Quantum corrections to Schwarzschild black hole
Eur. Phys. J. C
Quantum corrections to Schwarzschild black hole
Xavier Calmet 0
Basem Kamal ElMenoufi 0
0 Department of Physics and Astronomy, University of Sussex , Falmer, Brighton BN1 9QH , UK
Using effective field theory techniques, we compute quantum corrections to spherically symmetric solutions of Einstein's gravity and focus in particular on the Schwarzschild black hole. Quantum modifications are covariantly encoded in a nonlocal effective action. We work to quadratic order in curvatures simultaneously taking local and nonlocal corrections into account. Looking for solutions perturbatively close to that of classical general relativity, we find that an eternal Schwarzschild black hole remains a solution and receives no quantum corrections up to this order in the curvature expansion. In contrast, the field of a massive star receives corrections which are fully determined by the effective field theory.

With the recent discovery of gravitational waves, we have
entered a new era in astrophysics in which we will be
probing black hole physics directly. Besides gravitational waves,
the Event Horizon Telescope will soon be in a situation to
probe the horizon of Sgr A∗, the black hole at the center of
our galaxy, by studying its shadow directly. Recent progress
in observational astronomy thus prompts us to try and
understand quantum effects on black holes, as we should soon be
in a situation able to confront theory with observations. Some
of the questions that we can hope to probe with such
observations are whether the horizons of black holes are more
violent regions of spacetime, in contrast to what is expected
in general relativity, and what physical effect resolves the
curvature singularity at r = 0. Both questions are crucial
if we want to understand the information paradox linked to
Hawking radiation [1].
Black holes are amongst the simplest and yet most
mysterious objects in our universe. According to the nohair
thea email:
b email:
orem, they are described by only a few parameters: their
masses, angular momenta and charges. Despite this apparent
simplicity, they are incredibly challenging as understanding
their physics requires merging quantum mechanics and
general relativity, which is one of the remaining holy grails of
theoretical physics (see e.g. [2] for a recent review).
While we are far away from having a consistent theory
of quantum gravity valid at all energy scales, effective field
theory (EFT) techniques applied to general relativity lead to
a consistent theory of quantum gravity valid up to an energy
scale close to the Planck scale [3, 4]. Several universal
features of quantum gravity can be identified using these EFT
methods. The most intriguing of these features is the
dynamical nonlocality of spacetime induced at short distances by
quantum effects.
This nonlocality has been shown to have interesting
features in cosmology and could indeed avoid the big crunch
singularity in a collapsing universe [5]. Investigating the effects
of this nonlocality in black hole physics is our main
motivation to consider quantum corrections to spherically
symmetric solutions in general relativity. In particular, we revisit the
issue of quantum corrections to the Schwarzschild black hole
solution which have been studied in the past [6, 7]. We
identify a complication which has not been realized previously,
namely that of how to define a black hole.
A mathematically consistent way to define a black hole
is to define it as a static vacuum solution, i.e., an eternal
black hole. If this definition is adopted, we obtain a result
that differs from previous investigations. In particular, we
show that the classical black hole solution remains a
solution in quantum gravity up to quartic order in the nonlocal
curvature expansion. This result is obtained by looking for
perturbative solutions around the Schwarzschild black hole
metric. Nevertheless, higher curvature operators give rise
to nontrivial corrections. The nonlocal operators that led
to singularity avoidances in [5] thus do not affect the
singularity in the case of the Schwarzschild black hole
solution.
While eternal black holes are mathematically well defined,
they may not capture the full physical picture. A real,
astrophysical, black hole is the final state of the evolution of a
matter distribution, for example of a heavy star, after it has
undergone gravitational collapse. This process is certainly
not happening in vacuum. This raises the question of how to
define a real astrophysical black hole and of how to calculate
quantum corrections to its metric. A nonvanishing
energymomentum tensor could be used to model a collapsing star.
At a time when the star has not yet collapsed into a black
hole, the star can be described as a static source at a specific
time in its evolution.
Another complication appears due to the nonlocality: we
are forced to integrate the modified equations of motion over
regions of Planck size curvature. One may thus worry about
the sensitivity of the EFT to regions of spacetime with high
curvature and, in particular, to the singularity at r = 0. This is
investigated in a paper by Donoghue and Ibiapina Bevilaqua
[8]. Clearly the effective field theory breaks down in regions
of large curvature, which in turn raises the question if the
latter could offer a reliable picture in our case. However, the
ultimate ultraviolet physics that dominates regions of large
curvature should not affect observables at long distances, i.e.
the exterior region of a black hole. Indeed, in an EFT, one
expects short distance physics to decouple at low energies [9].
We will be making this conservative assumption throughout
this paper.
The paper is organized as follows. In Sect. 2, we derive the
modified nonlocal equations of motion. We then consider, in
Sect. 3, an eternal Schwarzschild black hole and prove that
there are no correction up to quadratic order in curvatures
while subsequently showing that the leading order
correction appears at cubic order. In Sect. 4, we briefly discuss
singularities. In Sect. 5, we turn our attention to the quantum
correction around a static star. In Sect. 6, we compare our
findings with previous work and then conclude in Sect. 7.
2 Quantum gravity corrections to the equations of
motion
In analogy to chiral perturbation theory, the effective
Lagrangian for quantum general relativity is arranged as a
derivative expansion that respects general covariance [3,10].
Quantum fluctuations of the metric, or any other massless
field,1 result in a nonlocal effective action organized as an
expansion in curvatures [5,11–13]. The final outcome is
composed of two parts,
[g] =
1 This is certainly a valid approximation if one is working at energies
between the weak scale and the Planck mass.
where the first piece comprises the local effective Lagrangian
with renormalized constants. The local part of the Lagrangian
contains information about the unknown ultraviolet
portion of the theory. The second piece is the nonlocal portion
encoding the infrared effects. To second order in curvatures
[5], we have2
d4x √g
The various coefficients are given in [5]. In fact, we can
invoke the Gauss–Bonnet theorem to express the local action
in terms of two independent invariants. We choose to
eliminate the Riemann tensor in Eq. (2) which changes the
coefficients to
c¯1 = c1 − c3, c¯2 = c2 + 4c3, c¯3 = 0.
The resulting equations of motion are
where Gμν is the Einstein tensor, Hμν and Hμqν comprise,
respectively, the local and nonlocal parts of the quantum
correction to the field equations. The local contribution is
given by
Hμν = c¯1(μ) 2 R Rμν − 21 gμν R2 − 2gμν
+ c¯2(μ) ∇α∇μ Rνα + ∇α∇ν Rμα −
q
Let us now consider the remaining tensor Hμν which is the
result of varying the nonlocal action. A consistent method
to vary the logarithm has been constructed in [14] and have
been shown to contribute to the equation of motion terms
cubic in curvatures. The result is indeed quite complicated,
however, we do not consider such terms as our aim is to
study the theory only up to quadratic order.3 The nonlocal
contribution to the equations of motion reads
2 We dropped a total derivative R as it does not affect the equations
of motion.
3 Note, nevertheless, that these contributions to the equations of motion
are of the same order in derivatives as the ones considered here.
(∇α∇τ + ∇τ ∇α)Rανμτ = 0
where the linearized Einstein tensor reads
gμqν − gμν gq + ∇μ∇ν gq + 2 Rαμβν gαqβ
− ∇μ∇β gνqβ − ∇ν ∇β gμqβ + gμν ∇α∇β gαqβ .
Clearly, all the terms in Eq. (6) vanish identically, i.e.
Hμν [gSch.] = 0. This is true as well for all terms
proporq
tional to the Ricci tensor or Ricci scalar in Hμν . Moreover,
using the Bianchi identity one can easily show that
for the Schwarzschild solution.4 At this stage one has to
specify what the operator ln( ) means in curved space. It has been
shown in [5] that the latter is in fact a nonlocal (bilocal)
tensor. Formally, a convenient way to resolve a nonlocal form
factor is as follows [5]:
The quantum corrections to the equations of motion then take
the form
δμα Rνβσ τ + δνα Rμβσ τ − 21 gμν Rαβσ τ
d4x √g L(x , x ; μ)Rαβσ τ ,
Note that there is no way to place the indices on the
Riemann tensor such that the last piece is manifestly symmetric.
Hence, we choose to vary each tensor separately. The
variation of both the Ricci tensor and the Ricci scalar is simple
but we choose not to display since these terms do not effect
our analysis in the next section. The variation of the last two
terms in Eq. (7) yields
We are now in a position to investigate corrections to the
Schwarzschild solution.
3 Absence of perturbative correction to Schwarzschild
black hole
In this section we look for spherically symmetric black holes
in the vacuum of the theory including the full set of quantum
corrections up to quadratic order in curvatures. Indeed, this
is quite complicated to do in full generality since the
equations of motion are integrodifferential. Analytic solutions
are almost impossible to find and ultimately one must resort
to numerical methods. Here, we instead look for linearized
solutions around the Schwarzschild black hole solution. This,
on one hand, gives us analytic handle on the problem and it
conforms to the expectation of the effective theory
framework in the sense that quantuminduced corrections should
be small compared to the classical solution.
Precisely, we write the metric as follows:
where gq represents the quantum correction to Schwarzschild
solution. Linearizing Eq. (5) around gμScνh. one finds
GLμν [gq] + Hμν [gSch.] + Hμqν [gSch.] = 0,
where the righthand side is to be evaluated using the
Schwarzschild metric. One may worry that the nonlocal
structure forces us to work with coordinates which are regular
across the horizon since the integral probes all of spacetime.
One simple choice is the Eddington–Finkelstein coordinates
1 −
dv2 − 2dvdr − r 2d 2. (15)
We shall nevertheless see that our results do not depend
(as expected because of general covariance) on the choice
of coordinates and one can work as well with the standard
Schwarzschild coordinates.
Indeed, the function L(x , x ; μ) is very complicated to
write down for any particular background spacetime.
Nevertheless, looking at the metric in Eq. (15) one easily deduces
that
L(x , x ; μ) = Lflat(x − x ) + O(∂g).
4 Here, we freely commuted the derivative operators past the logarithm
in Eq. (8). This is admissible because the logarithm is precisely a
bilocal function that could only depend on the geodesic distance between
any two points.
(∇2 + k2)G(x − x , k) = δ(x − x )
eikx−x 
G(x − x , k) = − 4π x − x  .
1
Lflat(x − x ) = − 2π l→im0 P
x − x 3
+ 4π(ln(μ ) + γE − 1)δ(3)(x − x ) ,
r 3r14(2M G N − r )r − r1r + r1
where the principal value P is defined by
xi  f (xi )=0
xi −
h(r ) = c6
Armed with this expression, we can now show that the
righthand side of Eq. (14) vanishes when applied on the
solution. With appropriate placement of indices on the
Riemann tensor, all integrals which appear in Eq. (14) have the
following form:
d3x √g Lflat(x − x ) (r1)n = − r2n ln(μr ).
This is a crucial point in the derivation: there are no constant
terms accompanying ln(μr ). One can go ahead and evaluate
the righthand side of Eq. (14) for any pair (μ, ν) to find that
it vanishes identically. This is simple to understand given that
ln μ comprises a local contribution and must vanish when the
righthand side is evaluated on the Schwarzschild solution.
It is remarkable we obtain a finite result despite the integral
in Eq. (23) extends all the way down to r = 0. This is due to
the properties of the nonlocal function and is a sign of the
selfconsistency of the EFT.
Working in flat spacetime we find
Equivalently, one can also calculate the contribution of the
term Rμναβ ln( /μ2α)Rμναβ to the equations of motion by
varying this term with respect to the metric functions5 A(r )
and B(r ) = A(r )−1. Using the usual Schwarzschild metric
we find that this term gives the following contribution to the
field equations:
which vanishes since the integrant is not singular in the limit
r1 → r , i.e. the principal value integral is equal to zero.
This also shows that our result is coordinate independent as
expected.
While this shows that there are no corrections at
quartic order in curvature, which is in sharp contrast with
previous results, there will be corrections that are of higher
order, for example higher dimensional operators such as
c6 Rμανσ Rαδσγ Rδγμν , which will lead to quantum corrections of
the Schwarzschild solution. We find the following equation
of motion:
M¯ P2 r (r h (r ) + 2h (r ))
2(r h (r )+h(r )+1)3/2 +c624M 2G2N r968(Mr−G N2M−G45Nr) = 0
where we are doing perturbation around the standard
Schwarzschild solution A(r ) = 1 − 2Mr G + h(r ). Far away
from the hole, we find6
This simply demonstrates that the Schwarzschild solution is
not a solution of the field equations when higher dimensional
operators of dimensions d ≥ 6 are included.
4 Singularity avoidance?
An immediate consequence of our result is that the singularity
avoidance observed in [5] is nonuniversal as the very same
operators do not cure the curvature singularity of an eternal
Schwarzschild black hole. However, it is important to keep
in mind that these results are obtained in perturbation theory.
We actually have indications that perturbation theory will
break down below the reduced Planck mass. The nonlocal
operators (3) lead to a modification of the propagator for the
graviton given by
5 Here, we are using the standard static line element ds2 = A(r )dt2 − B(r )dr 2 − r 2d 2.
6 Note that there are several local as well as nonlocal operators at cubic
order that would contribute similarly.
2 p2 1 − 1210π G N N p2 log − p2
μ2
where N = Ns +3N f +12NV with Ns , N f and NV denoting
respectively the number of scalar, spinor and vector fields in
the theory. It has been pointed out [15] that this propagator
has a complex pole besides the usual massless pole
q12 = 0,
q32 = (q22)∗,
where W (x ) is the Lambert Wfunction. The complex pole
corresponds to new states with a mass and a width given by
q02 = (m − i /2)2. The position of this complex pole, i.e.,
the new scale NP = Re q0, which depends on the
number of particles in the theory, determines the energy scale at
which strong quantum gravitational effects are expected to
become relevant. At energies of the order of NP, we cannot
truncate the effective action at quadratic order and need to
consider the full effective action. It is very likely that this
nonperturbative phenomenon is responsible for singularity
avoidance. The physical picture that emerges is that the
complex states, having a width, are extended objects which will be
created when short distances, i.e., singularities, are probed.
These extended objects will screen the singularity from being
probed during physical processes and thus lead to singularity
avoidance.
5 Corrections to the gravitational field of a static source
It is well known that Birkhoff’s theorem7 holds only true
in classical Einstein’s gravity; see for example [16,17]. It
is quite interesting to understand the effect of the
quantum induced nonlocality on the field of a static spherically
symmetric object such as a star. One should expect a
nontrivial correction to arise in this case albeit the fact that the
Schwarzschild metric remains a solution to the vacuum
nonlocal equations of motion. This is a violation of Birkhoff’s
theorem even at the perturbative level as we show next.
We aim for a simplified treatment and thus only the Ricci
scalar term in Eq. (3) is considered. We use a perturbative
approach which is similar to that of Sect. 3. The solution
to Einstein equation for a constant density star is readily
obtained in closed form; see for example [18]. Outside the
7 Birkhoff’s theorem states that any spherically symmetric solution
of the vacuum field equations must be static and asymptotically flat.
This theorem implies that the exterior solution must be given by the
Schwarzschild metric.
GLμν = α(16π G N )2(∇μ∇ν − gμν )
where the integral extends only over the source region, T =
ρ0 − 3 P is the trace of the energymomentum tensor, ρ0 is
the mass density and P is the pressure. Note that here the
perturbative treatment is insensitive to the ultraviolet, the
local pieces in Eq. (2) drop out. Both the pressure and the
metric functions are known in the interior of the star [18], for
example,
(1 − 2G Mr 2/RS3)1/2 − (1 − 2G M/RS)1/2
P(r ) = ρ0 (1 − 2G M/RS)1/2 − 3(1 − 2G Mr 2/RS3)1/2
where RS is the radius of the star. To analyze the field far
away from the source, it is enough to expand the righthand
side in powers of G N . To lowest order, we have8
GLμν = (2πρ0α)(16π G N )2(∂μ∂ν − ημν ∂2)
where RS is the radius of the star and the pressure drops out
since it is O(G). We fix the gauge by looking for spherically
symmetric perturbations
Far away from the source, we find the leading correction
gtqt =
Note that it is not possible to recover our previous result for an
eternal Schwarzschild black hole by taking the limit RS = 0
as this limit is illdefined.
6 Comments on previous results
In Sect. 3, we established that the Schwarzschild black hole
solution furnishes an exact solution to the nonlocal equations
of motion accurate up to quadratic order in curvatures. The
expected breakdown of Birkhoff’s theorem led us to study the
field of a static star and we identified the leading nontrivial
quantum correction. In the current section we scrutinize the
previous results obtained in [6,7] and discuss them in light
8 Note that we are working with flatspace derivatives in spherical coor
of our findings. We will argue that studying quantum
corrections to black holes must be done using the effective action
formalism.
We start by linearizing the effective action Eq. (3) around
flat space
where we used the harmonic gauge, κ2 = 32π G and
We ignore the local action in Eq. (2) as we only need to
track the ln μ2 piece to determine their effect. To follow the
treatment in [6,7], one looks for solutions in the following
form:
hμν = h(μ0ν)(x ) + h(μ1ν)(x ), h(000) = −
which precisely agrees with [6,7]. One could easily transform
the above results to spherical polar coordinates but we do not
display this here.
We are interested to compare this result with ours that we
obtained using the full nonlinear action.
• The result is proportional to the combination ln(μr ). The
problem is not that the metric depends on the
renormalization scale,9 but rather the mere dependence on the term
ln(μr ). Our analysis in the previous sections confirms
that this dependence must be superfluous. It is precisely
the curvature expansion that remedies this problem.
• The Newtonian 1/r behavior of the lowest order solution,
i.e. h(μ0ν)(x ), is strictly valid for an idealized point mass.
9 The authors of [7] argued for dropping the ln μ piece as they consid
ered it problematic to have explicit dependence on an unphysical scale.
In [6] it is shown that this piece cancels out in gaugeinvariant
quantities. Working with the effective action, we know that this causes no
concern. Dimensional transmutation of the Wilson coefficients would
turn μ into a physical ultraviolet scale.
Indeed and due to the uniqueness of Schwarzschild metric
in Einstein gravity, a black hole looks like a point mass
to an asymptotic observer. This is not true anymore once
we deviate from classical general relativity and hence
one has to be careful when interpreting the meaning of
quantum corrections.
• The other pieces, i.e. the 1/r 3 power law in Eq. (37),
resemble the correction to the grqr component of a static
star found in Eq. (33). Note, however, that it is impossible
for the linearized analysis to pick the powerlaw behavior
in gtqt as the source is treated as a point mass.
We conclude that the previous results of [6,7] should be
reinterpreted with great care. They capture some of the quantum
corrections around a massive object such as a star. On the
other hand, as shown previously, the Schwarzschild black
hole solution remains a solution including nonlocal quantum
corrections up to quadratic order in curvatures.
7 Conclusions
Effective field theory techniques applied to general
relativity lead to a consistent theory of quantum gravity up to
energy scales close to the Planck scale. A key feature of
this effective theory approach is the appearance of nonlocal
effects which become appreciable close to the Planck scale
and could, in principle, resolve singularities. Using the EFT
approach, we have computed quantum corrections to
spherically symmetric solutions of Einstein gravity and focused
in particular on the Schwarzschild black hole solution. We
worked to quadratic order in curvatures simultaneously
taking local and nonlocal corrections into account. We searched
for solutions perturbatively close to that of classical general
relativity, and found that an eternal Schwarzschild black hole
remains a solution and receives no quantum corrections up
to this order in the curvature expansion. This is in contrast
to previous work. We have identified the reason for the
discrepancy which can be traced back to the fact that previous
results considered a linearized theory. We have also shown
that, while there are no quantum gravitational corrections to
an eternal black hole at quadratic order in curvature,
corrections will appear at higher order. In contrast, and due to
the breakdown of Birkhoff’s theorem, the field of a massive
star receives corrections which are fully determined by the
effective theory.
We have shown that while the nonlocality forces us to
integrate the equations of motion over regions of
spacetime with large curvature, we obtain finite results which is
a sign of the selfconsistency of the effective field theory.
Indeed in an effective field theory, one expects a decoupling
of scales and long distance corrections should be
calculable without requiring any knowledge of the full theory (i.e.
ultraviolet physics or singularity in our case). We have also
shown that the nonlocal terms do not soften the
singularity in the case of black holes, at least within a perturbative
framework.
Our findings emphasize the need to be very careful when
discussing quantum corrections to black holes which need
to be defined carefully. While, from a mathematical point
of view, an eternal black hole is a static vacuum solution,
astrophysical black holes are not. They are surrounded by
matter and are themselves the result of the gravitational
collapse of matter. Calculating quantum gravitational
corrections to real astrophysical black holes is thus a fantastically
difficult task which cannot be done easily analytically. This
investigation requires us to study a dynamical process where
a matter distribution, e.g., a star, collapses to form a black
hole and to follow quantum effects throughout the process.
Our work represents a first step in that direction. We have
found that an observer far away from a star experiences a
correction to Newton’s law that depends on the size of the
star. Long after the star has collapsed, the far field behavior
of the remaining object should approach that of an eternal
black hole. At this stage of the evolution, the observer would
find only corrections of cubic order in curvature to Newton’s
law.
Acknowledgements We would like to thank John Donoghue and
Leandro Ibiapina Bevilaqua for numerous discussions. This work is
supported in part by the Science and Technology Facilities Council (Grant
Number ST/L000504/1).
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