved spacetime, are shown in

solid, the dashed curves are the ones with a classically computed

DðrÞ. With parameters Λ0¼−0.1and M¼0.1;1;2;3;4;5;6;

7;8;9;10MPl. Curves of the same mass have the same color.

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106008-17

DðRÞ¼ZR

0

dr1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jE2−fðrÞj

p;ðC7Þ

with E¼fðRÞfor an observer initially starting at rest. It is

worth noting that for E¼0, the integral reduces to the one

in (C4). By fixing E, we equivalently specify the maximal

distance Rof the observer from the origin. Independent on

the particular value of E, the proper distance again exhibits

poles if E2−fðrÞ¼0, now shifted by E2away from the

horizons. Once more, (C7) gives rise to two different proper

distances, referred to as either classical or quantum

geodesic, depending on whether the underlying spacetime

is based on the constant or running versions of Gand Λ.

The lapse function for Schwarzschild-AdS based on the

classical and quantum radial geodesic are displayed

in Fig. 22.

The analogous expression for the proper distance

induced by an radial geodesic in the Kerr–(A)dS scenario

reads (see Appendix G)

DðRÞ¼ZR

0

drr2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jE2Ξ2½ðr2þa2Þ2−a2Δr−r2Δrj

p;

E2¼E2ðRÞ¼ R2ΔR

Ξ2½ðR2þa2Þ2−a2ΔR;ðC8Þ

and reduces to (C4) for E¼0. Again, we achieved that

there are no poles at the horizons. Once more, we have two

versions depending on whether Gand Λare running or not.

The numerical results can be found in Fig. 24; however, the

proper distance in the quantum Kerr scenario is again

identically zero.

FIG. 23. Left: proper distance along a radial path through a classical Schwarzschild–AdS spacetime for three different masses

M¼1;5;10MPl. Right: the same for a quantum Schwarzschild–AdS spacetime.

FIG. 24. Left: proper distance along a radial geodesic through a classical Schwarzschild–AdS spacetime for three different masses

M¼1;5;10MPl. Right: the same for a quantum Schwarzschild–AdS spacetime.

JAN M. PAWLOWSKI and DENNIS STOCK PHYS. REV. D 98, 106008 (2018)

106008-18

APPENDIX D: SHAPE AND DIVERGENCES OF

PROPER DISTANCES

As can be seen from Fig. 23–25, all functions DðrÞare

monotonously increasing, some proper distances display a

rapid increase. In order to understand these jumps and

possible divergences, we have to look at the integral

expressions for each proper distance (C4),(C7), and

(C8). The expression hðrÞunder each square root can

become zero, and if hðrÞhas just a single root at

r¼r0<R, the corresponding pole is integrable, causing

a jump in the proper distance,

DðRÞ¼ZR

0

dr1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

hðrÞ

p¼ZR

0

dr1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðr−r0Þ

˜

hðrÞ

q

∼ZR

0

drðr−r0Þ−1=2;ðD1Þ

where

˜

hðrÞhas no root at r¼r0. However, once the

multiplicity of r0is larger than one, the pole is not integrable

anymore and DðrÞexhibits a divergence at r¼r0.Inany

case, D0ðrÞis diverging for the radial path proper distances,

even at integrable poles of DðrÞ, as can be seen from (C6).In

case of the classical radial path, the position of these poles

has no direct physical significance; however, in the quantum

case, the poles are located precisely at the horizons, because

then, the function hðrÞis nothing other than the horizon

condition. Thus, for extremal black holes when at least two

horizons coincide, the quantum proper distance along a

radial path is ill defined. D0ðrÞis always diverging at the

horizons leading to a diverging Hawking temperature of the

horizon, as is shown in Sec. VIII.

For this reason, we introduce the scenario with an

infalling observer along a timelike, radial geodesic, in

order to remove these problems, only due to the poor choice

of the function hðrÞand absent in all other scenarios.

However, it turns out, that in both proper distance scenarios

for Kerr–(A)dS with an underlying quantum spacetime, the

proper distance must vanish identically, in order to satisfy

the condition Dð0Þ¼0. For instance, this can be seen by

solving (C6) in the limit r→0, satisfying the boundary

condition Dð0Þ¼ϵ, yielding

Dkerr

rad;UVðrÞ¼ ϵ

aﬃﬃ3

pðﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

r2þa2

pþrÞﬃﬃ3

p:ðD2Þ

Therefore, the solution vanishes identically in the limit

ϵ→0, which is confirmed also for the full, numerical

solution of (C4). The same behavior is found for Kerr–(A)

dS, when the scale matching is based on the geodesic in a

quantum-improved spacetime.

APPENDIX E: UV LIMITS OF D(r)

For statements about the curvature near the singularity

and also for the construction of the Penrose diagrams, the

UV-limit for each proper distance is needed.

The leading-order behavior in the UV for the classical

proper distances, i.e., constant G0and Λ0, can be obtained

from (C4),(C7) and (C8) by approximating the integral in

the limit r→0. For the identification based on the classical

Kretschmann scalar (10), the UV behavior can easily be

read off from (11).

In the quantum versions, the leading order of the

proper distance in the UV-limit can be obtained by

assuming a power law behavior of the form DðrÞ¼Arα,

with constants A>0and α>0in order to satisfy the

boundary condition Dð0Þ¼0. The constants Aand αcan

be determined by inserting this ansatz back into the above

equations, now being an integral, differential or functional

equation, respectively. All scenarios display monotonously

increasing functions satisfying Dð0Þ¼0, apart from the

quantum proper distance expressions for Kerr. They are

FIG. 25. Function DðrÞin classical Kretschmann matching (left) and for quantum Kretschmann scenario (right) for three different

masses M¼1;5;10MPl.

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identically zero, as an iterative algorithm for solving the

integral equations shows.

For each scenario, the analytical UV-expression is listed

in Table II. The numerical results for DðrÞare shown in

Fig. 23–25. Furthermore, the leading-order exponent αcan

be extracted numerically from the slope of the linear

relation between the proper distance DðrÞ¼Arαand its

integral function DðrÞ¼ A

αþ1rαþ1:

DðrÞ

DðrÞ¼r

αþ1:ðE1Þ

This cross-check confirms agreement between numerical

exponent and the one found analytically in Table II.

APPENDIX F: EIGENTIME OF AN INFALLING

OBSERVER IN A SCHWARZSCHILD–(A)DS

GEOMETRY

Another physically well-motivated choice for the inte-

gration path in (C2) is the curve determined by an observer

some distance away from the black hole, falling into the

black hole along a radial timelike geodesic. Because the

observer’s four-velocity uais conserved along geodesics,

we normalize it to be

−1¼

!uaua:ðF1Þ

Furthermore, we can choose the coordinate system such

that the motion takes place only in the equatorial plane

θ¼π=2. Using (6), the normalization condition of the

four-velocity in the equatorial plane reads:

−fðrÞ˙

t2þ

˙

r2

fðrÞþr2˙

ϕ2¼−1;ðF2Þ

where

˙

ðÞ denotes the derivative with respect to the

eigentime τ. We have also conserved quantities Eand L

corresponding to the Killing vector fields ξa¼ð∂

∂tÞaand

ψa¼ð∂

∂ϕÞa:

E¼−gabξaub¼fðrÞ˙

t; ðF3Þ

L¼gabψaub¼r2˙

ϕ:ðF4Þ

However, for simplicity, we will choose an observer with

L¼0. Inserting Eand Lback into (F2) to eliminate ˙

tand

˙

ϕ

leaves us with

E2¼˙

r2þfðrÞ:ðF5Þ

This is a type of energy equation for the observer, at least in

asymptotically flat spacetimes. We now have to specify the

initial conditions for the observer. In the asymptotically flat

spacetime, one usually places the observer initially at rest at

r¼∞, still leaving Efinite. However, we cannot do that in

the case of a nonvanishing cosmological constant, because

fðrÞis diverging for r→∞. Therefore, we take rather an

observer at rest ( ˙

r2¼0) at some finite distance Rto

determine E:

E2¼fðRÞ:ðF6Þ

The proper distance is then given by the eigentime the

observer needs to arrive at r¼0after starting at R, i.e., the

integral over the eigentime along the geodesic:

DðRÞ¼ZR

0

dr1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jE2−fðrÞj

p¼ZR

0

dr1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jfðRÞ−fðrÞj

p:

ðF7Þ

APPENDIX G: EIGENTIME OF AN INFALLING

OBSERVER IN A KERR–(A)DS GEOMETRY

Following the same procedure for a timelike geodesic in

the equatorial plane in Kerr–(A)dS, given by the metric (7),

the normalization of the four-velocity is

−1¼gtt

˙

t2þgϕϕ

˙

ϕ2þ2gtϕ

˙

t

˙

ϕþgrr

˙

r2;ðG1Þ

whereas the conserved quantities induced by the Killing

vector fields ξa¼ð∂

∂tÞaand ψa¼ð∂

∂ϕÞaread

E¼−gabξaub¼−gtt

˙

t−gtϕ

˙

ϕ;ðG2Þ

L¼gabψaub¼gϕϕ

˙

ϕþgtϕ

˙

t: ðG3Þ

Combining the equations and restricting again to L¼0

yields the following radial equation,

˙

r2¼E2Ξ2½ðr2þa2Þ2−a2Δr−r2Δr

r4:ðG4Þ

Subsequently, we arrive at the proper distance in a Kerr–(A)

dS geometry induced by an infalling observer in the

equatorial plane, initially starting at rest at r¼Rand

falling towards the singularity at r¼0:

DðRÞ¼ZR

0

drr2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jE2Ξ2½ðr2þa2Þ2−a2Δr−r2Δrj

p;

ðG5Þ

where Eis in this case then given by

E2¼E2ðRÞ¼ R2ΔR

Ξ2½ðR2þa2Þ2−a2ΔR:ðG6Þ

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[1] S. Weinberg, in General Relativity: An Einstein Centenary

Survey, edited by S. W. Hawking and W. Israel (Cambridge

University Press, Cambridge, England, 1979), p. 790.

[2] M. Reuter, Nonperturbative evolution equation for quantum

gravity, Phys. Rev. D 57, 971 (1998).

[3] A. Bonanno and M. Reuter, Quantum gravity effects near

the null black hole singularity, Phys. Rev. D 60, 084011

(1999).

[4] A. Bonanno and M. Reuter, Renormalization group im-

proved black hole spacetimes, Phys. Rev. D 62, 043008

(2000).

[5] M. Reuter and E. Tuiran, Quantum gravity effects in the

Kerr spacetime, Phys. Rev. D 83, 044041 (2011).

[6] B. Koch and F. Saueressig, Structural aspects of asymp-

totically safe black holes, Classical Quantum Gravity 31,

015006 (2014).

[7] K. Falls, D. F. Litim, and A. Raghuraman, Black holes and

asymptotically safe gravity, Int. J. Mod. Phys. A 27,

1250019 (2012).

[8] B. Koch and F. Saueressig, Black holes within asymptotic

safety, Int. J. Mod. Phys. A 29, 1430011 (2014).

[9] A. Bonanno, B. Koch, and A. Platania, Cosmic censorship

in quantum Einstein gravity, Classical Quantum Gravity 34,

095012 (2017).

[10] A. Bonanno and M. Reuter, Spacetime structure of an

evaporating black hole in quantum gravity, Phys. Rev. D 73,

083005 (2006).

[11] K. Falls and D. F. Litim, Black hole thermodynamics under

the microscope, Phys. Rev. D 89, 084002 (2014).

[12] M. Reuter and H. Weyer, Running Newton constant,

improved gravitational actions, and galaxy rotation curves,

Phys. Rev. D 70, 124028 (2004).

[13] H. Emoto, Asymptotic safety of quantum gravity and

improved spacetime of black hole singularity by cutoff

identification, arXiv:hep-th/0511075.

[14] M. Niedermaier, The asymptotic safety scenario in quantum

gravity: an introduction, Classical Quantum Gravity 24,

R171 (2007).

[15] D. F. Litim, Renormalization group and the Planck scale,

Phil. Trans. R. Soc. A 369, 2759 (2011).

[16] M. Reuter and F. Saueressig, Quantum Einstein gravity,

New J. Phys. 14, 055022 (2012).

[17] A. Bonanno and F. Saueressig, Asymptotically safe cos-

mology—A status reportLa cosmologie asymptotiquement

sûre: un rapport d'´

etape, C.R. Phys. 18, 254 (2017).

[18] R. Percacci, An Introduction to Covariant Quantum Gravity

and Asymptotic Safety, 100 Years of General Relativity,

Vol. 3 (World Scientific, Singapore, 2017).

[19] A. Eichhorn, Status of the Asymptotic Safety Paradigm for

Quantum Gravity and Matter, Found. Phys. 48, 1407

(2018).

[20] T. Denz, J. M. Pawlowski, and M. Reichert, Towards

apparent convergence in asymptotically safe quantum grav-

ity, Eur. Phys. J. C 78, 336 (2018).

[21] W. Dittrich and M. Reuter, Effective Lagrangians in

Quantum Electrodynamics (Springer Verlag, Berlin, 1985).

[22] G. W. Gibbons and S. W. Hawking, Cosmological event

horizons, thermodynamics, and particle creation, Phys. Rev.

D15, 2738 (1977).

[23] M. Hindmarsh and I. D. Saltas, fðRÞgravity from the

renormalization group, Phys. Rev. D 86, 064029 (2012).

[24] E. J. Copeland, C. Rahmede, and I. D. Saltas, Asympto tically

safe Starobinsky inflation, Phys. Rev. D 91, 103530 (2015).

[25] B. Carter, Global structure of the Kerr family of gravita-

tional fields, Phys. Rev. 174, 1559 (1968).

[26] E. Hackmann, C. Lammerzahl, V. Kagramanova, and J.

Kunz, Analytical solution of the geodesic equation in Kerr-

(anti-) de Sitter space-times, Phys. Rev. D 81, 044020 (2010).

[27] M. Dafermos, Black holes without spacelike singularities,

Commun. Math. Phys. 332, 729 (2014).

[28] L. Modesto, Disappearance of black hole singularity in

quantum gravity, Phys. Rev. D 70, 124009 (2004).

[29] Y.-F. Cai and D. A. Easson, Asymptotically safe gravity as a

scalar-tensor theory and its cosmological implications,

Phys. Rev. D 84, 103502 (2011).

[30] M. Reuter and H. Weyer, Quantum gravity at astrophysical

distances?, J. Cosmol. Astropart. Phys. 12 (2004) 001.

[31] B. Koch and I. Ramirez, Exact renormalization group with

optimal scale and its application to cosmology, Classical

Quantum Gravity 28, 055008 (2011).

[32] S. W. Hawking, Particle creation by black holes, Commun.

Math. Phys. 43, 199 (1975).

[33] G. W. Gibbons and S. W. Hawking, Action integrals and

partition functions in quantum gravity, Phys. Rev. D 15,

2752 (1977).

[34] S. Aretakis, Horizon instability of extremal black holes,

Adv. Theor. Math. Phys. 19, 507 (2015).

[35] P. Chen, Y. C. Ong, and D.-h. Yeom, Black hole remnants

and the information loss paradox, Phys. Rep. 603,1

(2015).

QUANTUM-IMPROVED SCHWARZSCHILD-(A)DS AND …PHYS. REV. D 98, 106008 (2018)

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