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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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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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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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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.
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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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