Quantum Corrections to Hawking Radiation

JID:PLB AID:137820 /SCO [m5G; v1.335] P.1 (1-5)
Physics Letters B ••• (••••) ••••••
Contents lists available at ScienceDirect
Physics Letters B
journal homepage: www.elsevier.com/locate/physletb
Quantum gravitational corrections to particle creation by black holes
Xavier Calmet a,∗, Stephen D.H. Hsu b
, Marco Sebastianutti a
a
Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, United Kingdom
b
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48823, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received 16 December 2022
Accepted 1 March 2023
Available online xxxx
Editor: A. Ringwald
We calculate quantum gravitational corrections to the amplitude for the emission of a Hawking particle
by a black hole. We show explicitly how the amplitudes depend on quantum corrections to the exterior
metric (quantum hair). This reveals the mechanism by which information escapes the black hole. The
quantum state of the black hole is reflected in the quantum state of the exterior metric, which in turn
influences the emission of Hawking quanta.
© 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3
.
1. Introduction
In this paper we compute quantum gravitational corrections to
the amplitudes for creation of particles by black holes, first dis-
cussed by Hawking [1].
We show that the Hawking amplitudes are modified by quan-
tum corrections to the exterior metric (i.e., quantum hair). This
reveals, explicitly, a mechanism by which information escapes from
black holes. The exterior metric state is itself determined by the
quantum state of the black hole, so there is a direct connection
between the particle states produced during evaporation and the
history of the black hole.
In a series of recent papers [2–9], it has been shown that quan-
tum gravitational corrections generically lead to quantum hair. In
other words, classical solutions to Einstein’s equations generically
receive quantum gravitational corrections. As long as one consid-
ers physical processes taking place at small curvature, which is the
case for all practical purposes in our universe except very close
to gravitational singularities, these corrections can be calculated
in a reliable manner using the unique effective action approach
[10–14].
We will first revisit Hawking’s calculation [1] and then consider
the tunneling method of Parikh and Wilczek [15]. The latter ac-
counts for energy conservation – i.e., that the black hole mass
and radius shrink in the evaporation process – whereas Hawk-
ing’s methods assume a purely static black hole. We first consider
Schwarzschild black holes before extending our results to a generic
collapse model.
* Corresponding author.
E-mail addresses: x.calmet@sussex.ac.uk (X. Calmet), hsusteve@gmail.com
(S.D.H. Hsu), m.sebastianutti@sussex.ac.uk (M. Sebastianutti).
2. Quantum gravitational corrections to Hawking radiation
It has been shown in [2,3,5] that the leading order quan-
tum gravitational correction to the Schwarzschild metric appears
at third order in curvature. It is generated by a local operator
c6GN R
μν
ασ Rασ
δγ R
δγ
μν, where GN is Newton constant. As this is
a local operator, its Wilson coefficient is not calculable from first
principles in the approach, but it must be extracted from experi-
ment or an observation. The leading order quantum gravitational
correction to Schwarzschild metric is [5]
ds2
= − f (r)dt2
+
1
g(r)
dr2
+ r2
d2
, (1)
f (r) = 1 −
2GN M
r
+ 640πc6
G5
N M3
r7
, (2)
g(r) = 1 −
2GN M
r
+ 128πc6
G4
N M2
r6

27 − 49
GN M
r

. (3)
We will now show how this quantum correction impacts the emis-
sion of Hawking particles.
In his seminal paper [1], Hawking considers the wave-function
of a scalar field = 0 in a curved space-time given by the
Schwarzschild metric. The field operator can be written as
=

i

fiai + f̄ia
†
i

=

i

pibi + p̄ib
†
i
+ qici + q̄ic
†
i

, (4)
where fi, f̄i (the bar denotes complex conjugation) are solutions of
the wave equation which are purely ingoing, whereas pi , p̄i and qi,
q̄i are respectively purely outgoing solutions and solutions which
do not contain any outgoing component. The ai, bi and ci are an-
nihilation operators and a
†
i
, b
†
i
and c
†
i
are creation operators.
https://doi.org/10.1016/j.physletb.2023.137820
0370-2693/© 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP3
.

X. Calmet, S.D.H. Hsu and M. Sebastianutti Physics Letters B ••• (••••) ••••••
Our aim is to show that the solutions fi, f̄i, pi, p̄i, qi and q̄i
depend on the quantum corrections. Indeed, we will show that
quantum gravitational corrections to the Schwarzschild metric lead
to modifications of Hawking’s solutions. This implies that Hawking
radiation carries information about the quantum system which is
the source of the quantum hair.
The classical Schwarzschild space-time and its quantum cor-
rections given above are spherically symmetric. One can thus de-
compose the incoming and outgoing solutions into spherical har-
monics. In the region outside the black hole, one can write the
incoming and outgoing solutions as
fωlm =
1
√
2πωr
Fω (r)eiω v
Ylm(θ,φ) , (5)
pωlm =
1
√
2πωr
Pω(r)eiωu
Ylm(θ,φ) (6)
where v and u are the advanced and retarded coordinates. In the
classical Schwarzschild case, they are given by
vc
= t + r + 2M log

r
2MGN
− 1

(7)
uc
= t − r − 2M log

r
2MGN
− 1

. (8)
Our aim is to find the leading quantum correction to these func-
tions. The simplest way to do so is to consider a geodesic path for
a particle moving in the background space-time, parametrized by
an affine parameter λ. The 4-momentum of the particle is given by
pμ = gμν
dx
dλ
ν
(9)
and is conserved along geodesics. Furthermore, the quantity
= −gμν
dx
dλ
μ
dx
dλ
ν
, (10)
is also conserved along geodesics. For massive particles we set
= 1 and λ = τ is the proper time along the path. For massless
particles, which is the case we are going to examine, = 0 and λ
is an arbitrary affine parameter.
Considering a generic stationary spherically symmetric metric
as in (1), and taking radial (pφ = L = 0) geodesics on the plane
specified by θ = π/2, from (9) we get
E = f (r)ṫ (11)
where E = −pt and a dot stands for d/dλ. From (10) we instead
obtain

dr
dλ
2
=
E2
f (r)g(r)−1
. (12)
Manipulating the two equations above we arrive at
d
dλ

t ∓ r∗
= 0 (13)
where r∗ is the tortoise coordinate defined as
dr∗
=
dr
f (r)g(r)
. (14)
The two conserved quantities in (13) are the advanced and re-
tarded coordinates, v and u respectively. Rewriting the expression
for this latter coordinate we finally obtain
du
dλ
=
2E
f (r)
. (15)
Along an ingoing geodesic C parametrized by λ, the advanced co-
ordinate u is expressed as some function u(λ). Only two steps are
needed to get an expression for such a function: we first write r
as a function of λ and then perform the integral in (15). The form
of u(λ) determines the final expressions of the Bogoliubov coeffi-
cients, which are ultimately responsible for the emission of quanta
by the black hole.
We now take f (r) and g(r) as in (2) and (3) and then perform
the integral of the square root of (12) in the interval r ∈ [rH ,r]
corresponding to λ ∈ [0,λ].1
Assuming that quantum corrections
are small compared to the classical Schwarzschild terms, and re-
maining close enough to the horizon, we find
r = rH − Eλ 1 +
1728πG4
N M2
r6
H
c6 + O

λ2

+ O

c2
6

, (16)
where rH is the modified Schwarzschild radius, defined by g(rH ) =
0:
rH = 2GN M 1 −
5π
G2
N M4
c6 + O

c2
6

. (17)
We can now use r(λ) in (16) to integrate (15), hence obtaining
u(λ) = −4MGN log(
λ
C
) +
8πc6
M3GN
log(
λ
C
) + O (λ) + O

c2
6

, (18)
with C a negative integration constant. Geometric optics then al-
lows us to relate the outgoing null coordinate to the ingoing one:
λ = (v0 − v)/D, where v0 is the advanced coordinate which gets
reflected into the horizon (λ = 0) and D is a negative constant.
We can now calculate the quantum corrected out-going solu-
tions to the original Klein-Gordon equation. They are given by
pω =
∞
0

αωω fω + βωω f̄ω

dω
, (19)
where αωω and βωω are the Bogoliubov coefficients given by
αωω = −iKeiω v0
e

2πMGN −
4π2c6
M3GN

ω
×
0
−∞
dx
ω
ω
1/2
eωx
exp

iω

4MGN −
8πc6
M3GN

ln

|x|
C D

(20)
and
βωω = iKe−iω v0
e
−

2πMGN −
4π2c6
M3GN

ω
×
0
−∞
dx
ω
ω
1/2
eωx
exp

iω

4MGN −
8πc6
M3GN

ln

|x|
C D

.
(21)
Note that these integrals can be represented in terms of Gamma
functions.
This demonstrates that the quantum amplitude for the produc-
tion of the particle depends on the quantum correction to the
metric. In other words, the quantum hair induces a quantum cor-
rection to the amplitude for production of Hawking radiation. This
is the mechanism by which information escapes the black hole.
1
Note that in the square root of (12) we pick the minus sign corresponding to
the ingoing geodesic C.
2

Note that while the quantum amplitude receives a quantum
gravitational correction, the power spectrum at this stage is still
that of a black body. To show this, it suffices to calculate
|αωω |2
= exp

8πGN M −
16π2
c6
M3GN
ω

|βωω |2
. (22)
From the flux of outgoing particles with frequencies between ω
and ω + dω [18]:
F(ω) =
dω
2π
1

αωω
βωω

2
− 1
. (23)
We finally arrive at:
F(ω) =
dω
2π
1
e

8πGN M−
16π2c6
M3GN
ω
− 1
. (24)
Comparing with the Planck distribution
F(ω) =
dω
2π
1
e
ω
T − 1
(25)
we find that
T =
1
8πMGN

1 − 2πc6
M4G2
N

=
1
8πMGN

1 + 2πc6
1
G2
N M4

+ O(c2
6), (26)
which is the same expression as the expression for the tempera-
ture obtained using the surface gravity in [5]. Eq. (25) implies that
for a quantum corrected Schwarzschild metric, a black hole emits
radiation exactly like a gray body of temperature T given by (26).
However, we have not yet imposed energy conservation of the
whole system. When the black hole radiates a particle its total
mass decreases and the black hole shrinks. In the next section we
follow the tunneling approach of Wilczek and Parikh [15], which
takes this effect into account.
3. Quantum gravitational corrections to the tunneling process
We now follow [15–17] to implement energy conservation in
the derivation of the spectrum of radiation for our quantum cor-
rected Schwarzschild metric. The quantum corrected Schwarzschild
metric in Painlevé-Gullstrand form is easily derived:
ds2
= − f (r)dt2
+ 2h(r)dtdr + dr2
+ r2
d2
, (27)
where
h(r) =

f (r)

g(r)−1 − 1 (28)
=

2GN M
r
1 − 864π
G3
N M
r5
+ 160π
G4
N M2
r6
c6
+ O(c2
6)

.
The tunneling rate is related to the imaginary part of the action
[15–17]. We can write the action for a particle moving freely in a
curved background as
S = pμdxμ (29)
with pμ as in (9). As we are computing Im S, the second term
in pμdxμ = ptdt + prdr does not contribute, since pt dt = −Edt is
real, hence
Im S = Im
r f
ri
pr dr = Im
r f
ri
pr
0
dp
r dr . (30)
Using Hamilton’s equation and the Hamiltonian of the system
which is given by H = M − E, we find dH = −(dE), with 0 ≤
E ≤ E where E is the energy of the quantum emitted. We thus
obtain:
Im S = Im
r f
ri
M−E
M
dH
dr/dt
dr = Im
r f
ri
dr
E
0
(−dE)
dr/dt
. (31)
Switching the order of integration and substituting,
dr
dt
= −h(r) +

f (r) + h(r)2 = 1 −

(r)
r
, (32)
where (r) = (
√
2GN M +
√
rδ(r))2
with the quantum correction
given by
δ(r) = 32π
G3
N M
r6
−54MGN +

2MGN
r
(27r + 5MGN ) c6
+ O(c2
6), (33)
we find
Im S = Im
E
0
(−dE
)
r f
ri
dr
1 −

(r, E)
r
. (34)
The function (r) is now a function of E following the substitu-
tion M → (M − E) in the original metric. This integral has a pole
at the new horizon r = rH , which is obtained by solving g(rH ) = 0.
Integrating along a counterclockwise contour we finally obtain:
Im S = −2π EGN (E − 2M) +
8π2
c6
GN

1
2M2
−
1
2(E − M)2

.(35)
Following the derivation in [16], the quantum corrected emission
rate of a Hawking particle is given by
∼ e−2 Im S
= e
−8π EMGN

1− E
2M

1−
2πc6
G2
N
M2(M−E)2

. (36)
In the limit E → 0 one recovers the usual Planckian spectrum as
first derived by Hawking. However, because of the non-trivial E
dependence of the action, the emission spectrum clearly deviates
from that of a black body.
The emission spectrum follows from the tunneling rate using
the method presented in [16] and it deviates from that of a black
body as can easily be seen from
F(ω) =
dω
2π
1
e
8πωMGN

1− ω
2M

1−
2πc6
G2
N
M2(M−ω)2

− 1
(37)
because of its additional dependence on ω. For small values of ω,
the above expression reduces to the Planck distribution with the
modified Hawking temperature given by Eq. (24).
These results show that there is information about the interior
state in the radiation from a black hole. Not only do the Hawk-
ing amplitudes depend on the quantum corrections to the metric
(quantum hair), but the power spectrum also depends on these
corrections and it does not match that of a black body once en-
ergy conservation and quantum gravitational effects are taken into
account.
3

4. Arbitrary quantum hair and Hawking radiation
It has been shown in [9] that quantum hair is a generic fea-
ture of quantum gravity. In particular, any collapse model will have
quantum hair that carries information about the system through-
out the gravitational collapse. The quantum hair captures the
memory of the collapse process and of all the matter that formed
the black hole. We can parametrize generic quantum hair as
ds2
= − f (t,r)dt2
+
1
g(t,r)
dr2
+ r2
d2
, (38)
d2
= dθ2
+ sin(θ)2
dφ2
,
f (t,r) = 1 −
2GN M
r
+ f (q)
(t,r),
g(t,r) = 1 −
2GN M
r
+ g(q)
(t,r),
where f (q)(r,t) and g(q)(r,t) are assumed to be subleading cor-
rections. Note that this metric applies to any space-time depen-
dent collapse model which admits the exterior Schwarzschild so-
lution as a classical solution outside the matter distribution. We
assume that the quantum correction can depend on time, but the
collapse is expected to happen swiftly and the black hole thus
evolves quickly to an equilibrium Schwarzschild-like configuration.
Although we retain explicit space-time dependence in the quan-
tum correction, we assume that the quantum correction is slowly
varying over the Hawking emission timescale, so that the deriva-
tion based on geodesics is a reasonable approximation.
To leading order in the quantum correction, the horizon radius
is given by
rH = rS

1 − g(q)
(t,rS )

, (39)
where rS = 2GN M.
Starting from Eqs. (12) and (15) and taking f (r) and g(r) as
in (38), we perform the integral of the square root of (12) in the
interval r ∈ [rH ,r] corresponding to λ ∈ [0,λ]. Using this proce-
dure, we can obtain the coordinate u(v) which depends on the
quantum corrections f (q)(r,t) and g(q)(r,t).
It is thus obvious that the quantum corrected Bogoliubov coef-
ficients depend on the quantum corrections as well as can be seen
from the general expressions
αωω = K
v0
−∞
dv
ω
ω
1/2
eiω v
e−iωu(v)
(40)
and
βωω = K
v0
−∞
dv
ω
ω
1/2
e−iωv
e−iωu(v)
. (41)
Thus the amplitude to emit a Hawking particle depends on the
quantum correction to the metric, i.e., the quantum hair. Informa-
tion about the collapsing matter is stored in the quantum correc-
tion to the gravitational field and it is then imprinted in Hawking
radiation. Clearly generic quantum corrections would also leave an
imprint in the modified emission spectrum obtained by imposing
the energy conservation constraint as done by Parikh and Wilczek
and applied to the case of a quantum corrected Schwarzschild met-
ric in section 3.
5. Conclusions
In this paper we have considered quantum gravitational cor-
rections to the emission of Hawking particles. We first describe
a black hole using a Schwarzschild metric as originally done by
Hawking. Extending his work to take into account known quantum
gravitational corrections to the metric, we find that the quantum
amplitude for the emission of a Hawking particle depends on the
quantum hair, i.e., on the quantum gravitational correction to the
classical Schwarzschild background.
We then apply the method of Parikh and Wilczek who demon-
strated that, when energy conservation is applied to the evapora-
tion of a black hole, the thermal spectrum deviates from that of a
black body. We calculate quantum gravitational corrections to the
power spectrum and again demonstrate that the spectrum depends
on the quantum hair.
Finally, we study a generic quantum gravitational correc-
tion which can parametrize the quantum corrections to any
space-time dependent collapse model which admits the exterior
Schwarzschild solution outside the matter distribution. We show
that the quantum hair generically affects the quantum Hawking
amplitude. Information about the collapsing matter is stored in
the quantum correction to the gravitational field and it is then
imprinted in Hawking radiation.
In order to perform explicit calculations we considered quan-
tum hair which manifests as a semiclassical correction to the exte-
rior metric. In general, as discussed in [6–8], the quantum state of
the black hole interior is reflected in the quantum state g of its
exterior graviton field. g will affect radiation amplitudes: i.e., it
represents the background or environment in which the Hawking
radiation originates. This is the mechanism by which information
escapes the black hole and the reason why its evaporation is uni-
tary.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Data availability
This manuscript has no associated data. Data sharing not ap-
plicable to this article as no datasets were generated or analyzed
during the current study.
Acknowledgements
The work of X.C. is supported in part by the Science and
Technology Facilities Council (grants numbers ST/T00102X/1 and
ST/T006048/1).
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5

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